Normalization (Statistics) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Normalization (Statistics).

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

Concept Recap

Normalization rescales data to a standard range or distribution — such as $[0,1]$ or zero mean and unit variance — to make different variables comparable.

Converting to a standard reference so you can compare apples to apples.

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: Normalization rescales different quantities to a common reference so you can compare them fairly.

Common stuck point: The procedure for normalization (statistics) is the easy part; the trap is comparing raw counts from groups of different sizes. Asking "Am I dividing by group size or rescaling so different-sized quantities can be compared fairly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I dividing by group size or rescaling so different-sized quantities can be compared fairly?

Worked Examples

Example 1

easy

City A has 500 crimes with population 100,000. City B has 300 crimes with population 50,000. Which city is safer? Calculate crime rates per 100,000 people.

Answer

City A: 500/100K rate; City B: 600/100K rate. City A is safer despite having more total crimes.

First step

City A crime rate:

\frac{500}{100,000} \times 100,000 = 500

per 100,000

Full solution

2
City B crime rate: $\frac{300}{50,000} \times 100,000 = 600$ per 100,000
3
City B has fewer total crimes (300 < 500) but a HIGHER crime rate (600 > 500 per 100,000)
4
Safer city by rate: City A (500 per 100,000) — normalization reveals the true comparison

Normalization (rate = count/population) allows fair comparison across groups of different sizes. Raw counts favor larger cities (more of everything); rates per capita give a meaningful comparison. Always normalize when comparing populations of different sizes.

Example 2

medium

Test scores: Raw score 85/100. Class mean=70, SD=10. Z-score normalize this score and explain what it means relative to classmates.

Example 3

medium

Two athletes are compared. Athlete A's 100m time:

11.5

s in a population with

\mu = 12

\sigma = 0.5

s. Athlete B's high jump:

1.95

m in a population with

\mu = 1.85

\sigma = 0.1

m. Use z-scores: who performed more exceptionally?

Example 4

medium

A study reports country GDPs in different currencies. Explain a simple normalization step that makes them comparable.

Example 5

hard

Why does k-Nearest Neighbors typically REQUIRE feature normalization but a decision tree does not?

Example 6

hard

A 'per-capita GDP' metric divides national GDP by population. Give one concrete situation where this normalization is MISLEADING.

Example 7

challenge

Prove that min-max normalization maps the SECOND-LARGEST value to a number in

(0,1)

strictly less than

1

, given all data points are distinct.

Example 8

challenge

Show that if all data points are identical (say all equal to

c

), min-max normalization is UNDEFINED. What's the typical workaround?

Practice Problems

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

Example 1

easy

Country A has 1000 hospital beds; Country B has 5000. Country A's population is 500,000; Country B's is 10,000,000. Calculate beds per 1000 people for each country and compare.

Example 2

hard

A student scored 75 in Math (

\mu=65, \sigma=8

) and 80 in English (

\mu=78, \sigma=3

). In which subject did they perform better relative to their class?

Example 3

easy

City A has

40

crimes among

20{,}000

people. What is the crime rate per

1{,}000

people?

Example 4

easy

Rescale the value

30

[0,1]

given the data range is

[10, 50]

using min-max normalization.

Example 5

easy

A z-score normalizes a value

x=85

with mean

\mu=75

and standard deviation

\sigma=5

. Compute it.

Example 6

easy

Store A sold

200

units; Store B sold

150

. But A had

1000

customers and B had

500

. Which has the higher units-per-customer rate?

Example 7

easy

To compare test scores from a

50

-point quiz and a

200

-point exam, what normalization makes them comparable?

Example 8

easy

Which is the more appropriate normalization to compare disease counts between a small town and a big city: per-capita rate or raw count?

Example 9

easy

Min-max normalize the maximum value of a data set. What does it always map to?

Example 10

easy

A report gives 'revenue per employee.' For a firm with $2M revenue and

40

employees, compute it.

