Standard deviation

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.

(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.

contributed by Frank LaBanca, EdD

Standard Deviation Formula

The formula for a sample standard deviation is:

Worked Example

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

The formula for a population standard deviation is:

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):

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). [1] ‘ ‘contributed by Jerry Velez’ ’