Practical Statistics for Educators - User contributions [en]

Visualizing Data

2022-05-11T17:51:09Z

Ciskowskid: /* Stem-and-leaf displays */

Introduction

==Data Visualization==

The table below gives some guidelines for which types of data visualization to use:

[[File:Data_Visualization_Table.PNG]]

''contributed by David Ciskowski''

== Stem-and-leaf displays ==
A stem and leaf plot is a representation of data in which each data value is separated into two parts -- a stem and a leaf. For example, if the data are two-digit numbers, then the stems are commonly the tens digits, and the leaves would be the units digits. The stems are listed vertically (from smallest to largest), and the corresponding leaves for the data values are listed horizontally beside the appropriate stem. On the final version of the stem and leaf plot, the leaves are usually ordered within each stem. Note that the stems on a stem and leaf plot provide a mechanism for grouping numeric data.

''contributed by Emily Rhew''

== Frequency table ==
Frequency tables created in the SPSS program allow one to calculate the mean, median, mode, standard deviation, range, quartiles and more when analyzing a data set.

Using a frequency table is the first step in analyzing data. It provides a snapshot of the presented data. Once a frequency table is created, one can then create box and whisker plots, as well as other valuable graphs that will assist in analyzing data chosen from several different variables.

Using SPSS in conjunction with Microsoft Excel is user-friendly and saves time when analyzing data. What used to be solved with only a calculator can now be solved much faster using the SPSS program.

''contributed by Chris Longo''

'''Frequency table,''' as the name implies allows users to track the number of times something occurs or reoccurs. The most simplistic frequency table can be done by hand. The first column denotes the category of numbers, written in ascending order, the second room for tally marks and the third column is the numeric frequency for the following numbers:

80, 81, 85, 83, 83, 83, 85, 85, 80,81, 82, 82, 82, 86 and 85.

Mark Tally Frequency
80 II 2
81 II 2
82 IIII 4
83 IIII 4
84 0
85 II 2
86 I 1

Frequency tables can also accommodate more numbers and can be handled easier when they are placed into a frequency of a group, also known as '''class interval'''. In order to determine the class intervals, you have to find the difference between the highest and smallest data value. Once this is determined, the class interval is set to allow for at least five categories.

25, 50, 60, 75, 30, 39, 60, 100, 94, 50, 30

Class interval Tally Frequency
0 - 19 0
20-39 IIII 4
40-59 II 2
60-79 III 3
80-99 I 1
100-119 I 1

''contributed by Tina Hislop''

== Bar graphs ==

Bar Graphs are wonderful ways to encourage young children to view a mathematical idea in a visual way, offering them the opportunity to understand the relationships that exist between numbers. For example, information such as transportation can be interpreted in a creative way by the creation of a 3 column bar graph for kindergarten students. The top should read,
How Do You Get To School?
Each of the three columns may be labeled with pictures and words to read,
Car,Bus, Walk (or bike, taxi etc. depending upon your population of students)
Use pictures or student names to acquire each student's data, and then count the results. The Bar Graph is created with the pieces of paper, and is visually exciting for the students to read.
Some good questions might be:

How do most of our friends in class get to school?
Which type of vehicle is used most?
How many boys (girls) ride in a car(bus)?
How many people ride in the car with the student in our class?

The list is endless. This is an early way to jumpstart your students in discovering more practical uses for math curriculum, and it sure beats a worksheet!

''contributed by Deborah Mumford''

<big>'''Making a Bar Graph in Google Sheets'''</big>

1. Highlight the data you would like to include (with Variable labels/titles).
2. Click on "''Insert''".
3. Click on "''Chart''".

Use the ''Chart Editor'' that pops up on the right side of the screen for the following formatting:
1. Change the "''Chart Type''" to "''Bar''".
2. Click on "''Customize''" to change the following: ''Chart Style, Chart & Axis Titles, Series, Legends, Horizontal Axis, Vertical Axis, and Gridlines and Ticks''.

''contributed by Kaitlyn Kakadeles''

== Histograms ==
Like a bar graph, a histogram is a graphical representation of the distribution of data. Whereas a bar graph is used to represent the frequency count of a categorical variable, a histogram is used to represent the frequency count of a continuous variable (Meyers, Gamst, & Guarino, 2006).

The data in a histogram is represented by a series of rectangles. The height of each rectangle is determined by the tabulated frequencies of the data. The rectangles are drawn over a set intervals (bins), with an area equal to the frequency of the observations in the interval.

For example, a histogram could be used to represent the height for a given sample of people. On the X-axis you would have different ranges of height and on the Y-axis the frequency or number of people that fall into each range of heights.

One advantage of using SPSS to create a histogram is that you can superimpose a drawing of the normal curve so we can easily see how close our data are to a normal distribution.

''contributed by Michael Minzloff''

<big>'''Making a Histogram in Google Sheets'''</big>

1. Highlight the data you would like to include (with Variable labels/titles). 2. Click on "''Insert''". 3. Click on "''Chart''".

Use the ''Chart Editor'' that pops up on the right side of the screen for the following formatting: 1. Change the "''Chart Type''" to "''Histogram chart''" (found at the bottom - under "''Other''"). 2. Click on "''Customize''" to change the following: ''Chart Style, Chart & Axis Titles, Series, Legends, Horizontal Axis, Vertical Axis, and Gridlines and Ticks''.

''contributed by Kaitlyn Kakadeles''

== Pie Charts ==
Though pie charts have little place in educational research, they can express key pieces of data in a visual way, providing often-powerful representations of relative “pieces of the pie”, or percents of a certain whole.

For example, if there are 100 senators, and 25 of them are over six feet tall and the other 75 are either exactly six feet tall or less than six feet tall, it seems clear that one-quarter of the senators would be over six feet tall, and three-quarters of them would be less than or equal to six feet tall. In terms of a pie graph, one-fourth of the pie (i.e., a sector with a central angle of 90°) could represent the fraction of the whole representing over-six-foot-tall senators. The rest of the pie (i.e., a sector with a central angle of 270°) would represent the fraction of the whole representing the less than or equal to six foot senators. You could even color these sectors red and blue, respectively. It would be easy to see that there were many more less than or equal to six foot tall senators, since the blue sector would be much bigger than the red sector. If you looked closely, you might even see that the blue sector had three times the area of the red sector.

