Lesson 15: Data Displays: Mean, Median, Mode, and Graphs

A basketball coach glances at a list of points scored — 12, 14, 15, 16, 43 — and says "our typical score is about 20". A fan says "no, it's 15". Both are using real mathematics; they are just using different averages. Data only becomes useful when we can summarise it honestly, and that is what this lesson is about: mean, median, mode and range, plus reading the graphs that display data — and spotting when a summary is quietly misleading you.

The four summaries

Take the quiz scores 6, 8, 8, 9, 10.

• Mean: add everything and divide by how many values there are: (6 + 8 + 8 + 9 + 10) ÷ 5 = 41 ÷ 5 = 8.2.
• Median: the middle value when the data is in order: 8.
• Mode: the most frequent value: 8 (it appears twice).
• Range: the spread from smallest to largest: 10 − 6 = 4.

For an even count of values, the median is the mean of the two middle values. The data set 4, 7, 9, 9, 12, 15 has median (9 + 9) ÷ 2 = 9, mean 56 ÷ 6 ≈ 9.3, mode 9 and range 11.

When the mean misleads: outliers

An outlier is a value far away from the rest. In the coach's data — 12, 14, 15, 16, 43 — the 43 is an outlier (one amazing game). The mean is 100 ÷ 5 = 20, dragged upwards by that single game. The median, 15, ignores the outlier's size and better describes a typical game. Rule of thumb: with outliers or skewed data, the median is the fairer summary; without them, the mean uses every value and is usually best. News stories about "average" salaries use this trick constantly — one billionaire in a room of teachers makes the "mean salary" soar while the median barely moves.

Reading graphs

Bar graphs compare categories (favourite sports, pets owned); the tallest bar is the mode of the survey. Line graphs show change over time (temperature through a day). Pictographs use symbols, where each symbol may stand for several items — check the key before counting. Whatever the graph, read the title, the axis labels and the scale first: a bar graph whose vertical axis starts at 50 instead of 0 makes small differences look enormous, a classic persuasion trick.

In the real world: weather and sport

Weather forecasts quote the mean temperature for a week to smooth out daily wobbles, and gardeners compare it with last year's mean to spot climate shifts. In sport, a batsman's or player's scoring consistency is captured by the range: two players can share a mean of 20 points while one ranges from 18–22 (reliable) and the other from 2–43 (a gamble). Smart analysts report a centre (mean or median) and a spread (range) together — one number alone never tells the whole story.

Common mistakes

• Forgetting to order the data for the median. The median of 9, 3, 7 is not 3 — order first: 3, 7, 9, so the median is 7.
• Leaving two middle values un-averaged. With 6 values, the median is the mean of the 3rd and 4th values, not just "one of them".
• Reporting the mode as "the biggest". The mode is the most frequent value, not the largest. In 2, 3, 3, 5, 5, 5, 8 the mode is 5, not 8.
• Dividing by the wrong count for the mean. The mean of 4, 6, 10, 12 is 32 ÷ 4 = 8 — divide by how many values there are, not by the largest value or by 5 out of habit.

Exam Experience

1. What is the median of the data set 3, 5, 7, 10, 12? A) 5 B) 7 C) 7.4 D) 10

2. What is the mean of 4, 6, 10 and 12? A) 6 B) 8 C) 9 D) 32

3. What is the mode of 2, 3, 3, 5, 5, 5, 8? A) 2 B) 3 C) 5 D) 8

4. (Constructed response) The table shows the noon temperatures (°F) for five days: Monday 62, Tuesday 65, Wednesday 71, Thursday 68, Friday 74. (a) Calculate the mean temperature. (b) Find the median temperature. (c) Find the range. (d) A website headline says "A scorching week — temperatures averaged in the mid-70s!" Using your answers, explain whether the headline is a fair summary of the week.

