Comparing Variability of Data Sets: What to Know 4

Standard Deviation

[perspective video — a 3-instructor video showing how to think about standard deviation as a measure of variability. Cover the parts of the formula (go into why squaring, why df if desired) but emphasize the concept of variability from std dev and variance more so than the technical use of the formula.]

Standard Deviation

Standard deviation is a measure of how spread out observations are from the mean. The symbol we use to denote standard deviation differs depending on whether we are discussing a sample or a population. We use the Greek letter [latex]\sigma[/latex] (sigma) to denote the standard deviation of a population of observations. We use the Latin letter [latex]s[/latex] to denote the standard deviation of a sample of observations.

Standard Deviation

The following formulas are used to calculate the standard deviation of a population and a sample:

Standard deviation of a population: [latex]\sigma = \sqrt{\dfrac{\sum \left(x-\mu\right)^2}{n}}[/latex], where [latex]\mu[/latex] represents the population mean.

Standard deviation of a sample: [latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex], where [latex]\bar{x}[/latex] represents the sample mean.

The following steps can be applied to calculate a standard deviation by hand.

Calculate the mean of the population or sample.
Take the difference between each data value and the mean. Then square each difference.
Add up all the squared differences
Divide by either the total number of observations in the case of a population or by 1 fewer than the total in the case of a sample.
Take the square root of the result of the division in step 4.

example

A sample of observations is listed below. Find its standard deviation.

[latex]8, 7, 13, 15, 23, 18[/latex]

Show Solution

Here is a breakdown of the formula for standard deviation of a sample, [latex]s[/latex].

[latex]s=\sqrt{\dfrac{\sum \left(x-\bar{x}\right)^2}{n-1}}[/latex]

The distance from each observation to the mean is known as a deviation from the mean and is expressed as [latex]\left(x-\bar{x}\right)[/latex]
The deviations from the mean are squared in the formula because some observations are above the mean, thus [latex]\left(x-\bar{x}\right)>0[/latex] (the difference is positive), and some observations are below the mean, thus [latex]\left(x-\bar{x}\right)<0[/latex] (the difference is negative). Squaring ensures the differences will each be expressed as positive distances and won’t cancel each other out when summed up.
The [latex]\sum[/latex] symbol sums up the squared deviations for all [latex]n[/latex] observations.
The denominator in the formula for a sample standard deviation is [latex]\left(n-1\right)[/latex] rather than [latex]n[/latex] as in the formula for the population standard deviation.
- Why do we divide by 1 fewer than the sample size, [latex]\left(n-1\right)[/latex]?
  - The answer to that is complicated but here are some ideas that may help

Why do we divide by [latex]\left(n-1\right)[/latex]?

Because the sample standard deviation is an underestimation. Recall that a sample is representative of a population if the characteristics of the sample tend to be similar to the characteristics of the population from which it was obtained. But the sample standard deviation tends to underestimate the population standard deviation (this can be shown mathematically but its beyond the scope of what we need here). We can fix that by increasing the size of our sample standard deviation if we divide by [latex]\left(n-1\right)[/latex] in the sample standard deviation formula rather than by [latex]n[/latex].
Because we are using degrees of freedom in the denominator. You may have heard that the denominator in the standard deviation formula is called the degrees of freedom, abbreviated df. That’s true, and it helps us to compensate for the underestimation that crops up when we divide strictly by sample size. There’s a lot going on here mathematically, but we can think of it this way: dividing by [latex]\left(n-1\right)[/latex] instead of [latex]n[/latex] helps our sample standard deviation more closely resemble the true (usually unknowable) population standard deviation. This will help make our statistical analysis more reasonable.
What are degrees of freedom, anyway? A nice way to think of degrees of freedom, [latex]\left(n-1\right)[/latex] is to imagine a set of three numbers whose mean is [latex]5[/latex]: say, [latex]4, 5[/latex], and [latex]6[/latex]. If those three numbers were written on pieces of paper in a hat, and you chose two of them, say [latex]4[/latex] and [latex]5[/latex], first, then the only way to get a mean of [latex]5[/latex] from the numbers on three scraps of paper would be that the next choice must have a [latex]6[/latex] on it. We could say that the first two scraps were free to vary; they could have been [latex]4[/latex] or [latex]5[/latex] or [latex]6[/latex] as they pleased. But the third pick couldn’t vary. After choosing the [latex]4[/latex] and the [latex]5[/latex] freely first, there was no freedom for the choice of the third in order to obtain the desired mean. Only two of our choices had a degree of freedom, so we say that the degrees of freedom of a sample size of [latex]3[/latex] is [latex]\left(3-1\right)=2[/latex].
Insert video explanation of the idea of n-1 as degrees of freedom. DC suggested this one: For a more detailed discussion, see https://www.khanacademy.org/math/ap-statistics/summarizing-quantitative-data-ap/more-standard-deviation/v/review-and-intuition-why-we-divide-by-n-1-for-the-unbiased-sample-variance

Don’t worry. We will be using the data analysis tool to calculate standard deviation for us!

Let’s practice using the tool by finding the standard deviation of the variable Average Sleep in the Sleep Study data set.

On the next page, you will continue to explore the measures of variability (spread) by taking a look at variance and range.

Module 4

Standard Deviation

Standard Deviation

Standard Deviation

example

Use a data analysis tool to identify the standard deviation of a data set

question 6