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 Statistics.
Concept Recap
A histogram is a graph that groups numerical data into equal-width ranges (bins) and shows the frequency of values in each range using adjacent bars that touch. Unlike bar graphs, histograms display the distribution shape of continuous data.
Unlike bar graphs for categories (red, blue, green), histograms are for numbers grouped into ranges. Test scores 60-70, 70-80, 80-90... The bars touch because the data is continuous - there's no gap between 69.9 and 70.0.
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:Histogram organizes data so the right pattern is visible without distorting the counts or scale.
Common stuck point:Students often know a procedure related to histogram but skip the recognition step: Am I choosing or interpreting a display that matches the type of data and the question being asked? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I choosing or interpreting a display that matches the type of data and the question being asked?
Worked Examples
Example 1
medium
A histogram of 30 test scores has bins 50-60 (6), 60-70 (10), 70-80 (9), 80-90 (5). What percent of students scored 70 or higher?What percent of students scored 70 or higher?
Answer
≈46.7%
First step
1
At or above 70: 9+5=14.
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Setup·Key insight·Why it works·Common pitfall·Connection
A histogram of 50 ages has bins 0-10,10-20,20-30,30-40 with frequencies 10,15,20,5. Estimate the mean age using midpoints.Estimate the mean age using bin midpoints.
Example 3
medium
Test scores are grouped: 50–59 (4), 60–69 (8), 70–79 (12), 80–89 (6), 90–99 (2). Describe how to construct a histogram and identify the modal class.
Example 4
medium
A histogram of student heights shows bars: 140–149 cm (3), 150–159 cm (10), 160–169 cm (15), 170–179 cm (8), 180–189 cm (4). Describe the shape of the distribution.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A histogram has bins 0-10, 10-20, 20-30 with frequencies 4, 7, 2. How many data values are in the 10-20 bin?Read the frequency of the 10–20 bin.
Example 2
easy
A histogram has bins with frequencies 3, 8, 5, 2. How many data values are there in total?
Example 3
easy
A histogram's tallest bar is the 70-80 bin. Which bin contains the most data values?
Example 4
easy
A histogram has bins 0-5, 5-10, 10-15 with frequencies 6, 9, 5. How many values are less than 10?How many values are less than 10?
Example 5
easy
Why do histogram bars touch while bar-graph bars have gaps?
Example 6
easy
A histogram has equal-width bins of width 10 starting at 0. Which bin does the value 25 fall into (use the convention left-inclusive)?
Example 7
easy
Is a histogram appropriate for (A) 100 students' heights in cm, or (B) the names of 5 favorite movies?
Example 8
easy
A histogram bin 30-40 has frequency 5 and the total is 25. What fraction of the data is in that bin?
Example 9
medium
A histogram has bins 0-10, 10-20, 20-30, 30-40 with frequencies 2, 5, 8, 5. In which bin does the median lie (n=20)?In which bin does the median lie? (n = 20)
Example 10
medium
A histogram has bins of frequencies 4, 6, 10. What percent of the data is in the largest bin?
Example 11
medium
A histogram has bins 0-20, 20-40, 40-60 with frequencies 10, 15, 25. How many values are at least 20?How many values are at least 20?
Example 12
medium
A histogram is described as right-skewed (long tail of high values). Where is the median relative to the mean?
Example 13
medium
A histogram has bins of width 5 and frequencies 2, 4, 6, 8 for bins starting at 0. Estimate the mean using bin midpoints.Estimate the mean using bin midpoints.
Example 14
medium
Two histograms have the same bins. Histogram P has frequencies 8, 4, 2; histogram Q has 2, 4, 8. Which is right-skewed?
Example 15
medium
A histogram has frequencies 5, 5, 5, 5 across four bins. What shape is this, and what is the relative frequency of each bin?What shape is this distribution?
Example 16
medium
A histogram has bins 0-10 (freq 3), 10-20 (freq 7), 20-30 (freq x), and total 20. Find x and the relative frequency of the 20-30 bin.Find the missing frequency x in the 20–30 bin.
Example 17
medium
A histogram of wait times has bins 0-5 (12), 5-10 (8), 10-15 (4), 15-20 (1). What percent of customers waited 10 minutes or more?What percent of customers waited 10 minutes or more?
