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S4 Mathematics | Statistics prep guide (for tutors) | mean, median, mode and standard deviation
Students reach for the calculator the moment they hear statistics, then freeze when asked which average actually represents the data. This lesson separates centre from spread.
Objectives — Students can: (1) compute mean, median and mode from raw data and say when each is most reliable; (2) compute a standard deviation and explain it as a measure of spread; (3) estimate mean and standard deviation from a grouped frequency table and say why it is an estimate.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Definition of mean | total divided by count | 155 / 8 |
| Squares and roots | square-root key | root of 25.984 |
| Frequency tables | sum of f is the count | sum f = 20 |
02 Lesson flow (50 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-10 min | The three averages | 8 values | mean 19.375, median 19.0, mode 15 |
| 10-22 min | Standard deviation | sum of squared deviations 207.875 | s = 5.449 |
| 22-32 min | Adding an outlier | mean becomes 23.9 | the median moves less |
| 32-44 min | Grouped estimate | sum f = 20 | estimated mean 16.0, sd 7.0 |
| 44-48 min | Comparing two sets | same mean, different spread | spread drives risk |
| 48-50 min | Exit check | three questions collected | independent work |
03 Teaching script (read it out)
Scene 1: Three averages, three jobs
Data: 12, 15, 15, 18, 20, 22, 25, 28, so n = 8. Mean = 155 / 8 = 19.375; the median is the average of the 4th and 5th values, (18 + 20) / 2 = 19.0; the mode is 15 because it appears twice.
Now add 60: the mean jumps to 23.9, a rise of 4.5, while the median only moves to 20. With an extreme value the median describes most of the group better.
Say the rule out loud: the mean uses every value and is therefore sensitive, the median only uses position and is therefore robust, and the mode only makes sense when values repeat.
Now add 60: the mean jumps to 23.9, a rise of 4.5, while the median only moves to 20. With an extreme value the median describes most of the group better.
Say the rule out loud: the mean uses every value and is therefore sensitive, the median only uses position and is therefore robust, and the mode only makes sense when values repeat.
Student task: answer which average you would report to a parent when one student scored zero, with a reason.
Scene 2: Standard deviation measures spread
Start with deviations: 12 minus 19.375 is -7.375. They always sum to zero, so we square each one: the sum of squared deviations is 207.875. Population sd = root(207.875 / 8) = 5.097; the sample sd divides by 7 and gives 5.449.
Two points to land: it carries the same units as the data, so sets can be compared directly; and it can never be negative because of the squaring. A small sd means values hug the mean, a large sd means they are spread out.
Compare with a second set that has the same mean but sd 7.097 and ask which group is more consistent.
Two points to land: it carries the same units as the data, so sets can be compared directly; and it can never be negative because of the squaring. A small sd means values hug the mean, a large sd means they are spread out.
Compare with a second set that has the same mean but sd 7.097 and ask which group is more consistent.
Student task: enter the same data in the calculator statistics mode, confirm s = 5.449, and state whether the calculator used n or n-1.
Scene 3: Grouped data: midpoints stand in for true values
Table: 0-10 has 4 students, 10-20 has 10, 20-30 has 6, sum f = 20. Using midpoints 5, 15, 25, sum fx = 320 so the estimated mean is 16.0; the estimated sd is about 7.0.
Be explicit that this is an estimate: once data are grouped we no longer know individual scores, so the midpoint assumes each value sits at the centre of its class. Continuous classes need a 0.5 boundary correction, though HKDSE questions usually accept the midpoint method.
Close by joining the two ideas: the mean reports the typical level, the standard deviation reports how much individuals differ, and a claim needs both.
Be explicit that this is an estimate: once data are grouped we no longer know individual scores, so the midpoint assumes each value sits at the centre of its class. Continuous classes need a 0.5 boundary correction, though HKDSE questions usually accept the midpoint method.
Close by joining the two ideas: the mean reports the typical level, the standard deviation reports how much individuals differ, and a claim needs both.
Student task: change the frequencies, recompute, and explain why the estimate differs from a true mean.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Find three averages | 8 values | sort before taking the median | 19.375 / 19.0 / 15 |
| Find the sd | same data | square deviations then root | 5.449 |
| Compare two sets | same mean, different sd | compare the sd values | 5.097 vs 7.097 |
| Grouped estimate | sum f = 20 | midpoint times frequency | 16.0 |
Sanity check (teach this): The sd 5.097 must be smaller than half the range (8.0) to be plausible, and the mean 19.375 must lie between 12 and 28. The grouped estimate 16.0 must land between 5 and 25; an answer of 40 means boundaries were used instead of midpoints.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Data 4, 6, 6, 8, 10: find the mean and sd. | mean 6.8; s about 2.28 | sort first, then square deviations |
| Add 3 to every value: what happens to the sd? | unchanged | adding a constant only shifts position |
| Double every value: what happens to the sd? | it doubles | scaling multiplies spread |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| Mixing n and n-1 | divides by n-1 when the whole set is given | “If the question gives the whole set use n; use n-1 only for a sample.” |
| Median without sorting | takes the 4th value as given | “The median is a position, and there is no position without sorting.” |
| Using class boundaries | uses 0 or 10 for the class 0-10 | “The midpoint is (0+10)/2 = 5, not a boundary.” |
| Negative sd | forgets to square before rooting | “After squaring nothing is negative; a negative answer means a missing square.” |
07 Homework (paper, section and question numbers)
- Secondary mathematics chapter index (quadratics, logarithms, trigonometry, probability, integration, vectors) — statistics before probability
- Secondary prep guide hub — all subjects
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| Mean 19.375 of 8 values; add 60, new mean? | 23.9 | wrong: redo act one |
| Double every value: new sd? | twice the original | wrong: redo act two |
| Grouped table with sum f = 20: estimated mean? | 16.0 | two wrong: back to act three |
09 Tutor FAQ
Which year is this for?
S4 to S6; it covers the HKDSE compulsory statistics topic and the M1 extension.
How many formulae must be memorised?
Three: the mean, the sum of squared deviations, and the square root; the rest follow from the definition.
Is there practice attached?
The homework list links to the mathematics chapter index, pairing with probability and permutations.
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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.
