Comparing Variability of Data Sets: Learn It 3

Standard Deviation

In statistics, we are particularly interested in understanding how data are distributed and where each observation is in reference to the mean. This measurement of variability is called standard deviation, which tells us 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.

Recall

We’ll be using statistical formulas, symbols, and language to discuss measures of variability. Take a moment to recall the formula you learned to calculate the mean of a sample. What symbols do we use to represent sample mean, summation, and sample size?

Core skill:

Express the formula for calculating the mean of a sample

Standard Deviation VIDEO ANIMATED CONCEPT

video about standard deviation and why we find it and it’s purpose

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 step 4

example

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

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

Show Solution

DATASET CHOICE OPTION

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

Although it’s important to understand each part of the formula – don’t worry – we will be using the statistical technology tool to calculate standard deviation for us! Let’s practice using tool by finding the standard deviation of the variable Average Sleep in the Sleep Study data set.

Module 4