Overview
What you’ll learn
- Calculate the mean, mode and median of ungrouped data and explain when each average is the right choice.
- Estimate the mean of grouped data using class midpoints and Σfx ÷ Σf.
- Locate quartiles and percentiles of ordered data sets and compute the interquartile range.
- Calculate the standard deviation of ungrouped and grouped data with the statistics mode of a calculator.
- Compare two data sets using mean and standard deviation, commenting on centre and consistency.
In 30 seconds
- 1Mean = Σx ÷ n; it uses every value, so a single extreme value drags it.
- 2The median is the middle of the ordered list; extreme values cannot move it.
- 3The mode is the most frequent value — the only average that works for categorical data.
- 4Grouped-data means are estimates: each class is replaced by its midpoint before computing Σfx ÷ Σf.
- 5Quartiles sit near the n/4 and 3n/4 positions of ordered data; IQR = Q3 − Q1.
- 6Standard deviation measures spread about the mean; bigger SD, less consistent.
- 7Adding the same constant to every value shifts the mean but leaves the SD unchanged.
- 8Comparing data sets needs both a centre (mean) and a spread (SD) in the same sentence.
Notes · 1 of 5
Mean, mode and median of ungrouped data
Three averages, three different jobs.
For a data list of n values: the mean is Σx ÷ n; the median is the middle value once the list is ordered (average the two middle values when n is even); the mode is the most frequent value. A data set can have one mode, several modes (bimodal) or none.
| Average | Best when | Weakness |
|---|---|---|
| Mean | Data is fairly balanced with no extreme values | One huge value drags it away from the bulk |
| Median | Data has outliers or is skewed (e.g. house prices, incomes) | Ignores the exact size of every other value |
| Mode | Data is categorical, or one value clearly dominates | May not exist, or may not be unique |
Notes · 2 of 5
Mean of grouped data
Midpoints stand in for lost values: Σfx ÷ Σf.
Once data is grouped into classes the exact values are gone, so the mean can only be estimated: assume every value in a class sits at its midpoint x, multiply by the class frequency f, and compute mean ≈ Σfx ÷ Σf. The word estimate (or "estimated mean") belongs in your answer.
| Time t (min) | Frequency f | Midpoint x | f × x |
|---|---|---|---|
| 10 ≤ t < 20 | 5 | 15 | 75 |
| 20 ≤ t < 30 | 8 | 25 | 200 |
| 30 ≤ t < 40 | 4 | 35 | 140 |
| 40 ≤ t < 50 | 3 | 45 | 135 |
Frequency tables of discrete values work the same way without the word "estimate": values x with frequencies f give an exact mean of Σfx ÷ Σf. The midpoint trick is only needed when grouping has destroyed the values.
Notes · 3 of 5
Quartiles, percentiles and the interquartile range
Slicing ordered data into hundred, quarter and half pieces.
Order the data first, always. The lower quartile Q1 sits a quarter of the way in, the median Q2 halfway, and the upper quartile Q3 three-quarters of the way — at the (n + 1)/4, (n + 1)/2 and 3(n + 1)/4 positions of the ordered list. Percentiles generalise this: the 20th percentile is the value below which 20% of the data lies. The interquartile range is IQR = Q3 − Q1, the spread of the middle 50%.
Percentile reading: for 40 ordered values, the 20th percentile sits at position (20/100) × 41 = 8.2, i.e. the 9th value. Exam questions either give you the ordered list or a cumulative frequency curve to read from — Chapter 3 covered the curve version.
Notes · 4 of 5
Range, interquartile range and standard deviation
Three spreads: crude, resistant, and squared.
The range (maximum − minimum) is quick but hangs on two extreme values. The IQR ignores the outer quarters. The standard deviation SD measures the typical distance of values from their mean: SD = √( Σ(x − μ)² ÷ n ) for a full data set. In practice you compute it in the statistics mode of your calculator — but knowing what it measures is what exam questions test.
Two properties examiners love: adding a constant to every value shifts the mean by that constant but leaves the SD unchanged (the whole cloud slides, nothing spreads); multiplying every value by k multiplies both the mean and the SD by k. Outliers inflate the SD heavily, because deviations are squared.
Notes · 5 of 5
Comparing two data sets
One sentence on centre, one sentence on spread.
A fair comparison quotes both a measure of centre and a measure of spread, in context: "Machine A fills bottles with a higher mean volume (502 ml vs 498 ml) and a smaller standard deviation (1.8 ml vs 5.6 ml), so A fills more consistently." Smaller SD = more consistent, more reliable, more predictable — never "better" on its own.
When means differ, judge each claim separately: "B scores higher on average, but A is more consistent" is a complete answer. Adding an outlier raises the SD a lot and the mean a little, while the median barely moves — a favourite "justify which average to report" question.
Revise
Key terms
The words examiners expect you to use precisely.
- Mean
- Σx ÷ n; the balance point of the data, sensitive to extreme values.
- Median
- The middle value of the ordered data; resistant to outliers.
- Mode
- The most frequent value; the only average for categorical data.
