Statistics

Histograms with Unequal Class Widths

Year 10 · Year 11

✓By the end of this lesson students will be able to understand the purpose of a histogram for continuous data.
✓By the end of this lesson students will be able to calculate frequency density for given class intervals.
✓By the end of this lesson students will be able to construct a histogram with unequal class widths using frequency density.
✓By the end of this lesson students will be able to interpret information from a histogram, including estimating frequencies for sub-intervals.

Key concepts

Histograms

A histogram is a type of bar chart used to represent continuous data. Unlike bar charts, there are no gaps between the bars, as the data is continuous. The key feature of a histogram is that the area of each bar is proportional to the frequency of the data in that class interval. This is crucial when dealing with unequal class widths.

Class Width

The class width is the size of the interval for each class. It is calculated by subtracting the lower bound from the upper bound of the class interval. For example, for the class 10 ≤ x < 20, the class width is 20 - 10 = 10.

Class Width = Upper Bound - Lower Bound

Frequency Density

When class widths are unequal, simply plotting frequency on the y-axis would be misleading, as wider bars would appear to have more frequency than they actually do. To correct this, we use frequency density. Frequency density is plotted on the y-axis, ensuring that the area of each bar (height × width = frequency density × class width) correctly represents the frequency for that class interval.

Frequency Density = Frequency / Class Width

Key facts to remember

1Histograms are used to display continuous data.
2The area of each bar in a histogram represents the frequency of the data in that class interval.
3When class widths are unequal, the y-axis must be labelled 'Frequency Density'.
4Frequency Density = Frequency ÷ Class Width.
5Class Width = Upper Bound - Lower Bound of the interval.
6To find the frequency from a histogram, calculate the area of the bar (Frequency Density × Class Width).

Worked examples

Example 1

The table shows the time, t minutes, that 60 students spent completing a maths puzzle. Draw a histogram to represent this data.

IFirst, calculate the class width for each interval:

II0 ≤ t < 5: Class width = 5 - 0 = 5

III5 ≤ t < 10: Class width = 10 - 5 = 5

IV10 ≤ t < 20: Class width = 20 - 10 = 10

V20 ≤ t < 30: Class width = 30 - 20 = 10

VI30 ≤ t < 50: Class width = 50 - 30 = 20

VIINext, calculate the frequency density for each interval:

VIII0 ≤ t < 5: Frequency Density = 10 / 5 = 2

95 ≤ t < 10: Frequency Density = 20 / 5 = 4

1010 ≤ t < 20: Frequency Density = 18 / 10 = 1.8

1120 ≤ t < 30: Frequency Density = 8 / 10 = 0.8

1230 ≤ t < 50: Frequency Density = 4 / 20 = 0.2

13Now, draw the histogram. The x-axis should represent 'Time (t minutes)' and the y-axis should represent 'Frequency Density'. Plot bars with widths corresponding to the class widths and heights corresponding to the calculated frequency densities.

14Bar 1: Width 5 (from 0 to 5), Height 2

15Bar 2: Width 5 (from 5 to 10), Height 4

16Bar 3: Width 10 (from 10 to 20), Height 1.8

17Bar 4: Width 10 (from 20 to 30), Height 0.8

18Bar 5: Width 20 (from 30 to 50), Height 0.2

Answer

A histogram with the following bars: - 0 ≤ t < 5: width 5, height 2 - 5 ≤ t < 10: width 5, height 4 - 10 ≤ t < 20: width 10, height 1.8 - 20 ≤ t < 30: width 10, height 0.8 - 30 ≤ t < 50: width 20, height 0.2 (Note: A visual representation of the histogram would be drawn on graph paper in an exam.)

Frequency table for the problem: Time (t minutes) | Frequency -----------------|----------- 0 ≤ t < 5 | 10 5 ≤ t < 10 | 20 10 ≤ t < 20 | 18 20 ≤ t < 30 | 8 30 ≤ t < 50 | 4

Example 2

A histogram shows the weights of apples. The bar for 0 < w ≤ 50g has a frequency density of 0.2 and a frequency of 10. The next bar is for 50 < w ≤ 100g and has a frequency density of 0.6. Calculate the frequency for the 50 < w ≤ 100g class. Also, estimate the number of apples with a weight between 70g and 90g.

IFirst, verify the relationship between frequency, class width, and frequency density using the first bar:

IIFor 0 < w ≤ 50g: Class width = 50 - 0 = 50g. Frequency = 10. Frequency Density = 0.2.

IIICheck: Frequency Density = Frequency / Class Width ⇒ 0.2 = 10 / 50. This is correct, so the y-axis is indeed frequency density.

IVNow, calculate the frequency for the 50 < w ≤ 100g class:

VClass width = 100 - 50 = 50g. Frequency Density = 0.6.

VIFrequency = Frequency Density × Class Width = 0.6 × 50 = 30.

VIINext, estimate the number of apples with a weight between 70g and 90g. This interval falls within the 50 < w ≤ 100g class.

VIIIThe frequency density for this sub-interval is still 0.6.

9The width of this sub-interval is 90 - 70 = 20g.

10Estimated frequency = Frequency Density × Sub-interval Width = 0.6 × 20 = 12.

Answer

The frequency for the 50 < w ≤ 100g class is 30. The estimated number of apples with a weight between 70g and 90g is 12.

Example 3

A histogram represents the ages of people attending a concert. The class 10 ≤ age < 20 has a frequency of 40 and a bar height of 4. The next class is 20 ≤ age < 40 and has a bar height of 2. Find the frequency for the class 20 ≤ age < 40.

IFor the class 10 ≤ age < 20:

IIClass width = 20 - 10 = 10.

IIIFrequency = 40.

IVBar height = 4.

VWe know that Frequency Density = Frequency / Class Width. So, Frequency Density = 40 / 10 = 4.

VISince the bar height is 4, this confirms that the y-axis represents Frequency Density.

VIIFor the class 20 ≤ age < 40:

VIIIClass width = 40 - 20 = 20.

9Bar height (Frequency Density) = 2.

10Frequency = Frequency Density × Class Width = 2 × 20 = 40.

Answer

The frequency for the class 20 ≤ age < 40 is 40.

Common mistakes

✗Using frequency on the y-axis when class widths are unequal, leading to a misleading representation.
✗Incorrectly calculating the class width, especially for intervals that start from zero or have different scales.
✗Misinterpreting the area of the bars; forgetting that area, not height, represents frequency.
✗Not labelling the axes correctly, particularly the y-axis as 'Frequency Density'.
✗Assuming that the frequency density for a sub-interval is different from the frequency density of the larger interval it belongs to.

Exam tips

★Always check if the class widths are equal or unequal. This determines whether you plot frequency or frequency density on the y-axis.
★Clearly label both axes: the x-axis with the variable and units (e.g., 'Time (minutes)'), and the y-axis as 'Frequency Density'.
★Use a ruler and pencil for drawing histograms to ensure accuracy and neatness. Marks are often awarded for presentation.
★Show all calculations for class width and frequency density in your working, even if they seem straightforward.

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