CSEC MATHEMATICS · MODULE 2 · LESSON B2

Averages and Spread: Mean, Median, Mode, Quartiles and Standard Deviation

Calculating the mean, median and mode from raw data and frequency tables, choosing the best average, range, quartiles, interquartile range, and comparing spread.

Intermediate Secondary Level MathematicsM2 1.4M2 1.5M2 1.6M2 1.8M2 1.9≈ 40 min🎮 3 games
Before you start

By the end of this lesson you can…

1WatchExplainer video 2ReadKey notes & exam tips 3PractisePaper 1 style 4ApplyPaper 2 style
Step 1 · Watch

Explainer video

5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.

LESSON B2 · EXPLAINER

Averages and Spread: Mean, Median, Mode, Quartiles and Standard Deviation

Which is the “typical” salary — the mean or the median? The answer depends on the data, and CSEC wants you to say why.

Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.

Explore in 3D

Averages

Change one value and see which average moves.

Step 2 · Read

Key notes

Copy these into your notebook in your own words — that is how they stick.

Mean

Σx ÷ n; from a table Σfx ÷ Σf.

Median

Middle value of ordered data: position (n + 1)/2. Even n → mean of the two middle values.

Mode

Most frequent value (can be several, or none).

Choosing

Extreme values → median; nominal / most popular → mode; symmetric numeric → mean. Always justify.

Range

Highest − lowest.

Quartiles

Q₁ at (n + 1)/4, Q₂ (median) at (n + 1)/2, Q₃ at 3(n + 1)/4. IQR = Q₃ − Q₁; SIQR = IQR/2.

Standard deviation

Larger value = more spread around the mean; compare, not calculate.

Proportion / percentage

Count the values above/below and divide by n (× 100 for %).

EXAM TIPS & COMMON MISTAKES

  • Order the data before finding the median or quartiles — the most common error.
  • The mean of a frequency table is Σfx ÷ Σf, NOT Σx ÷ number of rows.
  • When asked “which average is most appropriate and why”, a correct average with no reason gets half marks.
  • Standard deviation questions: use “more spread / less consistent” in your answer, with a reference to the data.
Play & practise

Games

3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Mathematics Arcade.

Step 3 · Practise

Paper 1 practice — multiple choice

8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.

Adapted from CSEC P1 May/June 2024 (mean item)

1. If the mean of the FOUR numbers 4, 8, x and 12 is 10, then x is

Answer: D. Sum = 40: 4 + 8 + x + 12 = 40 → x = 16.
Adapted from CSEC P1 May/June 2024 (median item)

2. Ms Clarke arranged the 15 test scores of her students in order of size and selected the 8th score. Which statistical measure did she obtain?

Answer: C. The middle score of 15 ordered values is the 8th: the median.
CSEC Paper 01 style · SO 1.4

3. The median of 3, 7, 9, 12, 15, 20 is

Answer: B. Middle two values 9 and 12: (9 + 12)/2 = 10.5.
CSEC Paper 01 style · SO 1.4

4. The mean of the data in the table (x: 1, 2, 3, 4, 5; f: 4, 7, 9, 6, 4) is closest to

Answer: B. Σfx = 89, Σf = 30; 89/30 = 2.97.
CSEC Paper 01 style · SO 1.5

5. A set of salaries includes one very large value. The most appropriate average to describe the typical salary is the

Answer: D. The median is not affected by extreme values.
CSEC Paper 01 style · SO 1.6

6. For the data 2, 4, 5, 7, 8, 9, 12, 14, 15, the interquartile range is

Answer: B. Q₁ = 4.5, Q₃ = 13, IQR = 8.5.
CSEC Paper 01 style · SO 1.8

7. Two classes have the same mean mark. Class P has a standard deviation of 4 and Class Q has a standard deviation of 11. Which statement is TRUE?

Answer: A. A larger standard deviation means the marks are more spread out.
CSEC Paper 01 style · SO 1.9

8. In a class of 40 students, 12 scored above 8. The percentage of students who scored above 8 is

Answer: C. 12/40 × 100 = 30%.
Step 4 · Apply

Paper 2 practice — structured questions

Write full answers, then reveal the mark scheme and tick the points you earned (19 marks in total). Show all your working — method marks count. Your answers are saved on this device.

CSEC Paper 02 style · Module 2 (Averages)

Question 1 [9 marks]

The table shows the marks of 30 students in a quiz marked out of 5.

(a) Complete a table of fx and cumulative frequency for x = 1, 2, 3, 4, 5 with f = 4, 7, 9, 6, 4. [2]

MARK SCHEME — (a) (max 2)

(b) Calculate the mean mark, correct to 2 decimal places. [3]

MARK SCHEME — (b) (max 3)

(c) State the modal mark and the median mark. [2]

MARK SCHEME — (c) (max 2)

(d) What percentage of students scored more than 3? [2]

MARK SCHEME — (d) (max 2)

CSEC Paper 02 style · Module 2 (Spread and the best average)

Question 2 [10 marks]

The monthly salaries (in $000) of six workers are 3, 3, 4, 4, 5 and 40.

(a) Calculate the mean and the median salary. [3]

MARK SCHEME — (a) (max 3)

(b) Which average is more appropriate to describe the typical salary? Give a reason. [2]

MARK SCHEME — (b) (max 2)

(c) Find the range, and the quartiles of the data 2, 4, 5, 7, 8, 9, 12, 14, 15. Hence find the interquartile range and the semi-interquartile range. [4]

MARK SCHEME — (c) (max 4)

(d) Two classes have the same mean. State what a larger standard deviation tells you about a class’s marks. [1]

MARK SCHEME — (d) (max 1)

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Sources

Where this topic appears in CSEC papers

Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.