- Midpoint
- The representative value (lower + upper bound) ÷ 2 used for each class in grouped calculations.
- Estimated mean
- Σfx ÷ Σf using class midpoints, since grouped data no longer holds exact values.
- Quartile
- Values cutting ordered data into quarters: Q1 at n/4, Q2 the median, Q3 at 3n/4.
Show 5 more key termsShow fewer key terms
- Percentile
- The value below which a given percentage of the ordered data lies.
- Interquartile range
- Q3 − Q1; the spread of the middle 50% of the data.
- Range
- Maximum − minimum; the crudest measure of spread.
- Standard deviation
- The typical distance of values from the mean: √(Σ(x − μ)² ÷ n).
- Consistency
- Small standard deviation; values cluster tightly about the mean.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Averaging the data without ordering it first when finding the median.
The median is a position in the sorted list, not in the given order.
Instead: Sort the list, then pick the middle (or average the two middle values).
Mistake: Forgetting that mean × n gives the total in reverse-mean problems.
The mean is the total shared out, so the total must be rebuilt first.
Instead: Total = mean × n, then subtract the known values.
Show 10 more mistakesShow fewer mistakes
Mistake: Using class boundaries or widths instead of midpoints for grouped means.
The estimate assumes all values sit at the centre of the class.
Instead: midpoint = (lower + upper) ÷ 2, then Σfx ÷ Σf.
Mistake: Stating an exact mean for grouped data without the word "estimate".
Grouping destroyed the exact values, so the result is an approximation.
Instead: Write "estimated mean ≈ …" for grouped data.
Mistake: Computing the IQR as Q3 + Q1, or as max − min.
The IQR is the gap between the quartiles, not their sum or the full spread.
Instead: IQR = Q3 − Q1.
Mistake: Dividing the sum of squared deviations by the wrong n in the SD formula.
The averaging step uses the number of values (or the total frequency for grouped data).
Instead: Variance = Σ(x − μ)² ÷ n, then square-root it.
Mistake: Claiming adding 5 to every value increases the standard deviation.
A uniform shift slides the data without stretching it.
Instead: Mean shifts by +5; the SD stays exactly the same.
Mistake: Reporting only the means when asked to compare two data sets.
Centre without spread tells half the story.
Instead: Compare mean with mean, SD with SD, and conclude in context.
Mistake: Saying "smaller SD means better" with no context.
Consistency is not the same as size or quality; a small SD can describe consistently wrong answers.
Instead: Say "more consistent / more predictable", tied to the context.
Mistake: Using the mode as the average for a data set where no value repeats.
With no repeated value there is no mode to report.
Instead: Switch to the mean or median; state "no mode" if the mode is demanded.
Mistake: Reading quartile positions as n/4 of the largest value instead of the data count.
Quartile positions are positions in the ordered list, not value calculations.
Instead: Position ≈ (n + 1)/4 for Q1 and 3(n + 1)/4 for Q3 of the ordered values.
Mistake: Letting one extreme value decide the choice of average.
The mean chases outliers; the median does not.
Instead: With outliers present, report the median and say why.
Check yourself
Quick quiz
32 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
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Which average is affected most by a single extreme value?
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 7Foundation
- 7Standard
- 6Challenging
20 questions worth 74 marks, each with a model answer, mark scheme and examiner tips.
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Full papers for this subject. Each school sets its own scope — check your assessment outline.
Exam practice
Practice paper
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Question paper
Chapter 4 Practice Paper
30 marks · 40 min · 4 pages
6 multiple-choice questions and structured questions with answer lines. Chapter 4 (S1 averages and spread: mean, mode, median, grouped estimates, quartiles, percentiles, standard deviation, comparing data sets).
Mark scheme
PROChapter 4 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
Questions
Frequently asked questions
What does Sec 4 E-Math Chapter 4 cover?
The averages-and-spread half of syllabus sub-topic S1 of O-Level 4052 / SEC G3 K310: mean, mode and median, choosing between them, estimated means for grouped data, quartiles and percentiles, range, interquartile range and standard deviation, and comparing two data sets using mean and SD. Diagrams themselves live in Chapter 3.
Do I need to memorise the standard deviation formula?
No — the formula sheet does not even list it, because the expectation is calculator statistics mode. You should still know what SD measures (typical distance from the mean), its units, and how shifts and scaling move it, because those are the parts exams actually ask about.
When should I use the median instead of the mean?
When the data has outliers or is skewed — house prices, incomes, delivery times with one failure. Say it in context: "the mean is dragged up by the extreme value, so the median better represents a typical value."
How do I answer "compare the two data sets" questions?
Two sentences, both with numbers: one comparing the means (centre), one comparing the standard deviations (consistency), then a contextual conclusion. "Set B averaged higher (12.4 vs 10.1) but A was more consistent (SD 1.2 vs 3.5)" is the model shape.
When is this chapter usually tested in school?
Averages and spread appear in almost every Prelim statistics question, often merged with the Chapter 3 diagrams: estimate a mean from a grouped table, read a standard deviation off the calculator, then compare two sets. The estimated mean and the comparison sentences carry the marks.
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Chapter 5: Probability