Practical Statistics for Educators - User contributions [en]

Scatter Plots

2025-12-09T05:01:33Z

Velez105:

==How to create a scatter plot in Google Sheets==

{| class="wikitable"
|-
! Step
! Action
|-
| Step 1
| Select two columns of data (first column will be graphed on the x-axis and second column will be graphed on the y-axis)
|-
| Step 2
| Words in row 1 of the columns will be labels for x- and y-axis
|-
| Step 3
| Select Insert Graph
|-
| Step 4
| Under Chart Type Select Scatter
|-
| Step 5
| Select Customize tab to edit the chart
|-
| Step 6
| To add a trendline, select series and check trendline
|-
| Step 7
| To add Pearson Correlation Coefficient (r), select series and check show R^2
|-
| Step 8
| Change the min and max values on the axes by selecting Horizontal axis or Vertical axis and changing min and/or max values to new values
|}
''contributed by Sara Dalton''

==How to label the dots on a scatter plot in Google Sheets==
{| class="wikitable"
|-
! Step
! Action
|-
| Step 1
| Create a scatter plot which includes three columns--two columns of data and a third column of labels
|-
| Step 2
| X-axis is first column of data
|-
| Step 3
| Series is second column of data
|-
| Step 4
| Edit Series to add labels
|-
| Step 5
| Under series, select or type in the range of the third column that contains the labels
|}

Helpful hint:
If some of the labels do not show up at the top of the chart, change the y axis by a half or one point and the data tags will become visible.

''contributed by Sara Dalton''
''edited by Paula Connolly''

==Image of different types of correlation==

[[File:correlation.png||test]]

Image link: https://medium.com/@dipti.rohan.pawar/correlation-statistical-analysis-9471411f0431

''contributed by Sara Dalton''

==Image of scatter plots associated with different values of r==

[[File:Scatter_Plot_and_r.png||test]]

Image link: https://stackoverflow.com/questions/7631799/what-does-correlation-coefficient-actually-represent

''contributed by Sara Dalton''

==What is a scatterplot?==

A '''scatterplot''' shows the relationship between two '''quantitative variables''' measured on the same '''individuals'''. The values of one variable appear on the horizontal axis (x axis) and the values of the other variable appear on the vertical axis (y axis). Each individual in the data set appears as a point on the graph.

A '''quantitative variable''' takes number values that are quantities - counts or measurements. Number of people and household income are quantitative variables.

An '''individual''' is an object described in a set of data. Individuals can be people, animals, or things.

'''Scatterplots''' are the only choice for displaying the relationship between two quantitative variables. For a single quantitative variable, there are many choices for displaying its distribution, including dotplots, histograms, boxplots and stemplots.

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''

[[Scatter Plots]]
A type of graph that uses dots to represent values for two different numeric variables, showing the relationship (correlation, patterns, clusters, outliers) between them, with one variable on the x-axis (independent) and the other on the y-axis (dependent). It helps visualize whether variables increase together (positive correlation), decrease together (negative correlation), or show no clear pattern, revealing trends not evident in the raw data.

== Key Components & How it Works ==
1. Variables: Two different continuous variables are used, one for the horizontal (x) axis and one for the vertical (y) axis.
2. Data Points: Each pair of (x, y) values from your data becomes a single dot on the graph.
3. Axes: The x-axis usually shows the independent variable (what you change), and the y-axis shows the dependent variable (what’s affected).
What It Shows
Positive Correlation
Negative Correlation
No Correlation
Cluster/Gaps
Outliers
Trend Lines
== Main Purpose of Scatter Plots ==
1) To visually test relationships or associations between variables, notes Coursera.
2) To identify patterns, trends, and anomalies in data.
3) To support or disprove hypotheses about how variables interact.

‘ ‘contributed by Jerry Velez’ ’

Z-scores

2025-12-09T05:00:46Z

Velez105: /* Z scores */

'''A Brief Explanation of Z-Scores:'''
A z-score is a standard score that is used by researchers to add focus and clarity to data. Z-scores indicate how many standard deviations a raw score is from the mean. The mean is fixed at zero and standard deviations are fixed at 1. For example, suppose the mean test score for a sample is 80 with a standard deviation of 12 and you scored a 98 on that test. Your z-score is +1.5, indicating that you scored 1.5 standard deviations above the mean. If a z-score is close to zero the corresponding raw score is close to the mean for the test. If a z-score is -2 the corresponding raw score is 2 standard deviations below the mean.

