IGCSE Maths · Statistics · Core & Extended

Classifying Data, Averages and Range

This opens the final unit of the course — Statistics. This content is identical for Core and Extended. Before calculating anything, it helps to know what kind of data you’re dealing with; then mean, median, mode, and range summarise it in different, complementary ways.

1. Classifying Statistical Data

TypeMeaningExample
QualitativeDescribes a quality/category, not a numberFavourite colour, type of pet
QuantitativeNumerical dataHeight, number of siblings
DiscreteCountable, specific values only (often whole numbers)Number of goals scored, shoe size
ContinuousCan take any value in a range, usually measuredHeight, weight, time
Try it — Classifying Data
  1. Classify “the number of pets a student owns” — qualitative or quantitative? Discrete or continuous?
  2. Classify “favourite subject” — qualitative or quantitative?
  3. Classify “the time taken to run 100 m” — discrete or continuous?
Show answers
  1. Quantitative and discrete (whole number of pets).
  2. Qualitative (a category, not a number).
  3. Continuous (time can take any value, not just whole numbers).

2. Mean, Median, Mode and Range

Mean = (sum of all values) ÷ (number of values)
Median = the middle value when data is ordered
Mode = the most frequently occurring value
Range = largest value − smallest value
Worked example Find the mean, median, mode, and range of: 4, 7, 7, 9, 10, 13
Mean = (4+7+7+9+10+13) ÷ 6 = 50÷6 = 8.33 (2 d.p.)
Median: data is already ordered; with 6 values, average the middle two (7 and 9): (7+9)/2 = 8
Mode = 7 (it appears twice, more than any other value)
Range = 13 − 4 = 9
Try it — Mean, Median, Mode, Range
  1. Find the mean, median, mode, and range of: 3, 5, 5, 8, 9.
  2. Find the median of: 2, 6, 9, 11, 15, 18 (an even number of values).
  3. A data set has mean 12 and 5 values. Find the sum of the values.
Show answers
  1. Mean=(3+5+5+8+9)/5=6. Median=5. Mode=5. Range=9−3=6
  2. (9+11)/2 = 10
  3. 12×5 = 60

3. Choosing the Right Average

AverageBest used when…Watch out for…
MeanData is fairly evenly spreadVery sensitive to extreme values (outliers)
MedianData has outliers or is skewedIgnores the actual size of most values
ModeData is qualitative, or you need the most common categoryMay not exist, or there may be more than one
Worked example A small company has 6 employees earning $28,000, $30,000, $31,000, $29,000, $32,000, and one director earning $210,000. Which average best represents “typical” pay, and why?
Mean = (28+30+31+29+32+210)/6 thousand = 360/6 = $60,000 — but this is pulled far above what most employees actually earn, because of the one very high salary.
Median = middle value when ordered (28,29,30,31,32,210): average of 30 and 31 = $30,500 — this reflects typical pay much better, since it isn’t distorted by the director’s salary.
Try it — Choosing an Average
  1. Explain why the mode might not be useful for the data set: 2, 3, 5, 7, 11 (no repeated values).
  2. A shoe shop wants to know which shoe size to stock the most of. Which average should they use, and why?
Show answers
  1. There’s no repeated value, so there’s no mode at all — every value appears exactly once.
  2. Mode — they need the most commonly occurring size, not an average value that might not even correspond to a real shoe size.

4. Mean and Modal Class for Grouped Data

When data is grouped into class intervals, you can’t find an exact mean — instead, use the midpoint of each class as an estimate for every value inside it.

Estimated mean = Σ(frequency × midpoint) ÷ Σfrequency
Worked example
ClassFrequencyMidpoint
0–1055
10–20815
20–301225
30–40535
Find the estimated mean, and state the modal class.
Estimated mean = (5×5 + 8×15 + 12×25 + 5×35) ÷ (5+8+12+5) = (25+120+300+175)/30 = 620/30 = 20.7 (1 d.p.)
Modal class: the class with the highest frequency is 20–30 (frequency 12).
Try it — Grouped Data
  1. A grouped frequency table has classes 0–5 (f=4), 5–10 (f=9), 10–15 (f=7). Find the estimated mean.
  2. Using the same table, state the modal class.
Show answers
  1. Midpoints: 2.5, 7.5, 12.5. Mean=(4×2.5+9×7.5+7×12.5)/20=(10+67.5+87.5)/20=165/20=8.25
  2. 5–10 (highest frequency, 9)
Common mistakes to watch for
  • Forgetting to order the data before finding the median — the “middle value” only makes sense once values are sorted.
  • For an even number of values, forgetting to average the two middle values for the median.
  • Confusing the modal class (the class interval with the highest frequency) with an actual “mode” value — grouped data doesn’t have one exact mode.
  • Using the class width instead of the midpoint when estimating a mean from grouped data.
  • Assuming the mean is always the “best” average — it can be heavily skewed by outliers, when the median might describe the data more fairly.

