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AQA GCSE · Question 18 · Statistics
The table shows information about the height of 40 plants. Draw a histogram to represent the heights.
Height, h cm Frequency 0 < h ≤ 5 6 5 < h ≤ 15 15 15 < h ≤ 25 9 25 < h ≤ 50 10
The table shows information about the height of 40 plants. Draw a histogram to represent the heights.
| Height, h cm | Frequency |
|---|---|
| 0 < h ≤ 5 | 6 |
| 5 < h ≤ 15 | 15 |
| 15 < h ≤ 25 | 9 |
| 25 < h ≤ 50 | 10 |
How to approach this question
1. Understand that for a histogram with unequal class widths, the vertical axis represents Frequency Density, not Frequency.
2. The formula is: Frequency Density = Frequency / Class Width.
3. For each row in the table, calculate the class width (upper bound - lower bound).
4. For each row, calculate the frequency density using the formula. It's a good idea to add these two columns to the table.
5. The width of each bar on the histogram is its class width.
6. The height of each bar is its calculated frequency density.
7. Draw the four bars on the axes provided.
Full Answer
To draw a histogram, we need to calculate the frequency density for each class.
Frequency Density = Frequency / Class Width
1. **Class 0 < h ≤ 5:**
Class Width = 5 - 0 = 5
Frequency Density = 6 / 5 = 1.2
2. **Class 5 < h ≤ 15:**
Class Width = 15 - 5 = 10
Frequency Density = 15 / 10 = 1.5
3. **Class 15 < h ≤ 25:**
Class Width = 25 - 15 = 10
Frequency Density = 9 / 10 = 0.9
4. **Class 25 < h ≤ 50:**
Class Width = 50 - 25 = 25
Frequency Density = 10 / 25 = 0.4
Now, draw the bars on the graph:
- Bar 1: from h=0 to h=5, with height (frequency density) 1.2
- Bar 2: from h=5 to h=15, with height 1.5
- Bar 3: from h=15 to h=25, with height 0.9
- Bar 4: from h=25 to h=50, with height 0.4
A histogram is used to represent continuous data, especially when the class intervals (or widths) are unequal. The area of each bar is proportional to the frequency, which means the height of each bar must be the **Frequency Density**.
The formula is: **Frequency Density = Frequency / Class Width**
Let's calculate this for each group:
* **Group 1: 0 < h ≤ 5**
* Frequency = 6
* Class Width = 5 - 0 = 5
* Frequency Density = 6 / 5 = **1.2**
* **Group 2: 5 < h ≤ 15**
* Frequency = 15
* Class Width = 15 - 5 = 10
* Frequency Density = 15 / 10 = **1.5**
* **Group 3: 15 < h ≤ 25**
* Frequency = 9
* Class Width = 25 - 15 = 10
* Frequency Density = 9 / 10 = **0.9**
* **Group 4: 25 < h ≤ 50**
* Frequency = 10
* Class Width = 50 - 25 = 25
* Frequency Density = 10 / 25 = **0.4**
Now, we draw the histogram:
- The first bar goes from 0 to 5 on the x-axis and has a height of 1.2 on the y-axis.
- The second bar goes from 5 to 15 and has a height of 1.5.
- The third bar goes from 15 to 25 and has a height of 0.9.
- The fourth bar goes from 25 to 50 and has a height of 0.4.
Common mistakes
✗ Plotting frequency on the y-axis instead of frequency density. This is the most common mistake and results in a bar chart, not a histogram.
✗ Incorrectly calculating class widths.
✗ Making arithmetic errors when calculating frequency density.
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