''contributed by Helen Knudsen''

==Impact of converting raw scores to z-scores==

When each raw score is converted to a z score:

1. The distribution of standard scores is similar in shape to the distribution of raw scores
2. The mean of the distribution of z scores will always equal 0, regardless of the value of the mean in the raw score distribution.
3. Both the variance of the distribution and the standard deviation of z scores always equals 1.

[I would like to insert an image here and am unsure how]

It is helpful to see what this looks like in a side-by-side distribution of raw and standard scores. Notice that the mean raw scores are 6.0; whereas the standard score is set at a mean of zero. And whereas the standard deviation from the raw scores was 3.18, when converted to standard scores, the standard deviation is 1.00 (Hinkle, Wiersma, & Jurs, 2003, p. 72).

In sum, calculating a z score for each raw score in a distribution will transform the original distribution of scores into one with identical shape but a mean of 0 and a standard deviation of 1 (Hinkle, Wiersma, & Jurs, 2003, p. 71).

References:
Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

''contributed by Emily Kilbourn''

== Calculating Z-scores using SPSS ==

''I, among others were having a hard time calculating Z-scores using the SPSS program. Amy, Michelle and I brainstormed last week, but had no luck. The book is vague in terms of how to approach it. Thanks for the guidance, Frank.''

'''<big>When calculating Z-scores on SPSS, follow these directions:</big>'''

1) Once you have the data entered in SPSS, click on "Analyze", "Descriptive Statistics", "Descriptives".

2) Move the variable over that you want to analyze.

3) Click on the small box that states, "Save standardized values as variables".

4) Click on "Options" if you would like to calculate mean, median, mode, etc. in addition to Z-scores.

5) Click "OK".

6) The Z-scores will appear in a separate column in the data editor.

''contributed by Chris Longo''

'''<big>To copy z-scores from a spreadsheet to SPSS keeping the values, follow these directions:</big>'''

1) Copy the z-scores from the spreadsheet

2) Go to "paste special" and go down to "values only."

3) The z-score values will appear in SPSS.

''contributed by Paula Connolly''

== Video Tutorial: How to Calculate and Interpret Z-Scores in SPSS==

https://youtu.be/Soi1iXxpGmA

''contributed by Sandra Peña''

[[Z scores]]
In statistics, it measures how many standard deviations a data point is from the mean, transforming data into a standard normal distribution to compare values from different datasets, identify outliers, and determine probabilities using a z-table. A positive z-score means the value is above the mean, while a negative one means it's below.

== Formula==
z = (X – μ) / σ
X: The raw data point.
μ (mu): The population mean.
σ (sigma): The population standard deviation.
== Key Uses ==
Standardization: Converts any dataset to a standard scale, allowing for comparisons (e.g., comparing a test score in one class to another).
Probability: Helps find the percentage of data falling below or above a certain point using a z-table.
Outlier Detection: Values with large z-scores (e.g., beyond ±2 or ±3) are often considered outliers.
== Interpretation Example ==
If a student scores 1150 on the SAT (mean=500, std dev=100), their z-score is (1150-500)/100 = 1.5.
This means their score is 1.5 standard deviations above the average.
Looking up z=1.5 on a z-table shows about 93.32% of scores are below it, so only 6.68% scored higher.
== Key Uses in Education ==
Comparing Subjects
Outliers Identification
Percentiles
Composite Scores
Program Effectiveness Evaluation

[https://www.youtube.com/watch?v=OavI_GiQUtI]

‘ ‘contributed by Jerry Velez’ ’

Z-scores

2025-12-09T05:00:24Z

Velez105:

'''A Brief Explanation of Z-Scores:'''
A z-score is a standard score that is used by researchers to add focus and clarity to data. Z-scores indicate how many standard deviations a raw score is from the mean. The mean is fixed at zero and standard deviations are fixed at 1. For example, suppose the mean test score for a sample is 80 with a standard deviation of 12 and you scored a 98 on that test. Your z-score is +1.5, indicating that you scored 1.5 standard deviations above the mean. If a z-score is close to zero the corresponding raw score is close to the mean for the test. If a z-score is -2 the corresponding raw score is 2 standard deviations below the mean.