Now, pretend there is a big election where some of the senators are removed from office. The new senate includes 40% over-six-foot tall senators (represented in a new pie graph by a BIGGER red sector with a central angle of 144°) and 60% less than or equal to six foot tall senators (represented in the new pie graph with a SMALLER blue sector with a central angle of 216°). When you place the new pie graph next to the old one, comparing changes in sector sizes is easy (especially since you’ve colored them!). For example, the red sector has grown from having a central angle of 90° to having a central angle of 144°. You may even be impressed by the fact that this means the new senate has a greater percentage of over six foot tall senators, as can be seen in the relative sizes of the sectors.

The key word in this sentence is “seen”. The pie graph is, ultimately, a visual tool for the representation of data. Furthermore, since people have some experience with circles (e.g., eating pizza), the pie graph is often easily understood, and can be quite popular. Sadly, it carries little statistical significance.

''contributed by Chris Ruggiero''

<big>'''Making Pie Chart in Google Sheets'''</big>

1. Highlight the data you would like to include (with Variable labels/titles). 2. Click on "''Insert''". 3. Click on "''Chart''".

Use the ''Chart Editor'' that pops up on the right side of the screen for the following formatting:
1. Change the "''Chart Type''" to "''Pie''". You have three options: ''(1) Standard Pie Chart, (2) Doughnut Chart, and (3) 3D Pie Chart''. Choose the one that best tells a story with the data you have provided.
2. Click on "''Customize''" to change the following: ''Chart Style, Chart & Axis Titles, Series, Legends, Horizontal Axis, Vertical Axis, and Gridlines and Ticks''.

''contributed by Kaitlyn Kakadeles''

== Line graph ==
Line graphs provide an excellent way to map independent and dependent variables that are both quantitative. When both variables are quantitative, the line segment that connects two points on the graph expresses a slope, which can be interpreted visually relative to the slope of other lines or expressed as a precise mathematical formula.

Line graphs are like scatter plots in that they record individual data values as marks on the graph. The difference is that a line is created connecting each data point together. In this way, the local change from point to point can be seen. This is done when it is important to be able to see the local change between any to pairs of points. An overall trend can still be seen, but this trend is joined by the local trend between individual or small groups of points. Unlike scatter plots, the independent variable can be either scalar or ordinal.

''contributed by Emily Rhew''

<big>'''Making a Line Graph in Google Sheets'''</big>

1. Highlight the data you would like to include (with Variable labels/titles). 2. Click on "''Insert''". 3. Click on "''Chart''".

Use the ''Chart Editor'' that pops up on the right side of the screen for the following formatting: 1. Change the "''Chart Type''" to "''Line Graph''". You have three options: ''(1) Line Graph, (2) Smooth Line Chart, and (3) Combo Chart''. Choose the line graph that best represents the story you are trying to tell with the data. 2. Click on "''Customize''" to change the following: ''Chart Style, Chart & Axis Titles, Series, Legends, Horizontal Axis, Vertical Axis, and Gridlines and Ticks''.

''contributed by Kaitlyn Kakadeles''

== Scatterplot ==
Scatterplots were developed by Sir Francis Gaulton, who needed to find a manner to present data that was statistically correlated for use in his studies of anthroometry.

Scatterplots are comprised of a tile a horizontal axis and a vertical access. A scatterplot is typically utilized to plot data points for two variables. The independent variable also known as the control variable is placed on the horizontal axis. The dependent or variable being studied is usually placed on the y axis.

The graph provides a visual representation that establishes the extent to which the two variables are correlated. The scatterplots below represent a few of the various distinctions regarding the strength and direction of correlation representations.

[[File:Scatterplotholst.png]]

A. Strong Positive Correlation
B. Weak Positive Correlation
C. No correlation
D.String Positive Correlation
E. Weak Negative Correlatoin
F.No correlation

''contributed by Damien Holst''

<big>'''Making a Scatterplot in Google Sheets'''</big>

1. Highlight the data you would like to include (with Variable labels/titles). 2. Click on "''Insert''". 3. Click on "''Chart''".

Use the ''Chart Editor'' that pops up on the right side of the screen for the following formatting: 1. Change the "''Chart Type''" to "''Scatterplot''". You have two options: ''(1) Scatterplot and (2) Bubble Chart''. 2. Click on "''Customize''" to change the following: ''Chart Style, Chart & Axis Titles, Series, Legends, Horizontal Axis, Vertical Axis, and Gridlines and Ticks''.

''contributed by Kaitlyn Kakadeles''

File:Data Visualization Table.PNG

2022-05-11T17:50:44Z

Ciskowskid:

Some Probability Formulas

2022-05-11T17:40:33Z

Ciskowskid: /* Some useful probability formulas */

== Some useful probability formulas ==

'''The Addition Rule'''

Consider 2 events, ''A'' and ''B''. To find the probability that either ''A'' or ''B'' (or both) will occur, we can use the following formula, called the addition rule:

[[File:Addition_Rule_Formula.JPG]]

We subtract the probability of both ''A'' and ''B'' occurring, as to not "double count" them. This can seen better by looking at the following Venn Diagram:

[[File:A_and_B_Venn_Diagram.JPG]]

'''Conditional Probability'''

Sometimes we would like to know the probability of an event ''B'' occurring, given that another event ''A'' has already occurred. This in called conditional probability and we use the notation:

[[File:Condition_Probability_Notation.JPG]]

This is read as "The probability of B given A," and can be calculated as follows:

[[File:Conditional_Probability_Formula.JPG]]

''contributed by David Ciskowski''

Hypothesis testing

2022-05-11T17:34:21Z

Ciskowskid: /* Hypothesis Testing */

== Hypothesis Testing ==

A '''hypothesis test''' is a procedure for using observed data to decide between two competing claims, called hypotheses. The hypotheses are often statements about a parameter, like the population proportion ''p'' or the population mean ''μ''. A '''hypothesis test''' is sometimes referred to as a '''significance test'''.

