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 โ€” such as [0,1][0,1] or zero mean and unit variance โ€” to make different variables comparable.

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

Read the full concept explanation โ†’

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

1
City A crime rate: 500100,000ร—100,000=500\frac{500}{100,000} \times 100,000 = 500 per 100,000

Full solution

  1. 2
    City B crime rate: 30050,000ร—100,000=600\frac{300}{50,000} \times 100,000 = 600 per 100,000
  2. 3
    City B has fewer total crimes (300 < 500) but a HIGHER crime rate (600 > 500 per 100,000)
  3. 4
    Safer city by rate: City A (500 per 100,000) โ€” 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: 11.511.5 s in a population with ฮผ=12\mu = 12 s, ฯƒ=0.5\sigma = 0.5 s. Athlete B's high jump: 1.951.95 m in a population with ฮผ=1.85\mu = 1.85 m, ฯƒ=0.1\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)(0,1) strictly less than 11, given all data points are distinct.

Example 8

challenge
Show that if all data points are identical (say all equal to cc), 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 (ฮผ=65,ฯƒ=8\mu=65, \sigma=8) and 80 in English (ฮผ=78,ฯƒ=3\mu=78, \sigma=3). In which subject did they perform better relative to their class?

Example 3

easy
City A has 4040 crimes among 20,00020{,}000 people. What is the crime rate per 1,0001{,}000 people?

Example 4

easy
Rescale the value 3030 to [0,1][0,1] given the data range is [10,50][10, 50] using min-max normalization.

Example 5

easy
A z-score normalizes a value x=85x=85 with mean ฮผ=75\mu=75 and standard deviation ฯƒ=5\sigma=5. Compute it.

Example 6

easy
Store A sold 200200 units; Store B sold 150150. But A had 10001000 customers and B had 500500. Which has the higher units-per-customer rate?

Example 7

easy
To compare test scores from a 5050-point quiz and a 200200-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 4040 employees, compute it.

Example 11

medium
Crime data: City A has 300300 crimes / 100,000100{,}000 people; City B has 120120 crimes / 30,00030{,}000 people. Compute both rates per 100,000100{,}000 and say which is higher.

Example 12

medium
Choosing a denominator: traffic deaths could be normalized per 100,000100{,}000 people, per 100,000100{,}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 00โ€“100100) and income (range 00โ€“200,000200{,}000). Why normalize, and what does min-max scaling do to each?

Example 14

medium
Standardize the value 6060 using mean 5050 and standard deviation 88, 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 2,3,52, 3, 5. Give the normalized values.

Example 17

medium
Two products: A has 9090 five-star reviews out of 100100; B has 920920 out of 10001000. 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 100100. 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 00, SD 11. 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 2020 given the data range is [0,100][0, 100].

Example 24

easy
School A: 2525 teachers per 500500 students. School B: 4040 teachers per 10001000 students. Which has the better teacher-to-student ratio?

Example 25

easy
A test was originally scored out of 4040. Convert a score of 3030 to a percentage.

Example 26

easy
Country C has 6060 traffic deaths per 1,000,0001{,}000{,}000 people. Country D has 9,0009{,}000 deaths in a population of 300,000,000300{,}000{,}000. Which has the lower rate?

Example 27

medium
A salary of $70{,}000 has z=1.5z = 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}\{4, 6, 8, 10, 12\} with mean 88 and SD 8โ‰ˆ2.83\sqrt{8} \approx 2.83. Compute the z-score of 1212.

Example 29

medium
Index normalization: set 2020 sales of $200k to index 100100. 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}\{2, 5, 10, 20\}.

Example 32

medium
Convert raw values {30,50,70}\{30, 50, 70\} to proportions of their sum.

Example 33

hard
A new value x=6x = 6 arrives in a dataset originally with min 11, max 55 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 ฮผ=100\mu=100, ฯƒ=20\sigma=20. After z-score normalization, what raw value maps to z=โˆ’1.5z = -1.5?

Example 35

hard
A student's z-scores on five tests are 0.5,1.2,โˆ’0.3,0.8,โˆ’0.10.5, 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: xโ€ฒ=(xโˆ’median)/IQRx' = (x - \text{median})/\text{IQR}. Give one advantage over z-score normalization.
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Background Knowledge

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

ratiosproportional reasoning