''contributed by Helen Knudsen''

==Impact of converting raw scores to z-scores==

When each raw score is converted to a z score:

1. The distribution of standard scores is similar in shape to the distribution of raw scores
2. The mean of the distribution of z scores will always equal 0, regardless of the value of the mean in the raw score distribution.
3. Both the variance of the distribution and the standard deviation of z scores always equals 1.

[I would like to insert an image here and am unsure how]

It is helpful to see what this looks like in a side-by-side distribution of raw and standard scores. Notice that the mean raw scores are 6.0; whereas the standard score is set at a mean of zero. And whereas the standard deviation from the raw scores was 3.18, when converted to standard scores, the standard deviation is 1.00 (Hinkle, Wiersma, & Jurs, 2003, p. 72).

In sum, calculating a z score for each raw score in a distribution will transform the original distribution of scores into one with identical shape but a mean of 0 and a standard deviation of 1 (Hinkle, Wiersma, & Jurs, 2003, p. 71).

References:
Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

''contributed by Emily Kilbourn''

== Calculating Z-scores using SPSS ==

''I, among others were having a hard time calculating Z-scores using the SPSS program. Amy, Michelle and I brainstormed last week, but had no luck. The book is vague in terms of how to approach it. Thanks for the guidance, Frank.''

'''<big>When calculating Z-scores on SPSS, follow these directions:</big>'''

1) Once you have the data entered in SPSS, click on "Analyze", "Descriptive Statistics", "Descriptives".

2) Move the variable over that you want to analyze.

3) Click on the small box that states, "Save standardized values as variables".

4) Click on "Options" if you would like to calculate mean, median, mode, etc. in addition to Z-scores.

5) Click "OK".

6) The Z-scores will appear in a separate column in the data editor.

''contributed by Chris Longo''

'''<big>To copy z-scores from a spreadsheet to SPSS keeping the values, follow these directions:</big>'''

1) Copy the z-scores from the spreadsheet

2) Go to "paste special" and go down to "values only."

3) The z-score values will appear in SPSS.

''contributed by Paula Connolly''

== Video Tutorial: How to Calculate and Interpret Z-Scores in SPSS==

https://youtu.be/Soi1iXxpGmA

''contributed by Sandra Peña''

== [[Z scores]] ==
In statistics, it measures how many standard deviations a data point is from the mean, transforming data into a standard normal distribution to compare values from different datasets, identify outliers, and determine probabilities using a z-table. A positive z-score means the value is above the mean, while a negative one means it's below.

== Formula==
z = (X – μ) / σ
X: The raw data point.
μ (mu): The population mean.
σ (sigma): The population standard deviation.
== Key Uses ==
Standardization: Converts any dataset to a standard scale, allowing for comparisons (e.g., comparing a test score in one class to another).
Probability: Helps find the percentage of data falling below or above a certain point using a z-table.
Outlier Detection: Values with large z-scores (e.g., beyond ±2 or ±3) are often considered outliers.
== Interpretation Example ==
If a student scores 1150 on the SAT (mean=500, std dev=100), their z-score is (1150-500)/100 = 1.5.
This means their score is 1.5 standard deviations above the average.
Looking up z=1.5 on a z-table shows about 93.32% of scores are below it, so only 6.68% scored higher.
== Key Uses in Education ==
Comparing Subjects
Outliers Identification
Percentiles
Composite Scores
Program Effectiveness Evaluation

[https://www.youtube.com/watch?v=OavI_GiQUtI]

‘ ‘contributed by Jerry Velez’ ’

Standard deviation

2025-12-09T04:57:56Z

Velez105:

== 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]]