Example 11

medium

Crime data: City A has

300

crimes /

100{,}000

people; City B has

120

crimes /

30{,}000

people. Compute both rates per

100{,}000

and say which is higher.

Example 12

medium

Choosing a denominator: traffic deaths could be normalized per

100{,}000

people, per

100{,}000

vehicles, or per billion miles driven. Which best measures road SAFETY for drivers, and why?

Example 13

medium

Two features for a model: age (range

0

–

100

) and income (range

0

–

200{,}000

). Why normalize, and what does min-max scaling do to each?

Example 14

medium

Standardize the value

60

using mean

50

and standard deviation

8

, then interpret the sign and size.

Example 15

medium

When is normalizing to a rate WORSE than using raw counts? Give a context.

Example 16

medium

A scientist normalizes gene expression by dividing each value by the sample's total, giving proportions. Sample has values

2, 3, 5

. Give the normalized values.

Example 17

medium

Two products: A has

90

five-star reviews out of

100

; B has

920

out of

1000

. Normalize to proportions and decide which rates higher.

Example 18

medium

Convert $2{,}500 monthly income to an annual figure, then normalize two people earning $2{,}500/month and $28{,}000/year to a common annual basis. Who earns more?

Example 19

medium

Index normalization: set January sales of $80k to an index of

100

. If March sales are $100k, what is March's index value?

Example 20

challenge

Show that min-max normalization is a linear transformation and therefore preserves the relative ORDER and the relative spacing of data points.

Example 21

challenge

A dataset is standardized to mean

0

, SD

1

. Prove that the z-score is dimensionless (unit-free) and explain why this lets you compare height (cm) with weight (kg).

Example 22

challenge

You normalize per-capita but the subgroups have wildly different sizes; explain how a poorly chosen normalization can still produce a Simpson's-paradox-style reversal when subgroups are pooled.

Example 23

easy

Min-max normalize the value

20

given the data range is

[0, 100]

Example 24

easy

School A:

25

teachers per

500

students. School B:

40

teachers per

1000

students. Which has the better teacher-to-student ratio?

Example 25

easy

A test was originally scored out of

40

. Convert a score of

30

to a percentage.

Example 26

easy

Country C has

60

traffic deaths per

1{,}000{,}000

people. Country D has

9{,}000

deaths in a population of

300{,}000{,}000

. Which has the lower rate?

Example 27

medium

A salary of $70{,}000 has

z = 1.5

in a distribution with mean $55{,}000. Find the standard deviation.

Example 28

medium

A data set has values

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

with mean

8

and SD

\sqrt{8} \approx 2.83

. Compute the z-score of

12

Example 29

medium

Index normalization: set 2020 sales of $200k to index

100

. If 2025 sales are $260k, what is the 2025 index?

Example 30

medium

Choose the better normalization denominator for COMPARING air pollution between two cities: total emissions vs emissions per square kilometer. Justify briefly.

Example 31

medium

Min-max normalize the dataset

\{2, 5, 10, 20\}

Example 32

medium

Convert raw values

\{30, 50, 70\}

to proportions of their sum.

Example 33

hard

A new value

x = 6

arrives in a dataset originally with min

1

, max

5

used for normalization. After applying the SAME min-max scaler trained on the original data, what is the normalized value?

Example 34

hard

A dataset has

\mu=100

\sigma=20

. After z-score normalization, what raw value maps to

z = -1.5

Example 35

hard

A student's z-scores on five tests are

0.5, 1.2, -0.3, 0.8, -0.1

. What is the average z-score, and is it above or below the class average?

Example 36

challenge

Two distributions have identical means but DIFFERENT variances. After z-scoring each separately, are their distributions necessarily identical?

Example 37

challenge

Robust normalization uses median and IQR instead of mean and SD:

x' = (x - \text{median})/\text{IQR}

. Give one advantage over z-score normalization.

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Related Concepts

Ratios Proportional Reasoning Z-Score

Background Knowledge

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

ratiosproportional reasoning