Practice Normalization (Statistics) in Math

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

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

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

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 4

easy
Min-max normalize the value 2020 given the data range is [0,100][0, 100].

Example 5

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.

Example 6

medium
Min-max normalize the dataset {2,5,10,20}\{2, 5, 10, 20\}.

Example 7

easy
In min-max normalization, what does the data set's minimum value always map to?

Example 8

medium
Standardize the value 6060 using mean 5050 and standard deviation 88, then interpret the sign and size.

Example 9

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 10

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 11

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

Example 12

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

Example 13

medium
Index normalization: set 2020 sales of $200k to index 100100. If 2025 sales are $260k, what is the 2025 index?

Example 14

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

Example 15

medium
A salary of $70{,}000 has z=1.5z = 1.5 in a distribution with mean $55{,}000. Find the standard deviation.

Example 16

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 17

challenge
Show that if all data points are identical (say all equal to cc), min-max normalization is UNDEFINED. What's the typical workaround?

Example 18

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.

Example 19

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 20

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