== Worked Example ==
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 (SD) provides the amount of variation or the consistency of a dataset by giving the difference between the individual data points and the mean. In the case where the mean indicates the center of the point, standard deviation is employed to demonstrate how tight or loose the scores are around the mean. They may be combined to provide a more detailed picture of student or program performance. When SD is small, the scores are highly clustered, like the students have scored in a similar fashion. In order to give an example, where the mean score is 85 and the SD is 2, the majority of students fell within the range of 83-87.
On the other hand, when SD is large, the scores are more dispersed, whereby some students are very far below or above the mean. The mean of two classes can be equal to 85, but one with a SD of 3 is more homogeneous than SD of 10. Two classes can be equal in terms of 85 average, but one with SD of 3 is more homogeneous than one with SD of 10. Variability in education assists teachers in interpreting the differences in achievement and consistency in teaching. When the SD in student performance is high, it may indicate that teaching methods are effective with one group of students and not with other students, which defines the necessity of differentiation.
On the other hand, a small SD may indicate fair results or a ceiling effect in the design of the assessment. The SD is acquired in SPSS under Analyze - Descriptive Statistics - Descriptives. The data can be represented as visual tools like boxplots or histograms, which can be easier to interpret to determine the variability. Many inferential tests, including t-tests and ANOVA, are also based on the SD, as it has an effect on confidence intervals and effect sizes. Concisely, the standard deviation will convert numerical data into useful narratives on learning diversity, program consistency, and classroom equity.

''contributed by Casimir007''

[[Standard Deviation]]
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells you, on average, how much each data point deviates (differs) from the mean of the entire set.
Low SD indicates that the data points are clustered closely around the mean (highly consistent).
High SD indicates that the data points are spread out over a broader range (high variability).
== Why Standard Deviation Matters ==
SD is crucial because the mean alone can be misleading. It provides the necessary context to interpret the central value of a data set.
Consistency: It measures the consistency or reliability of data. For example, two classes might have an average test score (mean) of 75, but the one with a lower SD has more consistent scores (clustered around 75), while the one with a higher SD has scores that are widely spread (many high and many low scores).
Data Interpretation (Normal Distribution): For data that follows a normal distribution (bell curve), SD allows for quick interpretation using the Empirical Rule (68-95-99.7 Rule):
Approximately 68% of the data falls within 1 SD of the mean.
Approximately 95% falls within 2 SDs of the mean
Outlier Identification: It helps determine what constitutes a "normal" range of values and thus identifies potential outliers (data points that are statistically far from the mean).

== Main Elements and Calculation Steps ==
The Standard Deviation is the square root of the Variance. The calculation is based on four main elements:
Element Symbol Description
Individual Value xᵢ: Each data point in the set.
Mean μ: The simple average of all data points.
Number of Values N: The total count of data points.
Summation Σ: The command to add up all results of an operation.

== Calculation Steps ==
1. Calculate the Mean (μ): Add all the data points and divide by the total number of points (N).
2. Find the Deviation and Square It: For each data point (xᵢ), subtract the mean (μ), and then square the result: (xᵢ - μ)². (Squaring ensures all values are positive and emphasizes larger deviations)
3. Sum the Squared Deviations: Add up all the squared results from Step 2. This is called the Sum of Squares (Σ (xᵢ -μ)².
4. Calculate the Variance: Divide the Sum of Squares by the total number of data points (N). For a sample standard deviation, divide by n -1, which is known as Bessel’s correction.
5. Take the Square Root: Take the square root of the result from Step 4. This is the Standard Deviation (σ).
[https://www.youtube.com/watch?v=hMs9OeyIqD8]
‘ ‘contributed by Jerry Velez’ ’

Types of Data

2025-12-09T04:54:33Z

Velez105: Data types matter

''(based on Hinkle, Wiersma, & Jurs, 2003 [[refs]])''

Collected data are the results of the measurement of factors. For example, a student's knowledge of biology might be measured by a test or a written laboratory report. A grade on a test or lab represents measurement of knowledge. If a teacher examines the types of questions asked on a test, different levels of understanding are bound to be determined by the types of questions asked. Perhaps some questions are factual in nature, only requiring students to recall information. Some might be conceptual, which would utilize more higher-order thinking skills. Yet others might be analytical in nature, which, too, would be more higher-order computational skills. In any event, understanding is assessed, and assigned a numerical value which translates to a grade that depicts the measurement of mastery of information.

