Computing a sample standard deviation

Five steps, and only one of them catches people: the division is by \(n-1\), not \(n\). Get that wrong and the answer still looks entirely reasonable.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 Recognize the Question

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Red blood cell counts

The red blood cell counts of a healthy adult, measured on six days:

48, 53, 54, 52, 49, 50

Find the standard deviation of this sample. Round to two decimal places.

s =

Sample or population?

Same six counts: 48, 53, 54, 52, 49, 50. This time you are told these are every measurement ever taken on this adult — the whole population.

Find the standard deviation. Round to two decimal places.

σ =

A clean one, to isolate the method

A sample of four test scores: 6, 8, 10, 12.

Find the sample standard deviation. Round to two decimal places.

s =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

\(s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}\) sample — divide by one fewer

\(\sigma = \sqrt{\dfrac{\sum (x - \mu)^2}{N}}\) population — divide by all of them

Mean, subtract, square, add, divide, square-root. The only decision is the denominator, and the word sample in the question is what decides it. A sample is a handful drawn from something larger, and dividing by \(n-1\) compensates for the fact that a small handful tends to look less spread out than the thing it came from.

Common slips

(1) Divided by \(n\). That is the population formula, and it always gives a smaller answer. (2) Forgot the square root. That leaves you with the variance. (3) Rounded the mean. If the mean is not exact, carry the decimals rather than rounding before you subtract.

04 Practice — Step by Step

Where the n minus 1 lands

A sample of five delivery times, in minutes:

2 4 4 5 10

Find the standard deviation of this sample.

Step 1

The mean

What is the mean of the five values?

x̄ =

Step 2

The squared deviations

Subtract the mean from each value, square each result, and add them. What is \(\sum (x - \bar{x})^2\)?

Step 3

Divide by one fewer

This is a sample. Divide 36 by \(n-1\), then take the square root.

s =

Step 4

What the other denominator would have given

Had this been a whole population, you would divide by 5. That gives \(\sqrt{7.2} = 2.68\). Which is right here?

Q4 Q6

Computing a sample standard deviation

The one word to look for