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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?
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.
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.
| 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 |
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.
| 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.
After drawing a histogram, describe the shape of the distribution:
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.
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.
(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).
| Interval | Frequency | Cumulative |
|---|---|---|
| 0–<10 | 3 | 3 |
| 10–<20 | 7 | 10 |
| 20–<30 | 10 | 20 |
| 30–<40 | 8 | 28 |
| 40–<50 | 2 | 30 |
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.
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.