Not all measurement is the same. Some measures are more [[accurate]] than others. Saying a UConn basketball player is tall is different from saying that her height is six foot five inches (or 1.96 meters, if I am being a responsible, metric-oriented scientist). There is a level of accuracy associated with the quantified measurement that is not present in the qualitative description of tall.

It is reasonable to say that some measurements are more amenable to accuracy than others. We can much more easily measure the basketball player's wingspan that we can measure an affective trait, like anxiety before the big game against Tennessee.

When choosing a statistical method to evaluate data, it is important to consider the accuracy of the type of measurement used. Scales of measurement are hierarchically categorized based on their level of accuracy. From least accurate to most accurate, the scales are: i.) nominal, ii.) ordinal, iii.) interval, and iv.) ratio.

''contributed by Frank LaBanca, EdD''
''edited by Paula Connolly

== Nominal Scale ==

The least accurate measurement scale is termed nominal. This is sometimes referred to as categorical data. As the name implies, the measurements are classified by categories based on some defined characteristics. Generally, the number of objects in each category is counted for a total. Gender and ethnic background would be examples of nominal data that might be used in an educational setting.

Example:

Using gender as a nominal data source, the two categories (cases, or levels) are male and female. A tally of males and females can be counted to determine how many objects (in this case, individuals) fit into each of the two nominal cases.

''Nominal data has the following properties:''
* Data categories are mutually exclusive. An object can belong to one and only one category.
* There is no logical order (or reason for a logical order) for categories.

''contributed by Frank LaBanca, EdD''

== Ordinal Scale ==

One of the key features to nominal data is that there is no logical order for categories. However, in an ordinal scale, categories still exist, but there is a ''logical'' organization and ordering to the categories. Ordinal scale is sometimes referred to as rank data. Scores can be ranked from highest to lowest, and then categorized within that order framework. The letter grading system (A, B, C, D, F) is an example of ordinal scale data.

Examples:

1. We know that someone who gets a grade of A has a higher grade than a person with a grade of B. However, we cannot infer that the distance between students with grades of A and B respectively are equal from students with grades of B and C.

2. A cooperating teacher has had four student teachers over the years and is asked to rank them from best to worse. He assigns them values:
* Jim = 1
* Susie = 2
* Roberta = 3
* Carl = 4

We can't say for certain that a Jim (1) compared to a Susie (2) is equal distance from Susie (2) to Roberta (3). So although we often assign a numerical value to each, we must be '''''extremely cautious''''' when considering differences. The process of [http://www.marvelousessays.com essay writing] will be much easier with MarvelousEssays.Com as there are a lot of highly professional and talented writers who are always eager to help you out with any sort of academic assignments regardless of the complexity levels. I do know what I�m talking about! 1-2 may not equal 2-3 on the ordinal scale

''Ordindal data has the following properties:''
* Data categories are mutually exclusive
* Data categories have a logical order
* Data categories are scaled or ranked according to the amount of a particular characteristic present

''contributed by Frank LaBanca, EdD''

== Interval ==

Interval level data has all of the properties of nominal and ordinal with the addition of intervals between categories being equal. Sometimes the interval scale is referred to as the equal unit scale.

Examples:

1. We might ask someone if they agree or disagree with a statement. If the scale is 4-point, for example
* Strongly agree
* Agree
* Disagree
* Strongly disagree

We are assuming that the distance from strongly agree to agree is the same as agree is to disagree. This means that we can interpret differences in the distance along the scale. If we contrast this to an ordinal scale, we can only talk about differences in order, not differences in the degree of order. In this case, we must be very careful to ensure that our distances along the scale make logical sense. Sometimes we would term this as "equally appearing intervals."

2. Dates are also interval data. A 1-week treatment from September 1 to September 7 is half of a 2-week treatment from September 1 to September 14.

3. Although not applicable to Educational Research, temperature is also an interval data scale. Temperature is an important model to consider because there is a zero on a temperature scale, but notice that zero is NOT the absence of the trait or the start of the scale. Zero is still a temperature - it is just another value along the scale's continuum.

