Practice Distribution (Intuition) in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A distribution describes how data values are spread out across their range — which values occur, how often, and whether the data is symmetric or skewed.

If you took many measurements, where would most values fall? What's the shape?

Showing a random 20 of 50 problems.

Example 1

hard
If a dataset is right-skewed, rank the three centers (mean, median, mode) from smallest to largest.

Example 2

hard
Two cities both report 'average rainfall 3030 inches/year.' City A has yearly totals tightly between 2828 and 3232. City B has totals ranging from 55 to 8080. Which city's reported average is more useful for planning, and why?

Example 3

medium
A distribution of exam scores has mean 7272 and median 8080. Which way is it skewed, and which center better represents a typical student?

Example 4

medium
A company says 'average wait time is 5 minutes,' but most customers wait under 3 minutes while a few wait 30+. What shape is this and which center is more honest?

Example 5

challenge
A histogram of birthdays-by-month for 1200 people is nearly flat at ~100 per month, but December shows 150. Is the flat shape signal and the December bump signal or noise, given monthly random variation is about ±15\pm 15?

Example 6

easy
A dataset of test scores has most values near 75, with a few very low and very few very high. Is the bulk of the data near the center or the extremes?

Example 7

challenge
Data on city sizes spans 1,000 to 10,000,000 and is extremely right-skewed. After taking the logarithm of each value, the histogram becomes roughly symmetric. What does this reveal about the original distribution's shape?

Example 8

easy
In a symmetric, bell-shaped distribution, are the mean and median approximately equal?

Example 9

hard
A distribution has values from 00 to 100100 with mean 5050 but 95%95\% of values lie between 4848 and 5252. What does this say about the tails, and what shape is consistent?

Example 10

medium
Sketch-reasoning: counts for values 1,2,3,4,5 are 2,5,9,5,2. Is this distribution symmetric, left-skewed, or right-skewed?

Example 11

easy
A histogram of marathon finishing times rises quickly to a peak and trails slowly to the right. Is it skewed left or right?

Example 12

easy
For a right-skewed income distribution, is the mean typically greater than, less than, or equal to the median?

Example 13

medium
A histogram of human reaction times rises sharply to a peak then trails off slowly to the right. Which is larger, mean or mode, and why?

Example 14

challenge
Two independent uniform random numbers on [0,1][0,1] are added together. Describe the shape of the distribution of the sum.

Example 15

medium
Two classes both average 80 on a test. Class A scores are tightly bunched near 80; Class B has many near 60 and many near 100. Do equal means imply equal distributions?

Example 16

easy
True or false: knowing the shape of a distribution is just as important as knowing its mean.

Example 17

medium
Time between customer arrivals at a coffee shop tends to cluster at short waits with a long tail of occasional long waits. Name the shape.

Example 18

easy
A distribution has a long tail stretching to the right. Is it skewed left or skewed right?

Example 19

medium
A distribution of household sizes shows values 1,2,3,4,5 with frequencies 4,8,6,2,1. Is this distribution symmetric or skewed, and in which direction?

Example 20

medium
A dataset has a single tall spike at one value and almost nothing elsewhere. What does this say about its spread, and what is such a distribution called near the limit?
← Back to Distribution (Intuition) Worked Examples