Practice Histogram in Math

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

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Quick Recap

A histogram is a bar chart of a frequency distribution where bars represent count or density of data within consecutive equal-width intervals (bins).

Group data into bins and count how many fall in each. Shows the shape of data.

Showing a random 20 of 50 problems.

Example 1

hard
Estimate the median of a histogram with bins [0,10):20,[10,20):50,[20,30):30[0,10):20, [10,20):50, [20,30):30 using linear interpolation.

Example 2

challenge
Two histograms have identical means but different shapes (one symmetric, one strongly right-skewed). Compare their standard deviations qualitatively and explain.

Example 3

easy
A histogram has bars at intervals [0,10),[10,20),[20,30),[30,40)[0,10), [10,20), [20,30), [30,40) with frequencies 5, 12, 8, 3. What is the total number of data points, and which interval contains the mode?

Example 4

challenge
Two histograms have the same bins and total but A is concentrated in one bin while B is spread evenly. Which has larger variance?

Example 5

easy
A histogram of test scores is symmetric and unimodal. Which is roughly equal: mean and median, or median and IQR?

Example 6

medium
A histogram has n=40n = 40 observations with bins counts 4, 12, 16, 8. Estimate the median bin.

Example 7

easy
A histogram has bins of width 2 with counts 3, 5, 7, 5. What is the total number of observations?

Example 8

medium
A histogram is right-skewed. Is the mean greater or less than the median?

Example 9

easy
Why do histogram bars touch with no gaps?

Example 10

hard
A histogram has bin widths 2 and 4 with counts 8, 16, 8. Why is comparing bar heights misleading, and what fixes it?

Example 11

medium
Cumulative counts of a histogram reach 12, 30, 40 after three bins (total 40). How many are in the second bin?

Example 12

easy
A symmetric, bell-shaped histogram suggests roughly what relation between mean and median?

Example 13

easy
How many bins are required to cover scores from 0 to 100 using bin width 20?

Example 14

medium
A histogram has bins of width 5 and frequencies 4, 10, 6. Convert to density (frequency / bin width).

Example 15

hard
A histogram has 30 observations, mean 12, and bins with midpoints 5, 10, 15, 20. Counts are a,10,12,4a, 10, 12, 4. Find aa.

Example 16

medium
A histogram uses bin widths of 2 and 4. Bins: [0,2):10[0,2):10, [2,6):20[2,6):20, [6,8):10[6,8):10. Which bin has the highest density?

Example 17

medium
A histogram of waiting times shows a single tall left peak with a long right tail. Which is larger: mean or median?

Example 18

medium
A histogram with n=200n = 200 has bins [0,10):40,[10,20):100,[20,30):60[0,10):40, [10,20):100, [20,30):60. Estimate the proportion of data at or above 10.

Example 19

easy
What does the y-axis of a frequency histogram represent?

Example 20

easy
How many bins does a histogram with edges 0,5,10,15,20 have?
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Related Concepts

Normal Distribution