University of Michigan - Ann Arbor

The standard deviation is the most widely used measure of variability, representing the average distance that individual scores within a distribution deviate from the mean.

Standard Deviation

What does a standard deviation measure in a dataset?

Which of the following best describes what a high standard deviation indicates about a dataset?

If a dataset has a low standard deviation, what can be inferred about the data points?

Which of the following statements is true about standard deviation?

The pooled within-groups standard deviation is a type of combined average of two separate group standard deviations, typically used as the denominator when calculating Cohen's d. It is computed by adding the sum of squared differences for the first group to the sum of squared differences for the second group, dividing that total by the sum of both sample sizes, and extracting the square root.

Pooled Within-Groups Standard Deviation

The standard error is a measure of variability calculated by dividing the standard deviation ($$s$$) of a group by the square root of its sample size ($$n$$), expressed as $$SE = \frac{s}{\sqrt{n}}$$. In general, a difference between group means that exceeds two standard errors indicates statistical significance.

Standard Error

The standard deviation ($$SD$$) for describing the variability of a sample is computed by finding the difference between each score ($$X$$) and the mean ($$M$$), squaring each difference, finding the mean of these squared values, and taking the square root. The formula is: $$SD = \sqrt{\frac{\sum (X-M)^2}{N}}$$, where $$N$$ is the total number of scores.

Statistical Standard Deviation Formula

The variance, symbolized as $$SD^2$$, is a measure of variability defined as the mean of the squared differences between each score and the distribution's mean. While variance is a foundational measure in inferential statistics, its square root—the standard deviation—is typically used in descriptive statistics.

Variance

The standard deviation quantifies the average distance of scores from the mean. For example, when comparing two distributions with the same mean of $$10$$, a distribution where scores are tightly clustered near the center might have a standard deviation of $$1.69$$, meaning scores differ from the mean by about $$1.69$$ units on average. In contrast, a distribution with scores spread across a broader range might have a standard deviation of $$4.30$$, meaning scores differ from the mean by about $$4.30$$ units on average.

Example of Standard Deviations in Distributions

A $$z$$ score indicates how many standard deviations a particular raw score is above or below the mean of its distribution. It standardizes the distance of a score from the mean, providing a consistent way to describe a score's relative location.

z Score

The standard deviation ($$SD$$) is computed by determining the difference between each score ($$X$$) and the mean ($$M$$), squaring those differences, finding the mean of the squared differences, and finally taking the square root of that mean. The formula is: $$SD = \sqrt{\frac {\sum (X-M)^2}{N}}$$.

Formula for the Standard Deviation

Which statistical measure is defined as representing the average distance that individual scores within a distribution deviate from the overall average?

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