'''Hypothesis Testing: The Basics'''

*The claim that we weigh evidence ''against'' in a hypothesis test is called the '''null hypothesis''' (H0). The null hypothesis has the form H0: parameter = null value.
*The claim about the population that we are trying to find evidence for is the '''alternative hypothesis''' (Ha).
**A '''one-sided''' alternative hypothesis has the form Ha: parameter < null value or Ha: parameter > null value.
**A '''two-sided''' alternative hypothesis has the form Ha: parameter ≠ null value.
*Often, H0 is a statement of no change or no difference. The alternative hypothesis states what we hope or suspect is true.
*The '''P-value''' of a test is the probability of getting evidence for the alternative hypothesis Ha that is as strong or stronger than the observed evidence when the null hypothesis H0 is true.
*Small P-values are evidence against the null hypothesis and for the alternative hypothesis because they say that the observed result is unlikely to occur when H0 is true. To determine if a P-value should be considered small, we compare it to the '''significance level α'''.
*We make a conclusion in a hypothesis test based on the P-value.
**If P-value < α: Reject H0 and conclude there is convincing evidence for Ha (in context).
**If P-value > α: Fail to reject H0 and conclude there is no convincing evidence for Ha (in context).

Reference:

Daren, S. S., & Tabor, J. (2020). ''Updated version of the practice of Statistics (Teachers Edition)'' (Sixth Edition). W H FREEMAN & CO LTD.

''contributed by Katie Ciskowski''

==Test Statistic Formulas==

[[File:Test_Statistic_Formulas.PNG]]

''contributed by David Ciskowski''

== ==

The words probability and confidence seem to come up a lot. You should be getting the message that few things are definite in our discipline, or in any empirical science. Sometimes we get it wrong.

== Type I Error ==

A level of significance of 5% is the rate you'll declare results to be significant when there are no relationships in the population. In other words, it's the rate of false alarms or false positives. Such things happen, because some samples show a relationship just by chance.

The only time you need to worry about setting the Type I error rate is when you look for a lot of effects in your data. The more effects you look for, the more likely it is that you will turn up an effect that seems bigger than it really is. This phenomenon is usually called the inflation of the overall Type I error rate, or the cumulative Type I error rate. So if you're going fishing for relationships amongst a lot of variables, and you want your readers to believe every "catch" (significant effect), you're supposed to reduce the Type I error rate by adjusting the p value downwards for declaring statistical significance.

The simplest adjustment is called the Bonferroni. For example, if you do three tests, you should reduce the p value to 0.05/3, or about 0.02. This adjustment follows quite simply from the meaning of probability, on the assumption that the three tests are independent. If the tests are not independent, the adjustment is too severe. For example, Bonferroni-adjusted 95% confidence intervals for three effects would each be 98% confidence.

Why not use a lower p value all the time, for example a p value of 0.01, to declare significance? Surely that way only one in every 100 effects you test for is likely to be bogus? Yes, but it is harder to get significant results, unless you use a bigger sample to narrow down that confidence interval. In any case, you are entitled to stay with a 5% level for one or two tests, if they are pre-planned--in other words, if you set up the whole study just to do these tests. It's only when you tack on a lot of other tests afterwards (so-called post-hoc tests) that you need to be wary of false alarms.

''contributed by Karen Burke, EdD''

== Type II Error ==

The other sort of error is the chance you'll miss the effect (i.e. declare that there is no significant effect) when it really is there. In other words, it's the rate of failed alarms or false negatives. Once again, the alarm will fail sometimes purely by chance: the effect is present in the population, but the sample you drew doesn't show it.

The smaller the sample, the more likely you are to commit a Type II error, because the confidence interval is wider and more likely to overlap zero. The Type II error needs to be considered explicitly at the time you design your study. That's when you're supposed to work out the sample size needed to make sure your study has the power to detect anything useful. For this purpose, the usual Type II error rate is set to 20%, or 10% for really classy studies. The power of the study is sometimes referred to as 80% (or 90% for a Type II error rate of 10%). In other words, the study has enough power to detect the smallest worthwhile effects 80% (or 90%) of the time.

''contributed by Karen Burke, EdD''

== Bias ==

People use the term bias to describe deviation from the truth. That's the way we use the term in statistics, too: we say that a statistic is biased if the average value of the statistic from many samples is different from the value in the population. To put it simply, the value from a sample tends to be wrong.

The easiest way to get bias is to use a sample that is in some way a non-random sample of the population: if the average subject in the sample tends to be different from the average person in the population, the effect you are looking at could well be different in the sample compared with the population.

Some statistics are biased, if we calculate them in the wrong way. Using n instead of n-1 to work out a standard deviation is a good example.

''contributed by Karen Burke, EdD''

File:Test Statistic Formulas.PNG

2022-05-11T17:33:23Z

Ciskowskid:

Confidence Intervals

2022-05-11T17:05:51Z

Ciskowskid: /* Confidence Interval: Formulas */

== Creating Confidence Intervals ==

The use of confidence intervals is in part, due to the fact that the traditional and restricted framework of statistical significance testing has not been universally endorsed, therefore creating the need for confidence intervals.

This comes down to a simple question, "Is it possible to assert something positive and tangible about the means of the groups in an experimental study?"

Instead of using significance level in a study, it maybe more beneficial to use a confidence interval (which is the opposite of the significance level).

For example, saying "the 6 month survival rate wan increased by 30 percentage points with a 99% confidence interval" than by simple saying the difference between the control group and experimental group was significant at the .01 level.

The creation of the confidence interval then, becomes the percentage remaining from the significance level. In this this case 100-1= 99%

''contributed by Mykal Kuslis, WCSU Cohort 8''

Reference:

Meyers, S., Gamst, G., & Guarino, A.J. (2017). Applied multivariate research: Design and interpretation. Thousand Oaks, CA: Sage Publications. (p.24-25)

==Confidence Intervals: The Basics==

To estimate an unknown population parameter, start with a statistic that will provide a reasonable guess. The chosen statistic is a '''point estimator''' for the parameter. The specific value of the point estimator that we use gives a '''point estimate''' for the parameter.

A '''confidence interval''' gives an interval of plausible values for an unknown population parameter based on sample data. Plausible does not mean the same thing as possible. You could argue that just about any value of a parameter is ''possible''. ''Plausible'' means that we shouldn't be surprised if any one of the values in the interval is equal to the parameter.

