Measuring spread: standard deviation and the five-number summary

Two ways of describing how spread out a data set is. One uses every value, the other uses five positions. Both appear on Exam 3.

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 What This Looks Like on ALEKS

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

Five-number summary

Distances to the nearest airport for 14 families, already sorted:

11, 12, 13, 14, 18, 22, 23, 27, 27, 37, 37, 38, 39, 42

What is the interquartile range?

IQR =

The median of an even set

Same fourteen values: 11, 12, 13, 14, 18, 22, 23, 27, 27, 37, 37, 38, 39, 42.

What is the median?

Sample or population?

For the six values 49, 52, 50, 54, 55, 52 the squared differences from the mean total 26.

What is the sample standard deviation, 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 standard deviation divides by n − 1

five-number summary: min, Q1, median, Q3, max and IQR = Q3 − Q1

For the standard deviation: find the mean, subtract it from each value, square the differences, add, divide by one less than the count, take the square root. For the five-number summary: sort the data, split it at the median, and take the median of each half.

Common slips

(1) Divided by n instead of n − 1. A sample standard deviation uses n − 1. Your calculator has both keys; the sample one is usually s or σ_n-1. (2) Forgot the square root. That gives the variance. (3) Did not sort before finding quartiles. Positions mean nothing in an unsorted list.

04 Practice — Step by Step

Standard deviation, step by step

Red blood cell counts measured on six days:

49 52 50 54 55 52

Find the sample standard deviation, to two decimal places.

Step 1

The mean

Add the six values and divide by 6.

Step 2

The squared differences

Differences from 52 are −3, 0, −2, 2, 3, 0. Square them and add.

Step 3

Divide by n minus 1, then root

26 ÷ 5, then take the square root. Two decimal places.

s =

Q19 Exam 3 Review

Measuring spread: standard deviation and the five-number summary

Now the five-number summary