Standard Deviation

Continuing with our measures of spread, we arrive at the standard deviation. This is perhaps the most popular measure of spread, other than perhaps the range in more everyday contexts.

Definition. The standard deviation is simply the square root of the variance:

\[ \sigma_X = \sqrt{\frac{1}{N}\sum _{i=1}^{N} (X_i-\mu_X)^2} \]

where \(\mu_X\) is the population mean and \(N\) is the population size.

An important note...

As with the variance, there are two versions of the standard deviation: one for populations, and one for samples. For now we'll focus on the population version. We'll dive into the sample standard deviation when we study inferential statistics.

We'll turn now to a couple of examples.

Example: Test scores for a reasonably challenging chemistry test

Our favorite chemistry test scores are back:

83, 87, 61, 92, 38, 78, 73, 55, 98, 74, 86, 69, 40, 83

In the section on variance, we calculated the variance as \(\sigma^2 \approx 312.37\). So all we have to do is take the square root. That gives us \(\sigma \approx 18.34\).

Example: Test scores for an easy math test

Here are the test scores for the easy math test we've been measuring:

100, 100, 93, 92, 95, 98, 100, 100, 100, 95, 94, 88, 92

In the section on variance, we calculated the variance as \(\sigma^2 \approx 14.99\). Taking the square root gives us \(\sigma \approx 4.03\).

Understanding the standard deviation

You might be wondering why we need a measure of spread that's just like the variance, except that we take the square root. The reason is that taking the square root brings the measurement back into the scale of the original data. Recall that the variance is often one or more orders of magnitude larger than the data values, on account of squaring the deviations. Taking the root brings us back.

The measurement here is somewhat similar to the average deviation. The calculations and scales are similar. Which one is better? That's a matter of some debate, and you can read about those in the Related section below. We'll highlight some of the considerations in the following sections.

Strengths of the standard deviation as a measure of spread

Incorporates all data values. As with the variance, the standard deviation incorporates all data values into the final result.

Scale is roughly the same as the original data. By taking the square root of the variance, we (roughly) undo the squaring action as applied to the individual deviations. The "undoing" isn't exactly undoing the individual squares, as we're undoing the sum of squares and also the \(n\) in the denominator. But it's close enough to make it useful from a rescaling perspective.

Weaknesses of the standard deviation as a measure of spread

The calculation is less intuitive. Gorard argues in Revisiting a 90-year-old debate: the advantages of the mean deviation that the meaning of, and calculation for, the standard deviation is more difficult than that of the average deviation. That strikes me as a correct and fair objection.

The standard deviation is sensitive to outliers. Once again, we're dealing with a measure that isn't robust in the face of outliers.

Exercises

Exercise 1. Compute the standard deviation of the following data:

202, 102, 285, 98

Does it seem to be a reasonable measurement of the spread for this dataset?

Exercise 2. Compute the standard deviation of the following data:

185, 245, 205, 215, 3829, 190

Is it a reasonable measurement of the spread for this dataset?