The interval estimate has the form ''point estimate ± margin of error''.

When calculating a confidence interval, it is common to use the form ''statistic ± (critical value) ∙ (standard deviation of statistic).

To interpret a C% confidence interval, say "We are C% confident that the interval from ____ to ____ captures the [parameter in context]."

Reference:

Daren, S. S., & Tabor, J. (2020). ''Updated version of the practice of Statistics (Teachers Edition)'' (Sixth Edition). W H FREEMAN & CO LTD.

''contributed by Katie Ciskowski''

==Confidence Interval: Formulas==

[[File:CI_Formulas.PNG]]

''contributed by David Ciskowski''

Confidence Intervals

2022-05-11T17:04:16Z

Ciskowskid: /* Confidence Interval: Formulas */

== Creating Confidence Intervals ==

The use of confidence intervals is in part, due to the fact that the traditional and restricted framework of statistical significance testing has not been universally endorsed, therefore creating the need for confidence intervals.

This comes down to a simple question, "Is it possible to assert something positive and tangible about the means of the groups in an experimental study?"

Instead of using significance level in a study, it maybe more beneficial to use a confidence interval (which is the opposite of the significance level).

For example, saying "the 6 month survival rate wan increased by 30 percentage points with a 99% confidence interval" than by simple saying the difference between the control group and experimental group was significant at the .01 level.

The creation of the confidence interval then, becomes the percentage remaining from the significance level. In this this case 100-1= 99%

''contributed by Mykal Kuslis, WCSU Cohort 8''

Reference:

Meyers, S., Gamst, G., & Guarino, A.J. (2017). Applied multivariate research: Design and interpretation. Thousand Oaks, CA: Sage Publications. (p.24-25)

==Confidence Intervals: The Basics==

To estimate an unknown population parameter, start with a statistic that will provide a reasonable guess. The chosen statistic is a '''point estimator''' for the parameter. The specific value of the point estimator that we use gives a '''point estimate''' for the parameter.

A '''confidence interval''' gives an interval of plausible values for an unknown population parameter based on sample data. Plausible does not mean the same thing as possible. You could argue that just about any value of a parameter is ''possible''. ''Plausible'' means that we shouldn't be surprised if any one of the values in the interval is equal to the parameter.

The interval estimate has the form ''point estimate ± margin of error''.

When calculating a confidence interval, it is common to use the form ''statistic ± (critical value) ∙ (standard deviation of statistic).

To interpret a C% confidence interval, say "We are C% confident that the interval from ____ to ____ captures the [parameter in context]."

Reference:

Daren, S. S., & Tabor, J. (2020). ''Updated version of the practice of Statistics (Teachers Edition)'' (Sixth Edition). W H FREEMAN & CO LTD.

''contributed by Katie Ciskowski''

==Confidence Interval: Formulas==

[[CI_Formulas.PNG]]

''contributed by David Ciskowski''

File:CI Formulas.PNG

2022-05-11T17:04:04Z

Ciskowskid:

Confidence Intervals

2022-05-11T16:45:40Z

Ciskowskid: /* Confidence Interval Formulas */

Confidence Intervals

2022-05-11T16:45:26Z

Ciskowskid: /* Confidence Intervals: The Basics */

Some Probability Formulas

2022-02-11T18:33:30Z

Ciskowskid: /* Some useful probability formulas */

Some Probability Formulas

2022-02-11T18:32:23Z

Ciskowskid: /* Some useful probability formulas */

File:Conditional Probability Formula.JPG

2022-02-11T18:31:46Z

Ciskowskid:

Some Probability Formulas

2022-02-11T18:12:17Z

Ciskowskid: /* Some useful probability formulas */

File:Condition Probability Notation.JPG

2022-02-11T18:11:27Z

Ciskowskid:

Some Probability Formulas

2022-02-11T18:06:06Z

Ciskowskid: /* Some useful probability formulas */

File:A and B Venn Diagram.JPG

2022-02-11T18:05:34Z

Ciskowskid:

Some Probability Formulas

2022-02-11T17:58:32Z

Ciskowskid: Created page with "== Some useful probability formulas == '''The Addition Rule''' Consider 2 events, ''A'' and ''B''. To find the probability that either ''A'' or ''B'' (or both) will occur, w..."

File:Addition Rule Formula.JPG

2022-02-11T17:58:22Z

Ciskowskid:

An introduction to probability

2022-02-11T17:44:14Z

Ciskowskid: /* Introduction to Probability */

== Introduction to Probability ==

'''Probability of an Event'''

If all of the outcomes in an experiment are equally likely, then the probability of an event, ''E'', occurring is given by:

[[File:P(E)_definition.JPG]]

Note: the number of outcomes that result in event ''E'' occurring can never be negative and can never be greater than the total number of outcomes, so we know:

[[File:Range_of_P(E).JPG]]

'''Theoretical Probability vs. Empirical Probability'''

A probability computed by using a probability formula is called a '''''theoretical probability'''''.

A probability found by observing the actual outcomes of an experiment that is repeated many times is called '''''empirical probability'''''.

Consider rolling a 6-sided die.

We know that each outcome is equally likely, so the theoretical probabilities are as follows:

[[File:Theoretical_Probability_of_Dice.JPG]]

However, if we actually rolled a 6-sided die 600 times and recorded the outcomes, we may find that the empirical probabilities differ:

(Geogebra [https://www.geogebra.org/m/UsoH4eNl] is a great tool for simulating this experiment)

[[File:Empirical_Probability_of_Dice.JPG]]

Notice only one outcome (rolling a 5) matched the theoretical probability.

''Contributed by David Ciskowski''

Main Page

2022-02-11T17:43:42Z

Ciskowskid: /* Modules */

<big>'''Practical Statistics for Educators'''</big>
edited and maintained by Frank LaBanca, EdD

----

== Philosophy ==

Quantitative statistical analyses can be intimidating for many educators pursing an advanced academic degree. The thought of computational math can sometimes trigger unwarranted fears.

Here, we approach statistics from a straightforward conceptually-based perspective. Our goal is to collaborate and provide insight for statistics that make them meaningful tools in the educational arena.