Answer key and explanations

• 1. B — The data is already in order; the middle (3rd) value is 7.
• 2. B — (4 + 6 + 10 + 12) ÷ 4 = 32 ÷ 4 = 8.
• 3. C — 5 appears three times, more than any other value.
• 4. (a) Total = 62 + 65 + 71 + 68 + 74 = 340, mean = 340 ÷ 5 = 68°F. (b) Ordered: 62, 65, 68, 71, 74, so median = 68°F. (c) Range = 74 − 62 = 12°F. (d) Not fair: both the mean and median are 68°F, and only one day (Friday, 74°F) even reached the 70s. "Mid-70s" describes the single warmest day, not the week's typical temperature.

Scoring guide for question 4 (4 points)

• 1 point for the mean, 68°F, from a correct total of 340 divided by 5.
• 1 point for the median, 68°F, found after ordering the data.
• 1 point for the range, 12°F.
• 1 point for the judgement (not fair) backed by evidence — e.g., mean and median are both 68, or only one day exceeded 70°F.

Going further: working backwards and comparing groups

Worked example 5 (the missing value). Five numbers have a mean of 12. Four of them are 8, 10, 14 and 16. The total of all five must be 5 × 12 = 60. The four known values sum to 48, so the missing value is 60 − 48 = 12. "Find the missing score for a target average" is an exam favourite, and the method is always: total needed, minus total known.

Worked example 6. Data: 3, 5, 6, 8, 40 (a cricketer's scores, one huge innings). Median = 6; mean = 62 ÷ 5 = 12.4. The mean claims a "typical" score of 12.4 that the player reached only once; the median describes the usual day far better. Choosing the right summary is an argument, and you now have the vocabulary for it.

Worked example 7. A class survey of books read in a month: Ana 5, Ben 8, Cy 8, Dee 3. The mode is 8 books (two students), and the mean is 24 ÷ 4 = 6 books. If the school librarian reports "students read an average of 6 books", she is using the mean; if she says "the most common total was 8", that is the mode. Both are honest — they answer different questions.

Worked example 8 (same centre, different spread). Class A and Class B both have a mean test score of 70. Class A's range is 12 marks; Class B's is 30. Same average, very different classes: A is consistent, B is a mixture of stars and strugglers. A teacher planning revision helps B very differently from A. Centre plus spread — never one without the other.

Quick practice

• 1. Mean of 7, 9 and 14.
• 2. Median of 11, 4, 9, 4, 15.
• 3. Mode of 6, 2, 6, 9, 2, 6.
• 4. Range of 22, 9, 15, 31.

Answers. 1. 30 ÷ 3 = 10. 2. Ordered: 4, 4, 9, 11, 15 → median 9 (order first!). 3. 6 (appears three times). 4. 31 − 9 = 22.

Investigation: heights, and the sixth friend

Five friends measure their heights: 142, 150, 155, 148 and 160 cm. The total is 755 cm, so the mean is 755 ÷ 5 = 151 cm. Ordered (142, 148, 150, 155, 160), the median is 150 cm, and the range is 160 − 142 = 18 cm. Mean and median sit close together — this data has no outlier pulling them apart. Now the target problem: a sixth friend joins, and the group wants the mean height to become exactly 152 cm. Six people at a mean of 152 need a total of 6 × 152 = 912 cm. The newcomer must measure 912 − 755 = 157 cm. The method never changes: total needed minus total known. It solves "what must I score on the last test?", "what must the last player score for the team average?", and a hundred other target-average questions. One caution to carry forward: had the sixth friend been a basketball player of 195 cm, the mean would jump to (755 + 195) ÷ 6 ≈ 158.3 while the median barely moved (to 152.5) — your old friend the outlier, still at work.

Challenge corner

Here is a trap that fools newspapers. Class A has 10 students with a mean score of 70. Class B has 20 students with a mean score of 80. What is the mean of all 30 students? The tempting answer, 75 (halfway between 70 and 80), is wrong, because the classes are different sizes. Compute with totals: Class A scored 10 × 70 = 700 marks in total; Class B scored 20 × 80 = 1,600. Combined mean: 2,300 ÷ 30 ≈ 76.7. The larger class pulls the combined mean toward its own — means must always be combined through their totals, never averaged directly. This single idea protects you from a lifetime of misleading statistics: whenever someone quotes an "average of averages", ask for the group sizes. If they cannot tell you, the number is decoration, not information — and you now have the tools to say exactly why.