Sec 2 Math Data Analysis: Mean, Median, Mode Notes

Overview

What you’ll learn

Calculate the mean, median and mode of a data set.
Find a missing value or a new mean using the link total = mean × number of values.
Read averages directly from dot diagrams and stem-and-leaf diagrams.
Identify the modal class of grouped data shown in a histogram or table.
Choose a sensible average for a given context and justify the choice.
Compare two distributions by discussing their averages and spread.

In 30 seconds

1Mean = sum of values ÷ number of values — the share-if-equal average.
2Median = the middle value after ordering; with an even count, average the two middle values.
3Mode = the most frequent value; a set can have none, one, or several.
4The master link: total = mean × number of values — it solves reverse-mean and missing-value problems.
5Dot diagrams and stem-and-leaf diagrams keep every original value visible, so all three averages can be read exactly.
6Grouped data hides exact values: quote the modal class, and estimate means with class midpoints.

Notes · 1 of 6

Three averages, three ideas

Mean shares equally, median sits in the middle, mode is the crowd favourite.

The definitions
Average	How to find it	One-line feel
Mean	add all values, divide by how many	the fair-share value
Median	order the values, take the middle	the halfway value
Mode	pick the value occurring most often	the popular value

For the data 4, 7, 9, 12: the mean is (4 + 7 + 9 + 12) ÷ 4 = 8, the median is (7 + 9) ÷ 2 = 8 too, and there is no mode — no value repeats. Three different questions, and the data decides which answer matters.

Notes · 2 of 6

The mean and the total

One link solves every reverse question: total = mean × count.

The formula mean = total ÷ count rearranges into the exam favourite total = mean × count. If four numbers average 12, they total 48 — no matter what the numbers are. This link finds missing values, new means after joining members, and answers to “is the receipt consistent?”.

The link at work
Question	Setup	Answer
Mean of 5 values is 14.5. Total?	5 × 14.5	72.5
A 6th value joins, new mean 15. New total?	6 × 15	90
The joiner’s value?	90 − 72.5	17.5

Notes · 3 of 6

Median and mode: the fine print

Order first for the median; count carefully for the mode.

11. Order the data
The median does not exist until the values are lined up.
22. Count the values
Odd count → one middle value. Even count → average the middle two.
33. Position check
For n values the median sits at position (n + 1) ÷ 2.
44. Mode honestly
If nothing repeats, say “no mode”; if two values tie, both are modes.

Median positions
n	Position of median	Example
5 (odd)	3rd	ordered 1, 2, 6, 8, 11 → 6
6 (even)	between 3rd and 4th	3, 4, 6, 6, 8, 9 → (6 + 6) ÷ 2 = 6
8 (even)	between 4th and 5th	needs the average of two values

Notes · 4 of 6

Dot diagrams

Every value shows as a dot, so averages are visible, not just computable.

Number of pets per household (11 households)
Value	0	1	2	3	4
Dots	2	4	3	1	1

A dot diagram stacks one dot per observation above its value. Tallest column → mode. Counting dots to the middle position → median. Total of (value × dots) ÷ 11 → mean: (0×2 + 1×4 + 2×3 + 3×1 + 4×1) ÷ 11 = 13/11 ≈ 1.18 pets.

Notes · 5 of 6

Stem-and-leaf diagrams

Split each value into a stem and a leaf — the data orders itself.

Marks of 9 pupils (stem = tens, leaf = ones)
Stem	Leaves
1	2 5 8
2	0 3 3 7
3	1 5

Read row by row: 12, 15, 18, 20, 23, 23, 27, 31, 35. The leaves arrive already ordered, which is the diagram’s superpower. With 9 values the median is the 5th: 23. The repeated leaf 3 in the twenties row makes 23 the mode.

Notes · 6 of 6

Grouped data and comparing distributions

Histograms give the modal class; comparisons discuss average and spread in words.

Marks of 40 pupils
Marks	0–20	20–40	40–60	60–80	80–100
Pupils	4	10	16	8	2

The tallest bar marks the modal class — here 40–60. Exact means and medians are impossible from a grouped table (individual marks are hidden), but an estimate uses class midpoints: (4×10 + 10×30 + 16×50 + 8×70 + 2×90) ÷ 40 = 1880 ÷ 40 = 47.

To compare two distributions, say two things: which has the larger average, and which is more consistent (smaller spread). Equal means with different spreads tell different stories — a tight class at 70 is not the same as a class scattered around 70.