5. Quick-Fire Challenge Round

True or False?
  1. The median is always one of the actual data values in the set.
  2. The mean is more affected by outliers than the median.
  3. A data set can have more than one mode.
  4. The estimated mean from grouped data is usually exact, not an estimate.
Show answers
  1. False — if there’s an even number of values, the median is the average of the two middle ones, which might not be an actual data value.
  2. True.
  3. True — a data set can be “bimodal” (two modes) or have no mode at all.
  4. False — it’s genuinely an estimate, since we don’t know the exact values within each class, only their midpoint assumption.
Spot the Error
  1. “Data: 3, 8, 5, 9, 2. Median = 5 (the middle number as listed).”
  2. “Mean of 4, 4, 6, 10: (4+4+6+10)/3 = 8.”
  3. “The modal class of a grouped table is the class with the smallest frequency.”
Show answers
  1. The data must be ordered first: 2,3,5,8,9 — the median (5) happens to be correct here by coincidence, but the reasoning (“middle as listed”) is wrong; always sort first.
  2. There are 4 values, not 3: (4+4+6+10)/4 = 6, not 8.
  3. The modal class has the highest frequency, not the smallest.

6. Mixed Practice — Bringing It All Together

  1. Find the mean, median, mode, and range of: 6, 9, 9, 11, 15.
  2. Classify “number of cars passing a point in an hour” — discrete or continuous?
  3. A grouped table has classes 0–20 (f=6, midpoint 10), 20–40 (f=10, midpoint 30), 40–60 (f=4, midpoint 50). Find the estimated mean and modal class.
Show answers
  1. Mean=(6+9+9+11+15)/5=10. Median=9. Mode=9. Range=15−6=9
  2. Discrete (whole number of cars).
  3. Mean=(6×10+10×30+4×50)/20=(60+300+200)/20=560/20=28. Modal class=20–40

Practice at Home

A large set to open the Statistics unit — take it steadily.

Part A — Classifying data
  1. Classify “eye colour” — qualitative or quantitative?
  2. Classify “mass of a bag of flour” — discrete or continuous?
  3. Classify “number of children in a family” — discrete or continuous?
Part B — Mean, median, mode, range
  1. Find the mean, median, mode, and range of: 12, 15, 15, 18, 20, 20, 20.
  2. A data set has 8 values with mean 15. Find the total sum.
  3. Find the median of: 4, 10, 15, 22 (an even number of values).
Part C — Choosing the right average
  1. A small business’s weekly sales (in $) are: 400, 420, 410, 390, 405, 3200 (one exceptional week). Find the mean and median, and explain which better represents a “typical” week.
  2. Explain why the mode is the most useful average for finding the most popular ice cream flavour in a survey.
Part D — Grouped data
  1. A grouped table has classes 0–10 (f=3), 10–20 (f=7), 20–30 (f=6), 30–40 (f=4). Find the estimated mean.
  2. State the modal class for the table in question 1.
  3. A grouped table has classes 10–20 (f=5, midpoint 15), 20–30 (f=9, midpoint 25), 30–40 (f=6, midpoint 35). Find the estimated mean.
Part E — Applied / challenge
  1. A cricket team’s scores over a season are mostly around 150–200, but one match had an unusually low score of 20 due to bad weather. Explain which average (mean or median) would give a fairer summary of the team’s typical performance, and why.
  2. Six numbers have a mean of 14. Five of the numbers are 10, 12, 15, 16, and 18. Find the sixth number.
  3. A data set has mean 20 and range 15, with lowest value 12. Find the highest value, and explain what other information (if any) you’d still need to find the median.

Key Vocabulary Recap

Discrete data — countable, specific values only
Continuous data — can take any value within a range
Mean — the sum of values divided by how many there are
Median — the middle value when data is ordered
Modal class — the class interval with the highest frequency in grouped data
Outlier — a value far removed from the rest of the data

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C9.1 and C9.3 (Core subject content) and sections E9.1 and E9.3 (Extended subject content) — Classifying statistical data, and Averages and measures of spread. The wording of these sections is identical for both tiers. Statistical charts, diagrams, and scatter diagrams are covered in a second note; cumulative frequency diagrams and histograms (both Extended-only) are covered in a third.

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