Averages, Range and Comparing Distributions

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An average tells you a typical value — the centre of the data. The range tells you how spread out the data are. To compare two groups properly you need both: one average and the range, each written as a sentence in the context of the question.

Key words

1. Mode, median, mean and range

Finding the median position: for n values in order, the median is the ((n + 1) ÷ 2)th value. With 9 values it is the 5th; with 6 values it is the 3.5th — halfway between the 3rd and 4th values.

Mean from a frequency table. Multiply each value by its frequency (fx), add these, and divide by the total frequency. Example: goals scored by a team in 20 matches.

• Mean = 33 ÷ 20 = 1.65 goals (a mean does not have to be a possible value).
• Mode = 1 goal (highest frequency, 6) — not 6!
• Median: n = 20, so halfway between the 10th and 11th values. Running totals: 0 goals → 4 values, 1 goal → up to the 10th value, 2 goals → up to the 15th. 10th = 1 and 11th = 2, so median = 1.5 goals.
• Range = 4 − 0 = 4 goals.

Working backwards from the mean. Total = mean × number of values. If 5 numbers have a mean of 8, their total is 40. If four of them are 6, 9, 7 and 10 (total 32), the fifth is 40 − 32 = 8.

2. Choosing the best average

Five people are in a lift: four students aged 12, 13, 13, 14 and a teacher aged 45. Mean = 97 ÷ 5 = 19.4 years, median = 13 years, mode = 13 years. The mean is pulled up by the outlier 45, so it describes nobody in the lift. Here the median (or mode) is more typical.

• Categorical data (colours, sports) → mode only.
• Data with an outlier → median.
• Data without outliers, when every value should count (e.g. test marks) → mean.
• Always say why: “The median, because it is not affected by the outlier of 45.”

3. Comparing two distributions: centre and spread

Karim and Adam play basketball. Here are their points in their last seven games. Karim: 12, 15, 14, 13, 16, 14, 14. Adam: 6, 22, 10, 25, 8, 14, 13.

Two-sentence comparison (the Checkpoint way): 1. Average: “Karim and Adam scored the same mean number of points (14), so on average they score equally well.” 2. Spread: “Karim’s range (4) is much smaller than Adam’s (19), so Karim is more consistent; Adam’s scores vary a lot.”

Centre and spread together. The average alone hides the difference between the players. A coach who needs a reliable 12+ points might choose Karim; a coach who needs a chance of a very high score might choose Adam. In general: a higher average means “better/bigger on average”; a smaller range means “more consistent” — but say which is “better” only in context (for times in a race, a lower average is better!).

4. Interpreting data and discussing conclusions

1. Look for patterns and trends within one data set (most values between 30 and 50; sales rise every summer) and relationships between data sets (taller students tend to have longer arm spans).
2. Answer the statistical question in words, using numbers as evidence.
3. Check the prediction: does the evidence support it? Say “the data support the prediction” rather than “prove it”.
4. Discuss reliability and variation. Data vary naturally (people are different; measurements have errors). A different sample would give slightly different results — sampling variation. Small or biased samples make conclusions weaker. Suggest an improvement: a larger or more representative sample, more trials, repeating the survey at a different time.

Example. Prediction: “Year 9 students are taller than Year 7 students.” A sample of 10 students from each year gave: Year 7 mean 148 cm, range 20 cm; Year 9 mean 160 cm, range 24 cm. Conclusion: Year 9 students in the sample were taller on average (mean 160 cm compared with 148 cm), so the data support the prediction. The ranges are similar, so the heights are about equally spread in both years. However, only 10 students from each year were measured; another sample could give different means, so a larger random sample would make the conclusion more reliable.

Worked examples

Worked example 1 — All four measures [4 marks] Find the mode, median, mean and range of: 7, 3, 9, 4, 8, 13, 5, 10, 4.

1. Order: 3, 4, 4, 5, 7, 8, 9, 10, 13 (9 values).
2. Mode = 4 (twice).
3. Median = 5th value = 7.
4. Mean = (3 + 4 + 4 + 5 + 7 + 8 + 9 + 10 + 13) ÷ 9 = 63 ÷ 9 = 7.
5. Range = 13 − 3 = 10.

Answer: mode 4, median 7, mean 7, range 10. Common mistake: finding the middle of the unordered list (8) as the median.

Worked example 2 — Median of an even number of values [1 mark] Find the median of 12, 15, 9, 20, 17, 11.

Order: 9, 11, 12, 15, 17, 20. The middle two are 12 and 15. Median = (12 + 15) ÷ 2 = 13.5.

Worked example 3 — Missing value [2 marks] The mean of six numbers is 9. Five of the numbers are 7, 12, 5, 10 and 11. Find the sixth number.

Total of six numbers = 6 × 9 = 54. Total of five = 7 + 12 + 5 + 10 + 11 = 45. Sixth number = 54 − 45 = 9. Common mistake: finding the mean of the five numbers instead of using the total.

Worked example 4 — Comparing two classes [2 marks] Class 8A: median mark 62, range 15. Class 8B: median mark 58, range 34. Make two comparisons between the classes.

Answer: (1) Class 8A did better on average, because its median mark (62) is higher than 8B’s (58). (2) Class 8A’s marks were more consistent, because its range (15) is smaller than 8B’s (34). Common mistake: “8A’s median is 62 and 8B’s median is 58.” Just copying the numbers is not a comparison — say which is higher and what that means.

Worked example 5 — Choosing an average [2 marks] The weekly pocket money of seven friends is: 50, 60, 60, 70, 75, 80, 500 Egyptian pounds. Which average best represents a typical amount? Explain.

Mean = 895 ÷ 7 ≈ 127.9 pounds; median = 70 pounds. Six of the seven friends get 80 pounds or less, so the mean is not typical. Answer: the median (70 pounds), because the mean is pulled up by the outlier 500.

Worked example 6 — Checking a prediction [2 marks] Rana predicts that students who sleep more than 8 hours get higher test scores. In her sample of 12 students, the 5 students who slept more than 8 hours had a mean score of 71; the other 7 had a mean score of 69. Rana says: “This proves my prediction.” Explain why Rana should be careful.

Answer: the difference in the means (71 − 69 = 2) is very small, and the sample is very small (12 students), so a different sample could easily give the opposite result (sampling variation). The data only slightly support the prediction; they do not prove it. A larger sample is needed.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3