Each "module" corresponds with the topics presented each week, and will expand as the course progresses. A topical outline can be found @ [http://docs.google.com/Doc?id=dfqvtcqp_46hhzzcsgt ]

Comments and edits are welcome and encouraged! Please give yourself credit as you contribute. At the end of a section you insert please add the following in italics:
''contributed by <your name>''
If you are modifying content, add the following under the contribution line:
''modified by <your name>'' We are glad to accept as many modifications as necessary to give the most meaning to each section. As we asynchronously socially construct knowledge together, we can recognize the accomplishments and contributions of each writer.

''contributed by Frank LaBanca, EdD''

== Contributions ==

Our contributors [[contributions here]].

Please submit your contribution at [https://forms.gle/PBaVrxFffbk5CKgg8]

== Modules ==

1.1 [[The Greek Alphabet]] and its significance in statistics

1.2 [[An introduction to probability]] PowerPoint @[http://docs.google.com/Presentation?id=dfqvtcqp_97wcrbtsn]

1.3 [[Some Probability Formulas]]

2.1 [[Types of Data]]

2.2 [[Visualizing Data]]

2.3 Visually representing data PowerPoint @ [http://docs.google.com/Presentation?docid=dfqvtcqp_27dwth2zz2#]

2.3.1 Table 2 from LaBanca dissertation @ [http://docs.google.com/Doc?id=dfqvtcqp_25cb5pqcfw]

2.3.2 Cool graph of movie box office from NY Times [http://www.nytimes.com/interactive/2008/02/23/movies/20080223_REVENUE_GRAPHIC.html#]

2.3.4 [[Histograms]]

2.3.5 Scatterplots YouTube @ [http://youtu.be/HFuU1uxJ1tQ]

2.4 [[Shapes of distribution]]

2.5 Survey of Attitudes Toward Statistics (SATS) Data Set @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7PokF_oAoCRUg]

2.6 [[Data Screening]]

3.1 [[Central Tendency]]

3.1.1 Central Tendency and Normal Distribution PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_47dfhfr4nw ]

3.1.2 Central Tendency YouTube @ [http://youtu.be/Fn4z8RDpwDY]

3.2 [[Interquartile ranges]]

3.2.1 [[The Box Plot]]

3.2.2 Interpreting a Box Plot - video [https://www.youtube.com/watch?v=b2C9I8HuCe4]

3.3 [[Standard deviation]]

3.3.1 [[Identifying percentile ranks and scores based on standard deviation]]

3.3.1.a [[Practice Identifying percentile ranks and scores based on standard deviation]]

3.4 [[z-scores]]

4.1 [[Percentile Rank]]
4.1.1 Areas under the standard normal curve for z values @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7NstEjJ40jJOQ]

4.1.2 z scores corresponding to divisions of the area under the normal curve @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7P6SVNiBdWbEg]

4.2 Conversion of data PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_98chfzg2tf]

4.2.1 Descriptive analysis of USRT data @ [http://docs.google.com/Doc?id=dfqvtcqp_61db6smpdm ]

4.3 [[Normal Curve Equivalent scores]]

4.3 [[Standard Error of Measurement]]

4.4 [[Confidence Intervals]]

4.4 z score machine @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7N8oLJZwmr3Zw]

5.1 [[Pearson r]]

5.2 [[Rules of thumb for interpreting the size of a correlation coefficient]]

5.3 Critical values for the correlation coefficient @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7MkuRpIgceTRQ]

5.4 [[Spearman rho]]

5.5 Correlation PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_134hsxg7td7]

5.6 [[Writing samples for correlations]]

6.1 [[Inferential Statistics Definition]]

6.2 [[Sampling]]

6.3 [[Sampling distributions]]

6.4 t test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_153gr9f3hgd]

6.4.1 t -t test video [https://www.youtube.com/watch?v=N2dYGnZ70X0]

6.5 Sample data set @ [http://wolfweb.unr.edu/homepage/liu/stat/help/help.htm]

6.6 Critical values for t @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OZzyZeHg9MIA]

6.7 Helpful Tutorial for Running a t-Test in Excel @ [https://www.rwu.edu/sites/default/files/downloads/fcas/mns/running_a_t-test_in_excel.pdf]

7.1 [[Effect size]]

7.1.1 Effect size calculator @ http://www.campbellcollaboration.org/resources/effect_size_input.php

7.1.2 [[Rules of thumb for interpreting effect sizes]]

7.2 Effect size PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_204f42f67dx ]

7.3 [[Hypothesis testing]]

7.4 Hypothesis testing PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_23295tm65dx]

7.5.1 Hypothesis testing template for a correlation @ [http://docs.google.com/Doc?id=dfqvtcqp_174ccchz4ds]

7.5.2 Hypothesis testing template for a t test @ [http://docs.google.com/Doc?id=dfqvtcqp_175hjcdsjff]

8.1 [[Type I and Type II Errors]]

8.2 Type I and Type II Errors PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_258wcq56rdv]

8.3 [[Levene's p versus the test statistic p]]

8.4 [[Analysis of Variance]]

8.5 ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_265ckd9j9dv]

8.6 [[ANOVA Case study]]

8.7 ANOVA video [https://www.youtube.com/watch?v=ITf4vHhyGpc]

8.8 Critical values for the F statistic @ [http://www.sussex.ac.uk/Users/grahamh/RM1web/F-ratio%20table%202005.pdf]

8.9 [[Rules of thumb for interpreting effect sizes of ANOVAs]]

9.1 Post Hoc test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_284dbhghdc9]

9.2 [[Selecting a Post Hoc test]]

9.3 Hypothesis testing template for ANOVA @ [http://docs.google.com/Doc?id=dfqvtcqp_295ckngxdgj]

10.1 [[Chi square]]

10.1.1 Chi square video [https://www.youtube.com/watch?v=VskmMgXmkMQ]

10.2 [[Example for calculating chi square]]

10.3 Critical values for chi square @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OhBVHQWoHTIQ]

10.4 [[Chi square analysis description/sample writing]]

10.5 Chi square PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_297dhg685g8]

11.1 [[Beyond the ANOVA]]

11.2 Beyond ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_307cmrtpxg3]

11.3 2-way ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_325hrt86ggt]