Example 18
challenge
A histogram has bins of width 10 starting at 0 with frequencies 4, 6, 10, 5, 5 (n=30). Estimate the median by linear interpolation within the median bin.Estimate the median by linear interpolation.
Example 19
challenge
A histogram with frequencies 6, 9, 5 over bins 0-10, 10-20, 20-30 is claimed to have a mean of 14 using midpoints. Verify or correct it.Verify the estimated mean using bin midpoints.
Example 20
challenge
A histogram of 40 scores has bins 50-60, 60-70, 70-80, 80-90 with frequencies 6, 14, x, 8. The 70-80 bin holds 30% of the data. Find x and confirm the total.The 70–80 bin holds 30% of 40 scores. Find x.
Example 21
easy
A histogram has bins 0-10, 10-20, 20-30, 30-40 with frequencies 3,5,8,4. How many data points in total?How many data points in total?
Example 22
easy
A histogram has bins 0-5 and 5-10 with frequencies 4 and 6. What fraction is in the lower bin?What fraction of the data is in the lower bin?
Example 23
easy
A histogram has bin width 10 starting at 0. Which bin does the value 47 fall in (left-inclusive convention)?
Example 24
easy
A histogram's bins have frequencies 5,10,15,20. Which bin is the tallest?
Example 25
easy
A histogram has bins 0-10,10-20,20-30 with frequencies 3,7,10. How many values are below 20?How many values are below 20?
Example 26
easy
A histogram has 4 bins of frequency 2,4,6,8. What is the mode bin (range)?
Example 27
medium
A histogram has bins 0-10,10-20,20-30,30-40 with frequencies 6,8,4,2. Using bin midpoints, what is the estimated mean?Estimate the mean using bin midpoints.
Example 28
medium
A histogram has bins 0-10 (4), 10-20 (6), 20-30 (10). What is the modal bin and what percent of the data does it contain?What is the modal bin and what percent does it contain?
Example 29
medium
A histogram has bins 0-20,20-40,40-60,60-80 with frequencies 5,10,20,15. In which bin does the median lie (n=50)?In which bin does the median lie? (n = 50)
Example 30
medium
A histogram has bins of widths 10 with frequencies 4,8,10,6,2. What is the relative frequency of the third bin?What is the relative frequency of the third bin?
Example 31
medium
A histogram has bins 0-10,10-20,20-30 with frequencies 2,?,6, total 14. What is the missing frequency?The total is 14. Find the missing frequency in the 10–20 bin.
Example 32
medium
A histogram has bins of equal width 5 starting at 0, frequencies 1,3,5,4,2. Estimate the mean using midpoints.Estimate the mean using bin midpoints.
Example 33
medium
A histogram is bimodal with peaks at the 10-20 and 40-50 bins. What does this suggest about the data?
Example 34
medium
A histogram has bins 0-10,10-20,20-30 with frequencies 5,10,5. Is the distribution symmetric?Is this distribution symmetric?
Example 35
hard
A histogram has bins 0-10,10-20,20-30,30-40,40-50 with frequencies 2,4,6,3,1. Estimate the median using interpolation (n=16).Estimate the median using interpolation. (n = 16)
Example 36
hard
A histogram is left-skewed (long tail to the left). Where is the mean relative to the median?
Example 37
hard
A histogram of 60 data points has bin frequencies summing to 60. After adding 10 more data points spread evenly across 5 existing bins, what is the new total?
Example 38
hard
A histogram of incomes ($ thousands) has bins of width 20. The bin 40-60 has frequency 30 out of 100. What is the bin's frequency density (per $ thousand)?
Example 39
hard
A histogram has unequal-width bins: 0-10 (w=10, freq 20), 10-30 (w=20, freq 30). Which bin has higher frequency density?
Example 40
challenge
A histogram of 200 values has bins 0-25,25-50,50-75,75-100 with frequencies 40,60,70,30. Using interpolation, estimate the 80th percentile.Estimate the 80th percentile using interpolation.
Example 41
medium
Explain two differences between a histogram and a bar graph.
Example 42
medium
A histogram shows frequencies 0–9 (2), 10–19 (5), 20–29 (9), 30–39 (7), 40–49 (1). How many observations are at least 20, and what is the modal class?