Practice Variability in Math

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

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Quick Recap

Variability is the degree to which data points in a set differ from each other and from the center of the distribution.

How spread out or bunched up the data is. No variability = everyone is the same.

Showing a random 20 of 50 problems.

Example 1

hard
A class has scores: {60,70,70,80,100}\{60, 70, 70, 80, 100\}. Find the mean, range, and SD (population).

Example 2

medium
Two classes have mean 7575. Class A: SD =2= 2; Class B: SD =12= 12. Which has more consistent scores?

Example 3

medium
A quality team says 'lower variability is always better.' Give a case where some variability is natural and expected.

Example 4

medium
Which measure of variability is best when the data has extreme outliers?

Example 5

medium
A factory measures the diameter of bolts. Which is more important for quality control: the mean or the variability?

Example 6

challenge
Adding a constant 77 to every value of a data set changes its variability how? Justify.

Example 7

medium
Two datasets have the same range of 4040. Must they have the same overall variability?

Example 8

hard
A data set has SD =8= 8 in meters. What is its variance, with units?

Example 9

challenge
For the data {2,4,6,8}\{2, 4, 6, 8\}, compare the population variance and sample variance.

Example 10

medium
Find the IQR of {1,3,5,7,9,11,13}\{1, 3, 5, 7, 9, 11, 13\}.

Example 11

easy
Data set 5,5,5,5,55,5,5,5,5 has what amount of variability?

Example 12

hard
Find the IQR of {2,4,6,8,10,12,14,16}\{2, 4, 6, 8, 10, 12, 14, 16\}.

Example 13

challenge
Process X: outputs 99,100,10199,100,101 (target 100100). Process Y: outputs 100,100,100100,100,100 but the true target shifts daily. Which concept (variability vs accuracy) does each illustrate, and why is low variability not the whole story?

Example 14

challenge
Combine two groups: Group A: nA=4n_A = 4, mean 55, variance 22. Group B: nB=6n_B = 6, mean 1010, variance 33. Find the combined mean.

Example 15

medium
Why does a single measure (mean) miss the story of two data sets with the same mean but different spreads?

Example 16

medium
A teacher reduces the spread of test scores by reteaching weak topics; the mean stays at 8080. What happened to the variability?

Example 17

hard
Two data sets have the same range. Do they have the same standard deviation?

Example 18

easy
Which set has lower variability: 98,99,100,101,10298,99,100,101,102 or 20,60,100,140,18020,60,100,140,180?

Example 19

medium
Find the range of {−5,−2,0,3,8}\{-5, -2, 0, 3, 8\}.

Example 20

hard
A factory produces bolts. Machine A produces bolts with diameter mean 10 mm, SD = 0.1 mm. Machine B produces bolts with mean 10 mm, SD = 0.5 mm. The specification requires bolts between 9.8 mm and 10.2 mm. Explain which machine is preferable and why variability matters here.
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