11.4 2-way ANOVA template @ [http://docs.google.com/Doc?id=dfqvtcqp_372599pr6w6]

11.5 1-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1i0kIWgmXLCEIIIYCSqh2JS9Th3T_pyRC/view?usp=sharing]

11.6 2-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1wX4xhQa7KGCd1Hey7VfX33Y1uT6QHdEu/view?usp=sharing]

12.1 [[MANOVA]]

12.2 [[Homogeneity vs Homoscedacity]] (Levene vs Box's M)

12.3 [[Post Hoc ANOVAs for MANOVA]] (univariate)

12.4 [[Post Hoc Discriminant Analysis]] (multivariate)

12.5 [[Covariates]]

12.6 [[MANCOVA]]

12.7 MANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1GiErYfmCdiNQlCps3anF4Bu3C_iYS7oD/view?usp=sharing]

12.8 MANCOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1TodMQy4vQ4eHStSIAevATOuJuFg9KUTD/view?usp=sharing]

13.1 [[Multiple Regression Analysis]]

13.1.1 [[Collinearity]]

13.2 [[Multiple Linear Regression]]

13.3 Reading the MLR Output: An annotated output [https://drive.google.com/file/d/141RNyYNnuDDTNvHYi4fE8EF8qWmal1aa/view?usp=sharing]

13.4 MLR Annotated SPSS Output @ [https://drive.google.com/file/d/1SsPL1YD4VYguxLtwqM_R7GFBD5RsG9H6/view?usp=sharing]

14.1 Internal Consistency Reliability[https://docs.google.com/document/d/18K16I8u9sbwpUhW9nIENF4x9wDXAy-4wFuBp6a0AljA/edit]

14.1.1 [[Internal Consistency Reliability]]

14.2 Cronbach's Alpha[https://docs.google.com/document/d/1_eyXOcFrBcDSctM27a9T2kUlx9D8TidV_YTHk-wvTu0/edit]

14.2.1 [[Cronbach's Alpha Values]]

14.2.2 Cronbach's Alpha in SPSS [https://www.youtube.com/watch?v=Kz8OdR6lV44]

== Applied Research Designs ==

15.1 [[Instrumentation]]

15.2 [[Limitations]]

15.3 [[Practice determining the stat]]

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

Main Page

2022-02-11T17:43:30Z

Ciskowskid: /* Modules */

An introduction to probability

2022-02-11T17:35:42Z

Ciskowskid: /* Introduction to Probability */

File:Empirical Probability of Dice.JPG

2022-02-11T17:33:56Z

Ciskowskid:

An introduction to probability

2022-02-11T17:24:01Z

Ciskowskid: /* Introduction to Probability */

File:Theoretical Probability of Dice.JPG

2022-02-11T17:19:20Z

Ciskowskid:

An introduction to probability

2022-02-11T17:14:42Z

Ciskowskid: Created page with "== Introduction to Probability == '''Probability of an Event''' If all of the outcomes in an experiment are equally likely, then the probability of an event, ''E'', occurrin..."

File:Range of P(E).JPG

2022-02-11T17:09:04Z

Ciskowskid:

File:P(E) definition.JPG

2022-02-11T17:05:56Z

Ciskowskid:

Main Page

2022-02-11T17:01:48Z

Ciskowskid: /* Modules */

<big>'''Practical Statistics for Educators'''</big>
edited and maintained by Frank LaBanca, EdD

----

== Philosophy ==

Quantitative statistical analyses can be intimidating for many educators pursing an advanced academic degree. The thought of computational math can sometimes trigger unwarranted fears.

Here, we approach statistics from a straightforward conceptually-based perspective. Our goal is to collaborate and provide insight for statistics that make them meaningful tools in the educational arena.

Each "module" corresponds with the topics presented each week, and will expand as the course progresses. A topical outline can be found @ [http://docs.google.com/Doc?id=dfqvtcqp_46hhzzcsgt ]

Comments and edits are welcome and encouraged! Please give yourself credit as you contribute. At the end of a section you insert please add the following in italics:
''contributed by <your name>''
If you are modifying content, add the following under the contribution line:
''modified by <your name>'' We are glad to accept as many modifications as necessary to give the most meaning to each section. As we asynchronously socially construct knowledge together, we can recognize the accomplishments and contributions of each writer.

''contributed by Frank LaBanca, EdD''

== Contributions ==

Our contributors [[contributions here]].

Please submit your contribution at [https://forms.gle/PBaVrxFffbk5CKgg8]

== Modules ==

1.1 [[The Greek Alphabet]] and its significance in statistics

1.2 [[An introduction to probability]] PowerPoint @[http://docs.google.com/Presentation?id=dfqvtcqp_97wcrbtsn]

2.1 [[Types of Data]]

2.2 [[Visualizing Data]]

2.3 Visually representing data PowerPoint @ [http://docs.google.com/Presentation?docid=dfqvtcqp_27dwth2zz2#]

2.3.1 Table 2 from LaBanca dissertation @ [http://docs.google.com/Doc?id=dfqvtcqp_25cb5pqcfw]

2.3.2 Cool graph of movie box office from NY Times [http://www.nytimes.com/interactive/2008/02/23/movies/20080223_REVENUE_GRAPHIC.html#]

2.3.4 [[Histograms]]

2.3.5 Scatterplots YouTube @ [http://youtu.be/HFuU1uxJ1tQ]

2.4 [[Shapes of distribution]]

2.5 Survey of Attitudes Toward Statistics (SATS) Data Set @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7PokF_oAoCRUg]

2.6 [[Data Screening]]

3.1 [[Central Tendency]]

3.1.1 Central Tendency and Normal Distribution PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_47dfhfr4nw ]

3.1.2 Central Tendency YouTube @ [http://youtu.be/Fn4z8RDpwDY]

3.2 [[Interquartile ranges]]

3.2.1 [[The Box Plot]]

3.2.2 Interpreting a Box Plot - video [https://www.youtube.com/watch?v=b2C9I8HuCe4]

3.3 [[Standard deviation]]

3.3.1 [[Identifying percentile ranks and scores based on standard deviation]]

3.3.1.a [[Practice Identifying percentile ranks and scores based on standard deviation]]

3.4 [[z-scores]]

