Histograms
Tell a histogram from a bar chart, build one from raw measurements using class intervals, identify the modal class, and describe a distribution as symmetric or skewed.
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Think First
How is a histogram different from a bar chart? Think about the type of data each one displays. Can you think of a real-world situation where you would need a histogram instead of a bar chart?
Histograms: Grouped Continuous Data
A histogram looks like a bar chart, but the bars touch each other. This signals that the data is continuous: there are no gaps between class intervals on the number line.
What You'll Master
- Explain why histograms have no gaps between bars
- Construct a histogram from a grouped frequency table
- Identify the modal class from a histogram or frequency table
- Describe the shape of a distribution (symmetric, skewed left, skewed right)
- Estimate the mean from grouped data using class midpoints
Words You Need
⚠ Spot the Trap
Gaps vs no gaps: In a bar chart, gaps between bars signal separate categories. In a histogram, no gaps signal continuous data. Drawing gaps in a histogram tells the reader the data is categorical, that is wrong.
Boundaries, not midpoints: Label the x-axis with class boundaries (140, 150, 160…), not midpoints (145, 155…). Bars span from boundary to boundary.
Histogram vs Bar Chart
| Feature | Bar Chart | Histogram |
|---|---|---|
| Data type | Categorical or discrete | Continuous (grouped) |
| Gaps between bars | Yes | No, bars touch |
| x-axis | Category labels | Number scale (boundaries) |
| Bar width | Equal (or varied) | Equal for equal intervals |
Drawing a Histogram, Step by Step
- Build a grouped frequency table with equal class intervals.
- Draw axes: x-axis = the measurement (class boundaries on number scale), y-axis = frequency.
- Draw bars from boundary to boundary, they must touch. Height = frequency.
- Label both axes with variable names and units.
- Add a title.
Interval notation: write "160–<170" (160 included, 170 excluded). A student of exactly 170 cm goes into the next bar, not this one. This prevents double-counting.
Worked Example: Heights of 30 Students
| Height (cm) | Frequency (f) | Midpoint (x) | f × x |
|---|---|---|---|
| 140–<150 | 3 | 145 | 435 |
| 150–<160 | 8 | 155 | 1240 |
| 160–<170 ★ | 12 | 165 | 1980 |
| 170–<180 | 6 | 175 | 1050 |
| 180–<190 | 1 | 185 | 185 |
| Total | 30 | 4890 |
Modal class: 160–<170 (★ highest frequency = 12).
Estimated mean: $$\bar{x} = \frac{4890}{30} = 163 \text{ cm}$$
This is an estimate we assumed all values in each interval are at the midpoint.
Describing Shape and Distribution
After drawing a histogram, describe the shape of the distribution:
- Symmetric (bell-shaped): frequencies rise then fall evenly. Mean ≈ median ≈ modal class midpoint.
- Skewed right (positive skew): tail stretches to the right. Most data is low; a few high values exist. Mean > median.
- Skewed left (negative skew): tail stretches to the left. Most data is high; a few low values exist. Mean < median.
The heights histogram above is slightly skewed right most students are 160–170 cm, but the few students at 180–190 cm pull the tail right.
Common Pitfalls
- Leaving gaps between bars, turns your histogram into a bar chart.
- Using midpoints on the x-axis instead of class boundaries.
- Estimating the mean without using midpoints, you cannot average the boundaries.
- Confusing the modal class with "the mode", in grouped data give the whole interval, not a single value.
- Forgetting to include units in axis labels.
Frequency Polygons
A frequency polygon is a line graph drawn from the same grouped data as a histogram. Instead of bars, you plot one point above the midpoint of each class interval at the height of that class frequency, then join the points with straight line segments.
The midpoint of a class interval is the average of its two boundaries. For the interval 160–<170 cm the midpoint is $(160 + 170) \div 2 = 165$ cm, so the point is plotted at $x = 165$.
To close the polygon, add a point at the midpoint of an empty class at each end, one below the first interval and one above the last, each at height 0. The line then starts and finishes on the x-axis, which is why the shape is called a polygon.
A frequency polygon can be drawn on its own or directly over the histogram of the same data. Drawn over the histogram, each plotted point sits at the top centre of its bar. The polygon is preferred when you want to compare two datasets, because two polygons can be drawn on one pair of axes without the bars hiding each other.
- Same data as the histogram, same frequencies, different display.
- Plot at the class midpoint, not at the class boundary.
- Add a zero-frequency point at each end so the line meets the x-axis.
- Use a polygon rather than a histogram when comparing two or more datasets.
Copy This Into Your Book
Histogram: bar graph for continuous grouped data. Bars touch (no gaps). x-axis = class boundaries. Height = frequency.
Modal class = class interval with highest frequency (tallest bar).
$$\text{Estimated mean} = \frac{\sum f \times \text{midpoint}}{\sum f}$$
Shape: symmetric (bell) / skewed right (tail right) / skewed left (tail left).
What is the key difference between a histogram and a bar chart?
A frequency table shows: 10–<15: 4, 15–<20: 7, 20–<25: 11, 25–<30: 14, 30–<35: 9. What is the modal class?
A histogram shows the interval 50–<60 with a bar reaching height 9 on the frequency axis. How many data values are in this interval?
A histogram of salaries shows most employees earning $40 000–$60 000, with a few earning over $200 000. How would you describe the shape?
Which of the following correctly describes a histogram?
Q6. Ages at a community centre: 10–<20: 5, 20–<30: 12, 30–<40: 18, 40–<50: 10, 50–<60: 5. (a) State the modal class. (b) Estimate the mean age using midpoints. Show all working including a table of f × midpoint.
Q7. A histogram has 5 bars of equal width 10. Bars have heights (frequencies): 3, 7, 10, 8, 2 for intervals starting at 0. Reconstruct the grouped frequency table with columns: class interval, frequency, cumulative frequency.
Q8. A teacher compares two classes' exam results. Group A has a bell-shaped (symmetric) histogram centred on 70%. Group B has a right-skewed histogram, most students scored below 50% but a few scored above 85%. Describe what this tells you about each group's performance and suggest which group needs more support.
Show Answers
Q6
(a) Modal class: 30–<40 (frequency 18, highest).
(b) Midpoints: 15, 25, 35, 45, 55. f × midpoint: 5×15=75, 12×25=300, 18×35=630, 10×45=450, 5×55=275. Sum of (f×x) = 1730. Total f = 50.
$\bar{x} = \dfrac{1730}{50} = \mathbf{34.6}$ years (estimate).
Q7
| Interval | Frequency | Cumulative |
|---|---|---|
| 0–<10 | 3 | 3 |
| 10–<20 | 7 | 10 |
| 20–<30 | 10 | 20 |
| 30–<40 | 8 | 28 |
| 40–<50 | 2 | 30 |
Q8
Group A is performing consistently, most students score around 70% with the spread even on both sides. This is typical of a well-prepared class. Group B is skewed right, the majority scored below 50% (low performance) with only a few high achievers pulling the tail right. Group B needs more support: most students are below expectations, and the skewed shape shows uneven understanding across the class.
Stretch Challenge
A frequency table with equal intervals of 10: 0–<10: 2, 10–<20: 5, 20–<30: 9, 30–<40: 8, 40–<50: 4, 50–<60: 2. Total = 30 values.
(a) Describe the shape of the histogram (symmetric? skewed?).
(b) The median is the average of the 15th and 16th values. Build the cumulative frequency column. Which class interval contains both the 15th and 16th values? Estimate the median.
(c) Estimate the mean using class midpoints.