''Interval data has the following properties:''
* Data categories are mutually exclusive
* Data categories have a logical order
* Data categories are scaled
* There are equal distances between characteristics and they are represented by equal distances in the numbers assigned to the categories.
* Zero is just a point along the scale.

''contributed by Frank LaBanca, EdD''

== Ratio ==

The highest level in the measurement scale hierarch is the ratio scale. Ratio-level data is generally considered the most precise method of measurment. Ratio data is similar to interval data with the added feature of having a true zero point. The true zero represents the absence of the characteristic that is being measured. Unfortunately, ratio data is not often available in social science/educational research.

Examples:

1. Physical science data is often available as ratio data. For example: mass, length, or energy.

2. In social research some ratio data examples: age, years of teacher experience, score on a 100-point test.

''Ratio data has the following properties:''
* Data categories are mutually exclusive
* Data categories have a logical order
* Data categories are scaled
* There are equal distances between characteristics and they are represented by equal distances in the numbers assigned to the categories.
* Zero is a point on the scale which represents the absence of a characteristic

''contributed by Frank LaBanca, EdD''

== Summary ==

The four levels of measurement are as follows:
{| class="wikitable"
|-
! Data Type
! Explanation
|-
| Nominal
| Categories without order
|-
| Ordinal
| Ordered categories
|-
| Interval
| Ordered categories with equal units between categories
|-
| Ratio
| Ordered categories with equal units between categories and contains a true zero point
|}
''contributed by Frank LaBanca, EdD''

Don't forget that the acronym for the levels of data is '''''NOIR''''', or black, in French. Very helpful hint from Dr. Delcourt.

''contributed by Susan Guertin''

[[Types of Data]]
Data types matter because they dictate the appropriate statistical tests and visualizations that can be applied. Using the wrong analysis for a data type can lead to misinterpretations and flawed conclusions about student performance or program effectiveness. For example, you can't calculate a meaningful average (mean) from qualitative data such as student feedback comments, nor can you accurately rank students by their favorite subjects. Correct data typing ensures validity and reliability in educational research and assessment.
== Main Types of Data ==
Data is categorized into two main types:
1. Qualitative (Categorical) Data: This type describes qualities or characteristics that cannot be measured numerically. It places individuals or items into groups or categories.
a. Nominal: Data that can only be named or labeled without a natural order (e.g., student ID number, type of school, yes/no survey responses).
b. Ordinal: Data that can be ranked or put in order, but the difference between the ranks is not uniform or meaningful (e.g., satisfaction ratings like Poor, Fair, Good, Excellent, letter grades like A, B, C).
2. Quantitative (Numerical) Data: This type represents counts or measurements where the numbers have mathematical meaning.
a. Interval: Data where the difference between values is meaningful, but there is no actual zero point (e.g., temperature in Celsius, dates).
b. Ratio: Data where the difference is meaningful, and there is an actual zero point, meaning zero signifies the absence of the quantity (e.g., number of correct answers on a test, student height, time spent studying).

[https://www.dataspire.org/our-data-literacy-resources]

‘ ‘contributed by Jerry Velez’ ’

Central Tendency

2025-12-05T01:36:20Z

Velez105: Central tendency is a single value that attempts to describe a set of data by identifying its central position. The key measures of central tendency presented in the table are the Mean and the Median.

Central Tendency is commonly referred to as the "measure of central tendency". A measure of central tendency is used to describe a data set by identifying the central position within that set of data. In statistics, the three most commonly used measures of central tendency are mean, median, and mode.

Mean: The mean is the average of all the numbers within a data set. To find the mean value, one would add up all the values in a data set and divide that sum by the total number of data points within the data set.
Example - 3+4+5+6+7 = 25. There are 5 values in this data set. 25/5 = 5. In this scenario the mean, or average, is 5.

Median: The median is the middle point in a sorted set of data. The median is identified by organizing the data set into order of magnitude (starting with the smallest number).
Once sorted the median is identified as the number directly in the middle of that sorted data.
Example - 6, 9, 23, 15, 2. If we put this data set in order by magnitude it is displayed as: 2, 6, 9, 15, 23. In this data set 9 is the median.