Revise

Key terms

The words examiners expect you to use precisely.

Mean: The sum of the values divided by the number of values.
Median: The middle value of the ordered data; the average of the two middle values when the count is even.
Mode: The value that occurs most often; there may be one, several, or none.
Modal class: The class interval with the highest frequency in grouped data.
Dot diagram: A graph stacking one dot per observation above its value.
Stem-and-leaf diagram: A display splitting values into a leading stem and trailing leaf, ordering the data automatically.

Show 6 more key terms

Histogram: A graph of touching bars showing frequencies of grouped data.
Class midpoint: The centre of a class interval, used to estimate means of grouped data.
Total = mean × count: The rearranged mean formula that unlocks reverse and missing-value problems.
Spread: How widely the values scatter; compared in words at this level.
Consistent: Describes a distribution whose values cluster closely around the average.
Outlier: An unusually large or small value that can drag the mean away from the typical values.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Finding the median without ordering the data first.
The median is a position in the sorted list, not in the messy original.
Instead: Order first, then take the middle: 3, 8, 5 orders to 3, 5, 8 with median 5.
Mistake: Dividing the mean by the count instead of multiplying (total = mean ÷ n).
The link runs total = mean × count.
Instead: Four values averaging 12 total 4 × 12 = 48, not 3.

Show 8 more mistakes

Mistake: Reporting a mode for data with no repeats.
Every value appearing once means no value is “most frequent”.
Instead: Say “no mode”; if two values tie for the top count, both are modes.
Mistake: Claiming an exact mean from a grouped table.
The individual values are hidden by the class intervals.
Instead: Quote the modal class, or estimate the mean using class midpoints — and label it an estimate.
Mistake: Saying a high scorer “raises the median”.
The median ignores how big the values are — only order matters.
Instead: One extreme value shifts the mean easily; the median barely moves.
Mistake: Comparing two distributions using only the means.
Equal means can hide very different spreads.
Instead: Comment on the average and on the consistency (spread) before concluding.
Mistake: Adding a value to a data set and dividing the new sum by the old count.
The count grows too.
Instead: A ninth joiner means divide by 9, not 8 — recount every time.
Mistake: Picking just one of the two middle values as the median of an even-sized data set.
With an even count, no single middle value exists — the median is the average of the two central ones.
Instead: For 4, 6, 10, 20 the median is (6 + 10) ÷ 2 = 8, not 6 or 10.
Mistake: Assuming the mode must be the biggest value in the list.
The mode is decided by frequency, not size — a small value that repeats most wins.
Instead: Find the value appearing most often, wherever it sits in the ordering.
Mistake: Slotting a new value into an unsorted list and reading off “the median”.
A new value often belongs in the middle of the order, so the old positions shift.
Instead: Re-sort the entire list first, then locate the middle position again.

Check yourself

Quick quiz

18 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Calculate the mean

What is the mean of 4, 7, 9 and 12?

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

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4Challenging

15 questions worth 61 marks, each with a model answer, mark scheme and examiner tips.

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Exam practice

Practice paper

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Question paper

Chapter 10 Practice Paper

30 marks · 40 min · 5 pages

10 multiple-choice questions and structured questions with answer lines. Chapter 10.

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Mark scheme

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Chapter 10 Practice Paper — mark scheme

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Model answers with a mark for each creditworthy point — for marking at home or in class.

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Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Term 4Coming soonEnd-of-Year Examination (EOY)

Questions

Frequently asked questions

What is Sec 2 Math Chapter 10 about?

Chapter 10 covers data analysis with averages: the mean, median and mode; reading them from dot diagrams and stem-and-leaf diagrams; grouped data and the modal class; choosing a suitable average; and comparing two distributions by average and spread.

When should I use the median instead of the mean?

When the data contains extreme values. One huge salary drags the mean upward while the median stays with the typical worker — that is why medians describe house prices and incomes.

Can I find the exact mean from a grouped table?

No — the table hides the individual values. You can name the modal class, and estimate the mean using class midpoints, but the estimate should always be labelled as one.

How do I compare two distributions?

Say two things: which distribution has the larger average, and which is more consistent (values packed closer together). Two classes with the same mean can still tell very different stories if one is tightly grouped and the other is scattered.

Sec 2 Mathematics Data Analysis: Averages notes