Measures of spread: range, IQR, standard deviation

Warm-up

Three quick questions from earlier lessons.

Discover

Data A: 48, 50, 52. Data B: 20, 50, 80. Same mean, 50. Which is more spread out?

B
Neither: the means are equal

Key idea

Spread measures how far values lie from each other or from the center.

Range = max − min. IQR =

Q_3 - Q_1

(spread of the middle half; resistant). Sample standard deviation

s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n - 1}}

(typical distance from the mean;

s^2

is the variance).

Worked example

$s$ for 2, 4, 6.

Squares: 4, 0, 4, sum 8

Mean 4; deviations −2, 0, 2.

s^2 = 4

Divide by n − 1 = 2.

s = 2

Square root.

Your turn, guided

Your turn: the IQR when $Q_1 = 12$ and $Q_3 = 30$ .

IQR

= 18

30 − 12.

Practice ★

Find the (sample) standard deviation of 33, 38, 43.
Find the (sample) standard deviation of 25, 34, 43.

Find the interquartile range (IQR) of the 8 values 3, 9, 14, 17, 24, 27, 33, 46.
Find the interquartile range (IQR) of the 8 values 4, 9, 13, 19, 24, 29, 32, 45.

Watch out!

Practice ★★

Two classes both have mean score 78. In class A the scores range from 76 to 80; in class B from 63 to 93. Which has the larger standard deviation?
1. Class B: its scores are more spread out
2. Class A
3. The same, because the means are equal

Boss problem. The data are 67, 68, 69, 70, 71.

Exit ticket

Three questions to check you have got it.

Find the (sample) standard deviation of 33, 38, 43.
Find the (sample) standard deviation of 25, 34, 43.
Find the (sample) standard deviation of 23, 32, 41.

Summary card

Spread

Range, IQR (resistant), standard deviation.
$s$ divides by $n - 1$ ; $s^2$ is the variance.

Watch out: the SD is the square root of the variance.

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