Mode: The mode is the number that occurs most frequently in a data set.
Example - 2, 3, 15, 3, 5, 7, 8, 3, 2, 1, 10, 9.
In this data set, 3, is the number that occurs most frequently and would be identified as the mode.

''contributed by Scott Trungadi''

''Having finally mastered the skills necessary to report my data, and thanks to the help from fellow students in my doctoral program, I was ready to write up a description of what all the numbers meant. I was excited to have reached this point in my central tendency assignment, as there is one thing I love doing, and that is write. Finally, something I might be good at! However, this also meant that I needed to understand and be able to explain what all the numbers meant.''

In October of 2007 during the kindergarten year, the Letter Naming Fluency
( LNF) portion of the Dynamic Indicators of Basic Early Literacy Skills (Dibels) was administered to a class of 18 kindergarten students, 10 males and 8 females.
The SPSS helped to process the following data regarding the testing administration. The mean score for the eighteen students for the Beginning LNF portion of Dibels was 32.33 with the median or midpoint being 35.0(Table 1). By gender, the boys in the class scored slightly lower with the mean score of 32.30 and the midpoint or median score of 34.50 as compared to the girls’ mean score of 32.38 and a median of 37.00(Table 5).

The standard deviation is based on all the scores in the group and is determined by how much each score deviates from the mean, or in other words, it is an estimate of what the range of scores probably was. The standard deviation tells me that in the Beginning and Ending LNF administration, all students tested, in a similar range-16.01 and 16.87(Tables 1 and 2).
However, in looking at the Beginning LNF scores analyzed by gender, there is a large discrepancy between how well the boys did when compared to the girls. There was a higher standard of deviation for the boys than the girls, respectively 16.34 for the boys and 6.71 for the girls. In trying to understand the possible reasons for this, one must consider birthdates. Although birthdates were not considered in this collection of data, it is important to note that there was a higher incidence of younger birth dates for the boys than the girls which might account for this wide range in the boys’ scores.

The Z scores in this statistical analysis refer to how many standard deviations a particular raw score lies above or below the group means. Table 6 indicates the range of Z scores for the students who had taken the Ending LNF portion of the Dibels test. The score range from 1.44, or 1.44 standard deviations above the group mean of 64.67 to -1.82, or 1.82 below the group mean of 64.67.

''I actually felt that I began to really understand what everything meant as I was scripting my report. It was very helpful. Hope this helps someone!''

''contributed by Debbie Mumford''

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Skewness and Central Tendency
[[File:Skeweness.jpg]]

Identifying where the mean, median, and mode are in relation to each other can help to determine the skew of a curve.

In a normal distribution, the mean, median, and mode will all be very close (mean = median = mode)

In a distribution that is skewed ''left'', otherwise known as a negatively skewed distribution, it is outliers on the lower end of the number line that is impacting the shape of the curve. In a curve that is skewed left the mean (which is the mathematical average) will be furthest to the left on the number line, the median remains at the mid point of the distribution on the number line, and the mode will be the farthest point to the right.

In a distribution that is skewed ''right'', otherwise known as a positively skewed distribution, it is outliers on the higher end of the number line that is impacting the shape of the curve. In a curve that is skewed right the mean (which is the mathematical average) will be furthest to the right on the number line, the median remains at the mid point of the distribution on the number line, and the mode will be the farthest point to the left.

''contributed by Ashley Brooksbank''

Image citation: Statistics: Basic Statistics II. (n.d.). Retrieved from ''https://guides.douglascollege.ca/c.php?g=408742&p=2970198.'''

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Descriptive Statistics

Descriptive statistics are very important because if we simply presented our raw data it would be hard to visualize what the data was showing, especially if there was a lot of it. Descriptive statistics therefore enables us to present the data in a more meaningful way, which allows simpler interpretation of the data.

Measures of central tendency: these are ways of describing the central position of a frequency distribution for a group of data. In this case, the frequency distribution is simply the distribution and pattern of marks scored by the 100 students from the lowest to the highest.

Measures of spread: these are ways of summarizing a group of data by describing how spread out the scores are. For example, the mean score of our 100 students may be 65 out of 100. However, not all students will have scored 65 marks. Rather, their scores will be spread out. Some will be lower and others higher. Measures of spread help us to summarize how spread out these scores are.

