Practice Center vs Spread in Math

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

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Quick Recap

Center and spread are two complementary ways to describe a data distribution. Center (mean, median, mode) tells you where values cluster; spread (range, interquartile range, standard deviation) tells you how far values are from that center. Together they give a complete picture of any dataset.

Where is the data located? How spread out is it around that location?

Showing a random 20 of 50 problems.

Example 1

medium
A dataset has mean 2020 and every value is increased by 55. What are the new mean and the new standard deviation compared to before?

Example 2

hard
Two factories produce nails of target length 5050 mm. Factory P: mean 50.050.0, SD 0.20.2. Factory Q: mean 50.050.0, SD 1.01.0. Without computing any probabilities, explain which factory's nails are more likely to meet a tolerance of ±0.5\pm 0.5 mm.

Example 3

medium
A dataset's MAD is 00. What does that tell you about the data?

Example 4

medium
Class A test scores: 70,70,70,70,7070, 70, 70, 70, 70. Class B: 50,60,70,80,9050, 60, 70, 80, 90. Compute the mean and SD (population) of each and explain what they tell a teacher.

Example 5

challenge
Two datasets have the same mean 5050 and same range 4040, but dataset X is tightly bunched near 5050 with two extreme values, while Y is evenly spread. Can range distinguish their spreads, and what measure would?

Example 6

easy
For the sorted data 2,4,6,82, 4, 6, 8, find the median.

Example 7

medium
A dataset has variance 3636. What is its standard deviation?

Example 8

easy
Is the range a measure of center or spread?

Example 9

easy
Which value (mean or median) is more affected by an extreme outlier?

Example 10

easy
A dataset has every value equal to 1212. What are its mean and standard deviation?

Example 11

medium
Dataset: 10,20,30,40,5010, 20, 30, 40, 50. Compute the population variance.

Example 12

medium
Compute the IQR of 5,7,9,11,13,15,17,19,215, 7, 9, 11, 13, 15, 17, 19, 21.

Example 13

hard
Show that adding a single very large outlier always increases the range but may not increase the IQR.

Example 14

easy
Two datasets both have mean 5050. One has SD 22, the other SD 2020. Which is more spread out?

Example 15

easy
For the data {2,4,6,8,10}\{2, 4, 6, 8, 10\}: calculate the mean (center) and standard deviation (spread), then explain why both are needed to describe the data.

Example 16

easy
Find the range of 14,9,22,7,1814, 9, 22, 7, 18.

Example 17

medium
Data: 10,12,14,16,10010, 12, 14, 16, 100. Compare the mean and median, and state which better represents the typical value.

Example 18

medium
For 1,2,3,4,51, 2, 3, 4, 5, the mean is 33. Compute the variance (average squared deviation).

Example 19

medium
For 4,8,6,5,74, 8, 6, 5, 7, find the mean and the mean absolute deviation (MAD).

Example 20

easy
Find the range of 5,8,2,10,65, 8, 2, 10, 6.
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