Practice Standard Deviation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The standard deviation measures the average distance of data values from the mean, giving a typical spread around the center.

The typical distance from the average. Low SD = clustered. High SD = spread out.

Showing a random 20 of 50 problems.

Example 1

easy
Population data 4,64, 6 has mean 55. Find the SD.

Example 2

medium
A data set has SD 66. Every value is increased by 2020. What is the new SD?

Example 3

medium
Find the population standard deviation of 2,4,4,4,5,5,7,92, 4, 4, 4, 5, 5, 7, 9.

Example 4

medium
Two data sets have SD 33 and SD 44. Is the SD of the combined data necessarily 32+42=5\sqrt{3^2+4^2}=5?

Example 5

medium
Find the sample standard deviation of {5,7,9}\{5, 7, 9\}.

Example 6

medium
Why isn't average absolute deviation (1nโˆ‘โˆฃxโˆ’ฮผโˆฃ\frac{1}{n}\sum |x-\mu|) the same as the SD?

Example 7

hard
A population {2,4,4,6}\{2, 4, 4, 6\} has mean 44 and SD ฯƒ1\sigma_1. The set {2,4,4,6,4}\{2, 4, 4, 6, 4\} has SD ฯƒ2\sigma_2. Which is larger and why?

Example 8

medium
A data set has mean 2020 and SD 33. Roughly what range contains values within one SD of the mean?

Example 9

hard
Three population values aโˆ’d,a,a+da-d, a, a+d have mean aa and SD ฯƒ\sigma. Find dd in terms of ฯƒ\sigma.

Example 10

medium
A population SD is 44. What is the variance?

Example 11

medium
Find the population standard deviation of {2,4,4,4,5,5,7,9}\{2, 4, 4, 4, 5, 5, 7, 9\}.

Example 12

medium
Population data 1,3,5,7,91, 3, 5, 7, 9 has mean 55. Find the SD.

Example 13

challenge
Two data points aa and bb form a population with SD 55. Find โˆฃaโˆ’bโˆฃ|a-b|.

Example 14

easy
All five values in a data set equal 88. What is the standard deviation?

Example 15

challenge
A population of nn values has SD ฯƒ\sigma. Each value is transformed by y=3xโˆ’5y=3x-5. Find the new SD.

Example 16

challenge
A population of size 44 has values 0,0,0,k0, 0, 0, k with k>0k>0. Find the SD as a function of kk, and the value of kk that makes SD =3= 3.

Example 17

medium
Compare the population standard deviations of A={4,4,4,4}A = \{4, 4, 4, 4\} and B={2,4,4,6}B = \{2, 4, 4, 6\}.

Example 18

hard
A population {x1,โ€ฆ,x5}\{x_1,\dots,x_5\} has mean 1010 and โˆ‘xi2=530\sum x_i^2 = 530. Find the population SD.

Example 19

easy
Can the standard deviation be negative?

Example 20

medium
Two classes have the same mean test score, but class A's SD is 44 and class B's SD is 1010. Which class is more consistent?
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