Both of these measures are important to calculate through SPSS prior to conducting other analyses as they can speak to you about what your data is saying and inform you of the next steps you should take.

''contributed by Sheri Prendergast''

'''Central Tendency'''

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One of the most basic statistical concepts in educational research is central tendency, which determines the median or the average of a data set, providing a single figure that can best represent a whole group of observations. The knowledge of central tendency enables teachers to analyze the data at the classroom and school level effectively, and draw conclusions regarding the performance of students and the effectiveness of the instructional process. The three significant measures include mean, median, and mode. The arithmetic mean is the average of all the data points, which is calculated by adding all the data points and dividing them by the number of cases. It suits interval or ratio data that are symmetrically distributed. Mean is, however, sensitive to the outliers with a few very high or very low scores having the ability to skew the mean. The median, the middle value that appears when all the values have been ranked, is not sensitive to extreme values and is applied in the case of skewed data. The mode, which is the most common score, is particularly useful when the data are nominal or categorical, such as the most popular grade or preferred learning style. In the educational sphere, both measures narrate a different tale. The mean reading score in a classroom can be 85, for instance, but the median can be 90, implying that some low achievers dragged the mean score. These measures can be calculated in SPSS under Analyze - Descriptive Statistics - Frequencies. With careful choice and interpretation of the appropriate measure of central tendency, teachers will be able to see trends and detect learning differences, and present findings in a manner that informs teaching and promotes equity.

''contributed by Casimir007''

[[Central Tendency]]
Central tendency is a single value that attempts to describe a set of data by identifying its central position. The key measures of central tendency presented in the table are the Mean and the Median.

The Intervention Group consistently shows higher measures of central tendency compared to the Comparison Group.
• Pretest Mean: 30.58 (Intervention) vs. 28.20 (Comparison)
• Posttest Mean: 33.55 (Intervention) vs. 26.12 (Comparison)
• Posttest Median (50th Percentile): 31.00 (Intervention) vs. 23.00 (Comparison)
The data strongly suggest that the intervention had a positive effect, as the Posttest Intervention Mean (33.55) and Median (31.00) are the highest values in the table. The sharp decline in the Posttest Comparison Mean (26.12) is also noteworthy, suggesting different learning outcomes across the groups.
== Learning Process ==
It is important to understand and report central tendency, which involves selecting the appropriate measure (mean, median, or mode) and then calculating and interpreting it in the context of the data. When the data is skewed (as indicated by the Skewness values, which are all positive), the Median often provides a better representation of the typical score than the Mean because it is less affected by extreme outliers.
• I learned that simply calculating the mean isn't enough; I must also check the skewness, like the Posttest Comparison Skewness of 1.562, which suggests the median score of 23.00 is a more reliable measure of central tendency than the mean of 26.12.
• I realized that the learning process requires not just computation, but comparative analysis, seeing how the Intervention Group's Mean increased by 2.97 points (33.55 - 30.58), while the Comparison Group's Mean decreased by 2.08 points (26.12 - 28.20), effectively highlighting the different impacts of the two approaches.
== Standard Deviation ==
Standard Deviation (SD) is a measure of the amount of variation or dispersion of a set of values. It quantifies how spread out the scores are from the mean. A low SD means scores are clustered closely around the mean, while a high SD means scores are more widely dispersed.
== Using the table as an example ==
• The Pretest Comparison group has an SD of 7.84. This is lower than the Posttest Intervention SD of 11.01.
• The Posttest Intervention SD of 11.01 is the largest, indicating that the scores in this group are the most spread out from their mean (33.55). The largest range (Max 61.00 - Min 21.00 = 40) also contributes to this large SD.
== Z-Score Analysis ==
This analysis helps put individual scores into context.
== Example of Z-Score Calculation ==
Let's find the Z-score for a student in the Posttest Intervention Group who scored 50.
This means that a score of 50.00 in the Posttest Intervention Group is 1.49 standard deviations above the mean, which is a very high score for that group.
[https://www.mygreatlearning.com/academy/learn-for-free/courses/measures-of-central-tendency]
‘‘contributed by Jerry Velez’’