4.1 [[Percentile Rank]]
4.1.1 Areas under the standard normal curve for z values @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7NstEjJ40jJOQ]

4.1.2 z scores corresponding to divisions of the area under the normal curve @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7P6SVNiBdWbEg]

4.2 Conversion of data PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_98chfzg2tf]

4.2.1 Descriptive analysis of USRT data @ [http://docs.google.com/Doc?id=dfqvtcqp_61db6smpdm ]

4.3 [[Normal Curve Equivalent scores]]

4.3 [[Standard Error of Measurement]]

4.4 [[Confidence Intervals]]

4.4 z score machine @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7N8oLJZwmr3Zw]

5.1 [[Pearson r]]

5.2 [[Rules of thumb for interpreting the size of a correlation coefficient]]

5.3 Critical values for the correlation coefficient @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7MkuRpIgceTRQ]

5.4 [[Spearman rho]]

5.5 Correlation PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_134hsxg7td7]

5.6 [[Writing samples for correlations]]

6.1 [[Inferential Statistics Definition]]

6.2 [[Sampling]]

6.3 [[Sampling distributions]]

6.4 t test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_153gr9f3hgd]

6.4.1 t -t test video [https://www.youtube.com/watch?v=N2dYGnZ70X0]

6.5 Sample data set @ [http://wolfweb.unr.edu/homepage/liu/stat/help/help.htm]

6.6 Critical values for t @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OZzyZeHg9MIA]

6.7 Helpful Tutorial for Running a t-Test in Excel @ [https://www.rwu.edu/sites/default/files/downloads/fcas/mns/running_a_t-test_in_excel.pdf]

7.1 [[Effect size]]

7.1.1 Effect size calculator @ http://www.campbellcollaboration.org/resources/effect_size_input.php

7.1.2 [[Rules of thumb for interpreting effect sizes]]

7.2 Effect size PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_204f42f67dx ]

7.3 [[Hypothesis testing]]

7.4 Hypothesis testing PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_23295tm65dx]

7.5.1 Hypothesis testing template for a correlation @ [http://docs.google.com/Doc?id=dfqvtcqp_174ccchz4ds]

7.5.2 Hypothesis testing template for a t test @ [http://docs.google.com/Doc?id=dfqvtcqp_175hjcdsjff]

8.1 [[Type I and Type II Errors]]

8.2 Type I and Type II Errors PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_258wcq56rdv]

8.3 [[Levene's p versus the test statistic p]]

8.4 [[Analysis of Variance]]

8.5 ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_265ckd9j9dv]

8.6 [[ANOVA Case study]]

8.7 ANOVA video [https://www.youtube.com/watch?v=ITf4vHhyGpc]

8.8 Critical values for the F statistic @ [http://www.sussex.ac.uk/Users/grahamh/RM1web/F-ratio%20table%202005.pdf]

8.9 [[Rules of thumb for interpreting effect sizes of ANOVAs]]

9.1 Post Hoc test PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_284dbhghdc9]

9.2 [[Selecting a Post Hoc test]]

9.3 Hypothesis testing template for ANOVA @ [http://docs.google.com/Doc?id=dfqvtcqp_295ckngxdgj]

10.1 [[Chi square]]

10.1.1 Chi square video [https://www.youtube.com/watch?v=VskmMgXmkMQ]

10.2 [[Example for calculating chi square]]

10.3 Critical values for chi square @ [http://spreadsheets.google.com/pub?key=pmUxljSzLg7OhBVHQWoHTIQ]

10.4 [[Chi square analysis description/sample writing]]

10.5 Chi square PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_297dhg685g8]

11.1 [[Beyond the ANOVA]]

11.2 Beyond ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_307cmrtpxg3]

11.3 2-way ANOVA PowerPoint @ [http://docs.google.com/Presentation?id=dfqvtcqp_325hrt86ggt]

11.4 2-way ANOVA template @ [http://docs.google.com/Doc?id=dfqvtcqp_372599pr6w6]

11.5 1-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1i0kIWgmXLCEIIIYCSqh2JS9Th3T_pyRC/view?usp=sharing]

11.6 2-way ANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1wX4xhQa7KGCd1Hey7VfX33Y1uT6QHdEu/view?usp=sharing]

12.1 [[MANOVA]]

12.2 [[Homogeneity vs Homoscedacity]] (Levene vs Box's M)

12.3 [[Post Hoc ANOVAs for MANOVA]] (univariate)

12.4 [[Post Hoc Discriminant Analysis]] (multivariate)

12.5 [[Covariates]]

12.6 [[MANCOVA]]

12.7 MANOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1GiErYfmCdiNQlCps3anF4Bu3C_iYS7oD/view?usp=sharing]

12.8 MANCOVA Annotated SPSS Output @ [https://drive.google.com/file/d/1TodMQy4vQ4eHStSIAevATOuJuFg9KUTD/view?usp=sharing]

13.1 [[Multiple Regression Analysis]]

13.1.1 [[Collinearity]]

13.2 [[Multiple Linear Regression]]

13.3 Reading the MLR Output: An annotated output [https://drive.google.com/file/d/141RNyYNnuDDTNvHYi4fE8EF8qWmal1aa/view?usp=sharing]

13.4 MLR Annotated SPSS Output @ [https://drive.google.com/file/d/1SsPL1YD4VYguxLtwqM_R7GFBD5RsG9H6/view?usp=sharing]

14.1 Internal Consistency Reliability[https://docs.google.com/document/d/18K16I8u9sbwpUhW9nIENF4x9wDXAy-4wFuBp6a0AljA/edit]

14.1.1 [[Internal Consistency Reliability]]

14.2 Cronbach's Alpha[https://docs.google.com/document/d/1_eyXOcFrBcDSctM27a9T2kUlx9D8TidV_YTHk-wvTu0/edit]

14.2.1 [[Cronbach's Alpha Values]]

14.2.2 Cronbach's Alpha in SPSS [https://www.youtube.com/watch?v=Kz8OdR6lV44]

== Applied Research Designs ==

15.1 [[Instrumentation]]

15.2 [[Limitations]]

15.3 [[Practice determining the stat]]

== Getting started ==
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]

