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 4,6 has mean 5. Find the SD.

Example 2

medium
A data set has SD 6. Every value is increased by 20. What is the new SD?

Example 3

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

Example 4

medium
Two data sets have SD 3 and SD 4. Is the SD of the combined data necessarily 32+42=5?

Example 5

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

Example 6

medium
Why isn't average absolute deviation (1n∑∣x−μ∣) the same as the SD?

Example 7

hard
A population {2,4,4,6} has mean 4 and SD σ1. The set {2,4,4,6,4} has SD σ2. Which is larger and why?

Example 8

medium
A data set has mean 20 and SD 3. Roughly what range contains values within one SD of the mean?

Example 9

hard
Three population values a−d,a,a+d have mean a and SD σ. Find d in terms of σ.

Example 10

medium
A population SD is 4. What is the variance?

Example 11

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

Example 12

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

Example 13

challenge
Two data points a and b form a population with SD 5. Find ∣a−b∣.

Example 14

easy
All five values in a data set equal 8. What is the standard deviation?

Example 15

challenge
A population of n values has SD σ. Each value is transformed by y=3x−5. Find the new SD.

Example 16

challenge
A population of size 4 has values 0,0,0,k with k>0. Find the SD as a function of k, and the value of k that makes SD =3.

Example 17

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

Example 18

hard
A population {x1,…,x5} has mean 10 and ∑xi2=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 4 and class B's SD is 10. Which class is more consistent?
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