Center vs Spread Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Center vs Spread.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Center says where values cluster; spread says how far they scatter — together they describe any data set.

Common stuck point: The procedure for center vs spread is the easy part; the trap is comparing groups by center alone. Asking "Have I reported both where the data sits and how spread out it is?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I reported both where the data sits and how spread out it is?

Worked Examples

Example 1

easy

For the data

\{2, 4, 6, 8, 10\}

: calculate the mean (center) and standard deviation (spread), then explain why both are needed to describe the data.

Answer

Mean = 6 (center); SD ≈ 2.83 (spread). Both are needed for a complete description.

First step

Mean:

\mu = \frac{2+4+6+8+10}{5} = \frac{30}{5} = 6

Full solution

2
Deviations from mean: $-4, -2, 0, 2, 4$ ; squared: $16, 4, 0, 4, 16$ ; sum = 40
3
$\sigma = \sqrt{\frac{40}{5}} = \sqrt{8} \approx 2.83$
4
Why both needed: mean tells us where data is centered, but two data sets could have mean 6 with very different spreads — the SD distinguishes them

Center and spread together form the minimum description of a distribution. Knowing only the mean is like knowing a city's average temperature without knowing the seasonal variation — incomplete and potentially misleading.

Example 2

medium

Three data sets all have mean = 10: Set A =

\{10, 10, 10, 10\}

, Set B =

\{8, 9, 11, 12\}

, Set C =

\{1, 5, 15, 19\}

. Calculate the SD of each and describe what the spread reveals.

Example 3

medium

Class A test scores:

70, 70, 70, 70, 70

. Class B:

50, 60, 70, 80, 90

. Compute the mean and SD (population) of each and explain what they tell a teacher.

Example 4

medium

Two basketball players average 20 points per game. Player X has SD

2

, Player Y has SD

9

. As a coach picking for a clutch game, what does each spread tell you?

Example 5

medium

A dataset has mean

50

and SD

10

. Convert the value

65

to a z-score and explain what it means.

Example 6

hard

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

Example 7

hard

Two factories produce nails of target length

50

mm. Factory P: mean

50.0

, SD

0.2

. Factory Q: mean

50.0

, SD

1.0

. Without computing any probabilities, explain which factory's nails are more likely to meet a tolerance of

\pm 0.5

mm.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A quality control manager says: 'Our bolts average 50 mm, which is the target.' Why might this still be a problem, and what additional information is needed?

Example 2

hard

For symmetric distributions, which pair (mean ± SD) or (median, IQR) is preferred? For skewed distributions? Justify with an example for each case.

Example 3

easy

Find the mean of the dataset

4, 6, 8, 10, 12

Example 4

easy

Find the median of

3, 7, 9, 12, 20

Example 5

easy

Find the range of

5, 8, 2, 10, 6

Example 6

easy

Does the mean describe center or spread?

Example 7

easy

Does the standard deviation describe center or spread?

Example 8

easy

Find the mode of

2, 4, 4, 4, 7, 9

Example 9

easy

Two datasets both have mean

50

. One has SD

2

, the other SD

20

. Which is more spread out?

Example 10

easy

For the sorted data

2, 4, 6, 8

, find the median.

Example 11

medium

Data:

10, 12, 14, 16, 100

. Compare the mean and median, and state which better represents the typical value.

Example 12

medium

Find the interquartile range (IQR) of

2, 4, 6, 8, 10, 12, 14, 16

Example 13

medium

A dataset has mean

20

and every value is increased by

5

. What are the new mean and the new standard deviation compared to before?

Example 14

medium

A dataset has SD

3

. If every value is multiplied by

4

, what is the new standard deviation?

Example 15

medium

For

1, 2, 3, 4, 5

, the mean is

3

. Compute the variance (average squared deviation).

Example 16

medium

Class A scores: all exactly

70

. Class B scores:

40,70,100

. Both can claim a center near

70

. How do their spreads differ and why does it matter?

Example 17

medium

Given

Q_1=10

and

Q_3=30

, what value separates the data such that 50% lies between which two numbers, and what is the IQR?

Example 18

medium

The five-number summary of a dataset is min 2,

Q_1

5, median 8,

Q_3

12, max 30. Which single number best summarizes its spread for skewed data, and what is it?

Example 19

medium

A dataset of all identical values 7,7,7,7 has what mean and what standard deviation?

Example 20

challenge

Dataset:

2, 4, 4, 4, 5, 5, 7, 9

. Compute the mean, then the standard deviation (population, divide by

n

Example 21

challenge

Two datasets have the same mean

50

and same range

40

, but dataset X is tightly bunched near

50

with two extreme values, while Y is evenly spread. Can range distinguish their spreads, and what measure would?

Example 22

challenge

A teacher adds

5

bonus points to every score, then doubles all scores. If the original mean was

60

and SD was

10

, find the new mean and new SD.

Example 23

easy

Find the mean of

3, 5, 7, 9, 11

Example 24

easy

Find the median of

1, 3, 8, 9, 10, 12

Example 25

easy

A dataset has every value equal to

12

. What are its mean and standard deviation?

Example 26

easy

Find the range of

14, 9, 22, 7, 18

Example 27

medium

For the data

1, 1, 2, 3, 8

, decide which is a better measure of center and justify briefly.

Example 28

medium

A dataset has variance

36

. What is its standard deviation?

Example 29

medium

If every value of a dataset is shifted up by

7

, what happens to the variance?

Example 30

medium

Compute the IQR of

5, 7, 9, 11, 13, 15, 17, 19, 21

Example 31

medium

For

4, 8, 6, 5, 7

, find the mean and the mean absolute deviation (MAD).

Example 32

medium

Dataset:

10, 20, 30, 40, 50

. Compute the population variance.

Example 33

medium

A dataset's MAD is

0

. What does that tell you about the data?

Example 34

hard

Population SD of

2, 4, 4, 6, 8, 10

rounded to two decimals.

Example 35

hard

Combined dataset

A\cup B

where

A=\{2,4,6\}

(mean

4

) and

B=\{10,12,14\}

(mean

12

). Find the overall mean and explain whether SD of

A\cup B

is bigger or smaller than SD of either set.

Example 36

hard

A dataset has mean

20

, SD

4

. Each value

x

is replaced by

y=2x-10

. Find the new mean and SD.

Example 37

hard

Construct a dataset of 5 distinct positive integers with mean

10

and range

20

Example 38

challenge

Prove that the mean minimizes the sum of squared deviations

\sum (x_i - c)^2

over all constants

c

← Back to Center vs Spread Practice Mode

Related Concepts

Mean Standard Deviation Distribution (Intuition)

Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation