Histogram Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Histogram.

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

Concept 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.

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: A histogram groups numeric data into equal-width bins and draws a bar for how many values fall in each.

Common stuck point: The procedure for histogram is the easy part; the trap is leaving gaps between bars. Asking "Am I grouping one numeric variable into intervals and showing the count in each?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I grouping one numeric variable into intervals and showing the count in each?

Worked Examples

Example 1

easy
The following data represent exam scores: {52,64,68,71,75,75,78,82,85,88,90,93}\{52, 64, 68, 71, 75, 75, 78, 82, 85, 88, 90, 93\}. Group them into intervals [50,70),[70,80),[80,90),[90,100][50,70), [70,80), [80,90), [90,100] and describe how to construct a histogram.

Answer

Histogram with bars of height 3, 4, 3, 2 over intervals [50,70),[70,80),[80,90),[90,100][50,70),[70,80),[80,90),[90,100].

First step

1
Count values in each interval: [50,70)[50,70): 52, 64, 68 → 3 values; [70,80)[70,80): 71, 75, 75, 78 → 4 values; [80,90)[80,90): 82, 85, 88 → 3 values; [90,100][90,100]: 90, 93 → 2 values

Full solution

  1. 2
    Draw horizontal axis with interval boundaries (50, 70, 80, 90, 100) and vertical axis for frequency
  2. 3
    Draw touching bars (no gaps) with heights 3, 4, 3, 2 for each interval respectively
  3. 4
    Label axes: 'Score' on x-axis, 'Frequency' on y-axis
Histograms display the distribution of quantitative data. Unlike bar charts, bars touch to show continuous data. The height represents frequency (or relative frequency). The shape reveals skewness, modes, and spread.

Example 2

medium
A histogram of household incomes is skewed right. Describe what this means and explain why the mean would be greater than the median for this distribution.

Example 3

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

Practice Problems

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

Example 1

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 2

hard
Two histograms of test scores have the same mean (75) but different shapes: one is symmetric and bell-shaped, the other is left-skewed. Explain what additional information the shapes provide that the mean alone does not.

Example 3

easy
A histogram has bins [0,10),[10,20),[20,30) with counts 4,7,2. How many data points total?

Example 4

easy
In a histogram, which bin is the mode if counts are 3, 9, 5 across three equal bins?

Example 5

easy
Why do histogram bars touch with no gaps?

Example 6

easy
A histogram's tallest bars are on the left with a long tail right. What is the shape?

Example 7

easy
Bins of width 5 with counts 2,4,6. What does the y-axis represent?

Example 8

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

Example 9

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

Example 10

easy
Two histograms of the same data: one with very narrow bins looks jagged. Why?

Example 11

medium
A histogram: [0,2):5, [2,4):10, [4,6):5. Estimate the median bin.

Example 12

medium
Bins [0,10):3, [10,20):5, [20,30):2. Estimate the mean using bin midpoints.

Example 13

medium
A histogram has counts 2,8,8,2 over four equal bins. Describe the shape.

Example 14

medium
Total 50 observations; bins have counts 10,20,15,5. What fraction is in the second bin?

Example 15

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

Example 16

medium
Counts 4,6,10 across bins of widths 2,2,4. Which bin has the highest density (count per unit width)?

Example 17

medium
A histogram has two separated peaks. What is this shape called?

Example 18

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

Example 19

medium
A histogram has bins with counts 6, 9, 5. What fraction of data is in the tallest bin?

Example 20

challenge
A histogram (total 100) has bins [0,10):20,[10,20):50,[20,30):30. Estimate the 60th percentile by linear interpolation.

Example 21

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 22

challenge
A histogram of exam scores is left-skewed. Order mean, median, mode from smallest to largest.

Example 23

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

Example 24

easy
A histogram has bins [0,5),[5,10),[10,15)[0,5),[5,10),[10,15) with counts 4, 8, 8. Which bin(s) contain the mode?

Example 25

easy
A histogram has bins of width 1 with counts 2, 4, 3. Estimate the mean using bin midpoints 0.5,1.5,2.50.5, 1.5, 2.5.

Example 26

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

Example 27

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

Example 28

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

Example 29

medium
A histogram has 60 observations across bins with frequencies 12, 24, 18, 6. What relative frequency is in the second bin?

Example 30

medium
A histogram has cumulative frequencies 5, 15, 35, 50 (total 50). How many observations fall in the third bin?

Example 31

medium
A histogram has bins [0,4):8[0,4):8, [4,8):16[4,8):16, [8,12):16[8,12):16, [12,16):8[12,16):8. Describe shape and the relationship between mean and median.

Example 32

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

Example 33

medium
A histogram has equal-width bins with counts 1, 2, 4, 2, 1. Identify the shape and modal bin.

Example 34

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 35

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 36

medium
A right-skewed histogram has mean 60 and median 50. Predict the relationship between mean, median, and mode.

Example 37

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 38

hard
A histogram of incomes (in thousands) has bins [0,20):50,[20,40):30,[40,60):15,[60,80):5[0,20):50, [20,40):30, [40,60):15, [60,80):5. Estimate the mean using midpoints.

Example 39

hard
A histogram with bin width 5 has densities 0.04, 0.10, 0.06. Compute the relative frequencies in each bin.

Example 40

hard
A histogram of 40 quiz scores has bins of width 10 with frequencies 4, 12, 16, 8. Estimate the variance using midpoints.

Example 41

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 42

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 43

challenge
A histogram with bin width hh approximates a density ff. If hh is too large, what visual artifact occurs and how does it affect mean estimates?

Example 44

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

Normal Distribution