Variability Examples in Math

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

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

Concept 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.

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: Variability is the degree to which data points differ from each other and from the center.

Common stuck point: The procedure for variability is the easy part; the trap is reporting only the mean and stopping. Asking "Am I describing how scattered the values are, separate from where they center?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing how scattered the values are, separate from where they center?

Worked Examples

Example 1

easy

Three measures of spread exist for the data

\{5, 10, 15, 20, 25\}

: range, IQR, and standard deviation. Calculate all three and compare what each captures.

Answer

Range=20, IQR=15, SD≈7.07. Each measures spread differently.

First step

Range:

25 - 5 = 20

— captures total spread including extremes

Full solution

2
$Q_1 = 7.5$ , $Q_3 = 22.5$ ; $IQR = 22.5 - 7.5 = 15$ — captures middle 50% spread
3
Mean: $\mu = 15$ ; deviations: $-10,-5,0,5,10$ ; $\sigma^2 = \frac{100+25+0+25+100}{5} = 50$ ; $\sigma = \sqrt{50} \approx 7.07$
4
Each captures different aspects: range is simple but sensitive to outliers; IQR is resistant; SD accounts for all deviations from mean

Variability can be measured in multiple ways depending on the context and the presence of outliers. Range is simplest; IQR is most resistant; standard deviation is used in most statistical inference procedures.

Example 2

medium

Two data sets have the same mean of 50 but different standard deviations: Set A has

\sigma=2

, Set B has

\sigma=15

. Describe what this means and sketch what their distributions would look like.

Example 3

medium

Two classes have mean

75

. Class A: SD

= 2

; Class B: SD

= 12

. Which has more consistent scores?

Example 4

medium

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

Example 5

hard

A class has scores:

\{60, 70, 70, 80, 100\}

. Find the mean, range, and SD (population).

Example 6

challenge

Show that adding the same constant

c

to every value leaves both variance and SD unchanged.

Practice Problems

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

Example 1

easy

Which data set has greater variability? Set A:

\{48, 49, 50, 51, 52\}

or Set B:

\{10, 30, 50, 70, 90\}

? Use range and explain.

Example 2

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.

Example 3

easy

Data set

5,5,5,5,5

has what amount of variability?

Example 4

easy

Set A:

10,10,10,10

. Set B:

4,8,12,16

. Which has more variability?

Example 5

easy

Variability describes which feature of a data set?

Example 6

easy

Two classes have the same average test score. Does that mean their score distributions are identical?

Example 7

easy

Which set has lower variability:

98,99,100,101,102

20,60,100,140,180

Example 8

easy

A factory wants every bolt to be exactly

5

cm. High variability in bolt length is good or bad here?

Example 9

easy

Is the range (max minus min) the same thing as variability?

Example 10

easy

A dataset of daily temperatures over a year has high variability. What does that tell you?

Example 11

medium

Set A:

48,49,50,51,52

. Set B:

30,40,50,60,70

. Both have mean

50

. Which has greater variability and how can you tell?

Example 12

medium

Investment A returns

5\%,5\%,5\%

; Investment B returns

-10\%,5\%,20\%

. Both average

5\%

. Which is riskier and why?

Example 13

medium

A teacher reduces the spread of test scores by reteaching weak topics; the mean stays at

80

. What happened to the variability?

Example 14

medium

A coach says 'all my runners finish within 2 seconds of each other.' Is the team's finish-time variability high or low?

Example 15

medium

Why might a manager prefer a supplier with slightly higher average delivery time but much lower variability?

Example 16

medium

Data:

7,7,8,8,30

. The value

30

is far from the rest. How does it affect the data's variability?

Example 17

medium

Two datasets have the same range of

40

. Must they have the same overall variability?

Example 18

medium

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

Example 19

challenge

Set A:

1,2,3,4,5

. Multiply every value by

10

to form Set B. Does B have more variability than A, and roughly by what factor?

Example 20

challenge

Adding a constant

7

to every value of a data set changes its variability how? Justify.

Example 21

challenge

Process X: outputs

99,100,101

(target

100

). Process Y: outputs

100,100,100

but the true target shifts daily. Which concept (variability vs accuracy) does each illustrate, and why is low variability not the whole story?

Example 22

medium

Class A scores:

70,75,80,85,90

. Class B scores:

79,80,80,80,81

. Both average

80

. Which class is more consistent?

Example 23

easy

Find the range of

\{4, 7, 9, 12, 15\}

Example 24

easy

Compare variability: Set A

\{1, 2, 3\}

vs Set B

\{1, 10, 100\}

Example 25

easy

Find the range of

\{0, 5, 5, 5, 10\}

Example 26

medium

Find the IQR of

\{1, 3, 5, 7, 9, 11, 13\}

Example 27

medium

Find the range of

\{-5, -2, 0, 3, 8\}

Example 28

medium

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

Example 29

medium

If every value of a data set is multiplied by

4

, what happens to the standard deviation?

Example 30

medium

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

Example 31

hard

Data:

\{4, 6, 8, 10, 12\}

. Find the standard deviation (population).

Example 32

hard

Find the IQR of

\{2, 4, 6, 8, 10, 12, 14, 16\}

Example 33

hard

Standardization: a score is

80

in a class with mean

70

and SD

5

. How many SDs above the mean is it?

Example 34

hard

A data set has SD

= 8

in meters. What is its variance, with units?

Example 35

hard

For

\{1, 1, 1, 9\}

, find the range and the IQR.

Example 36

hard

A boxplot has

Q_1 = 12

and

Q_3 = 28

. What is the IQR? What outlier boundaries does the

1.5 \times

IQR rule give?

Example 37

hard

A normal distribution has

\mu = 100, \sigma = 15

. Roughly what percent of data is within

1

SD of the mean?

Example 38

challenge

Combine two groups: Group A:

n_A = 4

, mean

5

, variance

2

. Group B:

n_B = 6

, mean

10

, variance

3

. Find the combined mean.

Example 39

challenge

For the data

\{2, 4, 6, 8\}

, compare the population variance and sample variance.

← Back to Variability Practice Mode

Related Concepts

Data (Abstract)Standard Deviation Range (Statistics)

Background Knowledge

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

data abstract