IGCSE Maths statistics: averages, cumulative frequency and histograms

What each tier expects

The key ideas

Mean, median and mode answer different questions. The mean uses every value but is pulled by outliers; the median is the middle value and resists outliers; the mode is the most common value and the only average for non-numerical data. The range shows total spread; the interquartile range shows the spread of the middle half.

For grouped data you do not know the exact values, so use the midpoint of each class as a representative value. The result is an estimate of the mean. The modal class is the class with the highest frequency, and only for equal-width classes is that also the tallest bar of a histogram.

A cumulative frequency diagram plots running totals against the upper class boundaries, joined with a smooth curve. Read the median at half the total frequency, the lower quartile at a quarter and the upper quartile at three quarters. Percentiles work the same way.

A histogram is used for continuous data with classes of different widths. The vertical axis is frequency density, and frequency = frequency density × class width, which is the bar's area. To compare two distributions, compare an average and a measure of spread, and say what each means in context.

Worked example 1: mean, median and mode from a frequency table

1. Scores: 1 (frequency 3), 2 (frequency 5), 3 (frequency 8), 4 (frequency 4).
2. Total frequency Σf = 3 + 5 + 8 + 4 = 20. Σfx = 3 + 10 + 24 + 16 = 53.
3. Mean = 53 ÷ 20 = 2.65.
4. Median: the 10th and 11th values. Running totals are 3, 8, 16, so both are scores of 3. Median = 3. Mode = 3 (frequency 8).

Worked example 2: grouped data and a histogram

1. Journey times t minutes: 0 < t ≤ 10 (4 people), 10 < t ≤ 20 (9), 20 < t ≤ 30 (5), 30 < t ≤ 50 (2).
2. Midpoints: 5, 15, 25, 40. Σfx = 20 + 135 + 125 + 80 = 360. Σf = 20. Estimated mean = 360 ÷ 20 = 18 minutes.
3. Modal class: 10 < t ≤ 20. The median is the 10.5th value, which lies in the same class.
4. Frequency densities: 4 ÷ 10 = 0.4, 9 ÷ 10 = 0.9, 5 ÷ 10 = 0.5, 2 ÷ 20 = 0.1. Check the last bar: area 0.1 × 20 = 2 people. Correct.

Worked example 3: quartiles and IQR from a list

1. Data (ordered): 3, 5, 7, 8, 9, 11, 12, 14, 15, 18, 20. There are 11 values.
2. Median = the 6th value = 11.
3. Lower quartile = the median of the lower half (3, 5, 7, 8, 9) = 7. Upper quartile = the median of the upper half (12, 14, 15, 18, 20) = 15.
4. IQR = 15 - 7 = 8. Range = 20 - 3 = 17.
5. Comparing with another class with median 13 and IQR 4: 'the second class scored higher on average (median 13 against 11) and their scores were more consistent (IQR 4 against 8).'

Common mistakes that cost marks

• Dividing Σfx by the number of classes instead of the total frequency.
• Using class boundaries or frequencies instead of midpoints for grouped means.
• Plotting cumulative frequency at the midpoint of each class instead of the upper boundary.
• Drawing histogram bars with height = frequency when classes have different widths.
• Giving the median as the middle class's frequency rather than the class or value.
• Comparing data sets with numbers only, and no sentence in context about average and spread.

Exam technique and how a tutor helps

Add an fx column and a running-total column to any frequency table you are given; the examiner credits them and they make the mean and median quick. On Cambridge and Edexcel calculator papers, a scientific calculator's statistics mode can find Σfx and the mean, but write the method too. When reading from a cumulative frequency curve, draw the lines across and down on the graph so the examiner can see your reading.

A comparison question nearly always wants two statements: one about an average and one about spread, each in the context of the question. 'Class B did better on average because their median was higher' earns more than 'B's median is higher'.

Statistics feels easy until the diagrams appear. In one-to-one lessons a tutor works through histograms and cumulative frequency together on a shared grid, so the student learns to see area as frequency and to read off quartiles accurately. Lessons then move to the wording of comparisons, which is where many otherwise correct answers drop marks.

Self-check: can you do these?

• Find the mean of 4, 7, 7, 9, 13. (Answer: 8)
• Five numbers have a mean of 6. Four of them are 4, 5, 7 and 9. Find the fifth. (Answer: 5)
• A class 20 < x ≤ 35 has frequency 12. Find its frequency density. (Answer: 0.8)
• A histogram bar for 10 < x ≤ 14 has frequency density 2.5. Find the frequency. (Answer: 10)
• Which average is best for data with one very large value, and why? (Answer: the median, because it is not pulled by the outlier)