Bars for grouped numbers
A histogram shows how numerical data is spread across intervals. Each bar represents a bin (interval), and its height shows the frequency: how many values fall in that bin.
Histogram vs bar graph
| Histogram | Bar graph |
|---|---|
| numerical data in intervals | categories |
| bars touch (no gaps) | bars have gaps |
| order matters on the x-axis | order of categories can change |
| shows a distribution | compares categories |
Histograms are ideal for continuous data like heights, times and test scores.
Making a histogram
Data: 20 students’ quiz times (minutes)
12, 15, 18, 22, 25, 27, 28, 31, 33, 34, 35, 36, 38, 41, 42, 45, 48, 52, 55, 58
- Choose equal-width bins: 10–19, 20–29, 30–39, 40–49, 50–59.
- Make a frequency table:
| Time (min) | Frequency |
|---|---|
| 10–19 | 3 |
| 20–29 | 4 |
| 30–39 | 6 |
| 40–49 | 4 |
| 50–59 | 3 |
- Draw bars with no gaps, heights matching the frequencies.
- Label the axes and add a title.
Choosing bins
- Usually 5 to 10 bins.
- Equal widths.
- Every value fits exactly one bin. Be clear about boundaries: a value of 20 goes in 20–29, not 10–19.
- Too few bins hide detail; too many make it bumpy.
Reading the shape
| Shape | Description | Example |
|---|---|---|
| Symmetric (bell-shaped) | peak in the middle, tails equal | heights of adults |
| Skewed right | long tail to the right | household incomes |
| Skewed left | long tail to the left | scores on an easy test |
| Uniform | bars about the same height | rolls of a fair die |
| Bimodal | two peaks | mixed groups |
In a skewed distribution, the mean is pulled toward the tail. For right-skewed data, mean > median.
Interpreting
From the quiz data:
- Most common interval: 30–39 min.
- Students finishing in under 30 min: 3 + 4 = 7 (35%).
- The distribution is roughly symmetric.
Worked examples
Example 1: How many students took 40 min or more? 4 + 3 = 7.
Example 2: A histogram of house prices has a long tail to the right. Shape? Skewed right.
Example 3: Why do histogram bars touch? The intervals are continuous: one ends where the next begins.
Common mistakes
- Leaving gaps between bars.
- Unequal bin widths without adjusting.
- Overlapping bins like 10–20 and 20–30.
Try it yourself
Ages at a party: 8, 12, 15, 21, 23, 25, 27, 29, 34, 36, 38, 44, 47, 52
- Make a frequency table with bins 0–9, 10–19, …, 50–59.
- Which bin has the highest frequency?
- Describe the shape.
Answers: 1) 1, 2, 5, 3, 2, 1 2) 20–29 3) roughly symmetric, peaking in the 20s
Quick recap
- A histogram shows grouped numerical data with touching bars.
- Use equal, non-overlapping bins.
- Describe the distribution: symmetric, skewed, uniform or bimodal.