Standard deviation

2022-02-11T16:04:24Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a '''''sample''''' standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a '''''sample standard deviation''''' using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

The formula for a '''''population''''' standard deviation is:

[[File:Population_Standard_Deviation_Formula.JPG]]

Here is a work example for finding a '''''population standard deviation''''' using a population of 10 test scores (notice this is the same data and process as above, but with the slight difference of dividing by ''n'' instead of ''n-1''):

[[File:Worked_example_of_population_standard_deviation.JPG]]

''Contributed by David Ciskowski''

Standard deviation

2022-02-11T16:03:11Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a '''''sample''''' standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

The formula for a '''''population''''' standard deviation is:

[[File:Population_Standard_Deviation_Formula.JPG]]

Here is a work example for finding a population standard deviation using a population of 10 test scores (notice this is the same data and process as above, but with the slight difference of dividing by ''n'' instead of ''n-1''):

[[File:Worked_example_of_population_standard_deviation.JPG]]

''Contributed by David Ciskowski''

File:Worked example of population standard deviation.JPG

2022-02-11T16:02:21Z

Ciskowskid: Ciskowskid uploaded a new version of "File:Worked example of population standard deviation.JPG"

File:Worked example of population standard deviation.JPG

2022-02-11T16:00:55Z

Ciskowskid:

File:Population Standard Deviation Formula.JPG

2022-02-11T15:57:28Z

Ciskowskid: Ciskowskid uploaded a new version of "File:Population Standard Deviation Formula.JPG"

Standard deviation

2022-02-11T15:55:48Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a '''''sample''''' standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

The formula for a '''''population''''' standard deviation is:

''Contributed by David Ciskowski''

File:Population Standard Deviation Formula.JPG

2022-02-11T15:54:50Z

Ciskowskid:

Standard deviation

2022-02-11T15:53:58Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a '''''sample''''' standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

The formula for a '''''population''''' standard deviation is:

[[File:Example.jpg]]

''Contributed by David Ciskowski''

Standard deviation

2022-02-11T15:39:08Z

Ciskowskid:

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

''Contributed by David Ciskowski''

Standard deviation

2022-02-11T15:38:34Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.JPG]]

Standard deviation

2022-02-11T15:33:28Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

[[File:Worked_example_of_sample_standard_deviation.jpg]]

File:Worked example of sample standard deviation.JPG

2022-02-11T15:32:00Z

Ciskowskid:

Standard deviation

2022-02-11T15:31:31Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Here is a worked example for finding a sample standard deviation using a sample of 10 test scores:

Standard deviation

2022-02-11T15:29:08Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:Sample_Standard_Deviation_Formula.JPG]]

Standard deviation

2022-02-11T15:23:54Z

Ciskowskid:

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[Image:sample_standard_deviation_formula.jpg]]

Standard deviation

2022-02-11T15:22:46Z

Ciskowskid:

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[Image:sample standard deviation formula.jpg]]

Standard deviation

2022-02-11T15:17:18Z

Ciskowskid: /* Standard Deviation Formula */

== Why bother finding standard deviation? ==

Standard deviation can be tedious to calculate by hand, but the value found can provide some very good insight into a set of data, particularly when considered in conjunction with one or more of the measures of central tendency.

Here is a good example. Suppose I had two sets of data. The first, which I will call Set 1, has these characteristics: n = 10 (there are ten pieces of data), mean = medium = mode = 6, the minimum value in Set 1 is 1, and the maximum value in Set 1 is 11 (so the range is 10). The second set of data, whcih I will call Set 2, has exactly the same characteristics (i.e., n = 10, mean = medium = mode = 6, min = 1, max = 11 and range = 10) as Set 1. Would you think that the sets of data had exactly the same numbers in them? They might, but they might not.

Now, consider this additional information regarding the sets: Set 1 has a standard deviation of 3.87, but Set 2 has a standard deviation of 2.24. Could the sets consist of the same data now? Hopefully, it is clear that they cannot. But what do those two values (i.e., 3.87 and 2.24) tell us about the sets, if anything?

Remember that standard deviation is a "measure of dispersion", so the numbers should communicate something about how dispersed the data are in each set. In this case, we would expect the data in the first set to be "more dispersed" than the the data of the second set (since 3.87 is greater than 2.24). In other words, if you were able to look at the data of the two sets side-by-side, the Set 2's data would look more clustered around the number 6 than Set 1's data did.

So let's do that. Here are the sets, in their entirety:

Set 1 = {1, 1, 1, 6, 6, 6, 6, 11, 11, 11}
Set 2 = {1, 6, 6, 6, 6, 6, 6, 6, 6, 11}

Though the two sets have some similar qualities, it is easy to see that Set 2 has more of its data bunched near 6 than Set 1 does, or, conversely, that the data associated with Set 1 are more spread out than the data of Set 2. This fact can be determined without having to look at the actual elements of the two sets, however, by finding and understanding the standard deviations of the two sets. Knowing that Set 1 has a standard deviation of 3.87 and that Set 2 has a standard deviation of 2.24 can provide a sense of these dispersions, particularly in a relative sense.

''contributed by Chris Ruggiero''

Standard deviation measures how much dispersion there is around a mean score. A statistical formula is used to calculate the standard deviation. The larger the standard deviation, the further the score is from the mean; this can be a negative standard deviation and go below the mean or a positive standard deviation and be higher than the mean. The image below shows a normal bell curve, and where the percent of scores would lie in relation to each standard deviation.

[[File:StandardDeviationBellCurve.jpg]]

(http://medical-dictionary.thefreedictionary.com/Normal+distribution+curve)

''contributed by Cassandra Cosentino''

== A great image to explain standard deviation ==

This data set comes from a high school student research project. He did a pretest data collection, treatment, and then a post test. Notice the pretest has a greater standard deviation than the postteest (spread) and that the posttest mean was greater than the pretest mean.

[[Image:Stnad crvs.jpg]]

''contributed by Frank LaBanca, EdD''

== Standard Deviation Formula ==

The formula for a sample standard deviation is:

[[File:sample-standard-deviation-formula.jpg]]

File:Sample Standard Deviation Formula.JPG

2022-02-11T15:11:41Z

Ciskowskid:

Standard deviation

2022-02-11T15:10:37Z

Ciskowskid: