GCSE Maths: Comparing distributions (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Comparing distributions
A distribution is all the values of a data set and how often they occur. To compare two distributions we use two kinds of measure: an average (what is typical) and a measure of spread (how consistent the data is). Comparing only an average, or only a spread, is not a full comparison.
Average. Use the mean if the data has no extreme values. Use the median if there are outliers, because the median is not affected by them. The group with the higher average is higher “on average”. Spread. The range (biggest − smallest) is the simplest measure of spread. A smaller range means the data is more consistent (closer together). Higher tier students also use the interquartile range (IQR), which ignores extreme values.
How to write a comparison. Always write your comparison in context, using the words of the question, and include the numbers. For example: “On average, the boys scored higher (mean 62 compared with 55), but the girls’ scores were more consistent (range 14 compared with 28).” A sentence such as “A has a bigger range” with no meaning is not enough.
Key ideas
- Compare an average (mean or median) and a spread (range, or IQR).
- Use the median if there are outliers.
- A smaller range (or IQR) means more consistent data.
- Write in context and include the numbers.
- A larger average means “higher on average”, not that every value is bigger.
| Team A | 3 | 5 | 8 | 4 | 6 | 4 |
|---|---|---|---|---|---|---|
| Team B | 2 | 9 | 1 | 8 | 7 | 3 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Team A. An average tells us what is typical.3 + 5 + 8 + 4 + 6 + 4 = 30; 30 ÷ 6 = 5
- Work out the mean for Team B. 2 + 9 + 1 + 8 + 7 + 3 = 30; 30 ÷ 6 = 5
- Work out the range for Team A. A spread tells us how consistent the data is.8 − 3 = 5
- Work out the range for Team B. 9 − 1 = 8
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Team A and Team B are the same (mean 5).
Team A is more consistent (smaller range: 5 compared with 8), so Team B is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
| Machine A | 41 | 38 | 44 | 40 | 42 | 39 |
|---|---|---|---|---|---|---|
| Machine B | 30 | 55 | 28 | 60 | 35 | 52 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Machine A. An average tells us what is typical.41 + 38 + 44 + 40 + 42 + 39 = 244; 244 ÷ 6 = 40.7
- Work out the mean for Machine B. 30 + 55 + 28 + 60 + 35 + 52 = 260; 260 ÷ 6 = 43.3
- Work out the range for Machine A. A spread tells us how consistent the data is.44 − 38 = 6
- Work out the range for Machine B. 60 − 28 = 32
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Machine B has the higher mass (mean 43.3 compared with 40.7).
Machine A is more consistent (smaller range: 6 compared with 32), so Machine B is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
| Runner P | 12 | 14 | 13 | 15 | 14 | 60 |
|---|---|---|---|---|---|---|
| Runner Q | 18 | 20 | 22 | 19 | 21 | 23 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Because of the outlier (60 seconds), use the median rather than the mean. Compare an average (the median) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the median for Runner P. An average tells us what is typical.In order: 12, 13, 14, 14, 15, 60; median = 14
- Work out the median for Runner Q. In order: 18, 19, 20, 21, 22, 23; median = 20.5
- Work out the range for Runner P. A spread tells us how consistent the data is.60 − 12 = 48
- Work out the range for Runner Q. 23 − 18 = 5
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Runner Q has the higher time (median 20.5 compared with 14).
Runner Q is more consistent (smaller range: 5 compared with 48), so Runner P is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
| Club A | 12 | 18 | 20 | 22 | 25 | 27 | 30 |
|---|---|---|---|---|---|---|---|
| Club B | 15 | 16 | 18 | 19 | 21 | 22 | 24 |
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the median) and a measure of spread (the interquartile range) for each group, then write sentences in context.
- Work out the median for Club A. An average tells us what is typical.In order: 12, 18, 20, 22, 25, 27, 30; median = 22
- Work out the median for Club B. In order: 15, 16, 18, 19, 21, 22, 24; median = 19
- Work out the interquartile range for Club A. A spread tells us how consistent the data is.Q1 = 18, Q3 = 27, IQR = 27 − 18 = 9
- Work out the interquartile range for Club B. Q1 = 16, Q3 = 22, IQR = 22 − 16 = 6
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Club A has the higher ages (median 22 compared with 19).
Club B is more consistent (smaller interquartile range: 6 compared with 9), so Club A is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Watch out
A comparison must include both an average and a spread, written in context, with numbers. Do not say “Team A is better” without saying what is better (higher average, more consistent).Exercise 24K Comparing distributions
Non-calculatorKey words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Amy. An average tells us what is typical.7 + 9 + 6 + 8 + 10 = 40; 40 ÷ 5 = 8
- Work out the mean for Ben. 4 + 10 + 5 + 9 + 7 = 35; 35 ÷ 5 = 7
- Work out the range for Amy. A spread tells us how consistent the data is.10 − 6 = 4
- Work out the range for Ben. 10 − 4 = 6
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Amy has the higher scores (mean 8 compared with 7).
Amy is more consistent (smaller range: 4 compared with 6), so Ben is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for The girls. An average tells us what is typical.150 + 154 + 148 + 152 + 156 + 150 = 910; 910 ÷ 6 = 151.7
- Work out the mean for The boys. 145 + 160 + 152 + 148 + 158 + 157 = 920; 920 ÷ 6 = 153.3
- Work out the range for The girls. A spread tells us how consistent the data is.156 − 148 = 8
- Work out the range for The boys. 160 − 145 = 15
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, The boys has the greater height (mean 153.3 compared with 151.7).
The girls is more consistent (smaller range: 8 compared with 15), so The boys is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Clinic A. An average tells us what is typical.12 + 15 + 9 + 14 + 10 = 60; 60 ÷ 5 = 12
- Work out the mean for Clinic B. 20 + 8 + 25 + 6 + 11 = 70; 70 ÷ 5 = 14
- Work out the range for Clinic A. A spread tells us how consistent the data is.15 − 9 = 6
- Work out the range for Clinic B. 25 − 6 = 19
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Clinic B has the longer waiting time (mean 14 compared with 12).
Clinic A is more consistent (smaller range: 6 compared with 19), so Clinic B is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Player X. An average tells us what is typical.1 + 0 + 3 + 2 + 1 + 0 + 2 = 9; 9 ÷ 7 = 1.3
- Work out the mean for Player Y. 0 + 5 + 0 + 0 + 1 + 6 + 0 = 12; 12 ÷ 7 = 1.7
- Work out the range for Player X. A spread tells us how consistent the data is.3 − 0 = 3
- Work out the range for Player Y. 6 − 0 = 6
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Player Y has the higher goals (mean 1.7 compared with 1.3).
Player X is more consistent (smaller range: 3 compared with 6), so Player Y is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Shop 1: 20, 22, 21, 23, 22, 65
Shop 2: 30, 32, 31, 33, 30, 34
Compare the ages using the median and the range. Explain why the median is better than the mean for Shop 1.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the median) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the median for Shop 1. An average tells us what is typical.In order: 20, 21, 22, 22, 23, 65; median = 22
- Work out the median for Shop 2. In order: 30, 30, 31, 32, 33, 34; median = 31.5
- Work out the range for Shop 1. A spread tells us how consistent the data is.65 − 20 = 45
- Work out the range for Shop 2. 34 − 30 = 4
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Shop 2 has the higher age (median 31.5 compared with 22).
Shop 2 is more consistent (smaller range: 4 compared with 45), so Shop 1 is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Team P
| Goals | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Matches | 4 | 7 | 6 | 2 | 1 |
| Goals | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Matches | 1 | 3 | 8 | 6 | 2 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Goals Frequency (f) f × goals 0 4 0 1 7 7 2 6 12 3 2 6 4 1 4 Total 20 29 - Add up the frequency column. This gives the number of values in the data set.4 + 7 + 6 + 2 + 1 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 7 + 12 + 6 + 4 = 29
- Divide the total by the number of values. mean = total ÷ number of values.29 ÷ 20 = 1.45
Check. The mean must be between the smallest and largest values (0 and 4). 1.45 is. Also 1.45 × 20 = 29. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Goals Frequency (f) f × goals 0 1 0 1 3 3 2 8 16 3 6 18 4 2 8 Total 20 45 - Add up the frequency column. This gives the number of values in the data set.1 + 3 + 8 + 6 + 2 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 3 + 16 + 18 + 8 = 45
- Divide the total by the number of values. mean = total ÷ number of values.45 ÷ 20 = 2.25
Check. The mean must be between the smallest and largest values (0 and 4). 2.25 is. Also 2.25 × 20 = 45. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 0
- Find the biggest value that occurs. Biggest = 4
- Subtract. Biggest minus smallest.4 − 0 = 4
Check. 0 + 4 = 4. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 0
- Find the biggest value that occurs. Biggest = 4
- Subtract. Biggest minus smallest.4 − 0 = 4
Check. 0 + 4 = 4. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Use the two means and the two ranges.
- Compare the means. Team Q: 2.25 is bigger than team P: 1.45
- Compare the ranges. Both teams scored between 0 and 4 goals.Range of each team = 4 − 0 = 4
- Write in context. Team Q scores more goals on average, and the spread of the number of goals is the same.
Check. Team Q has more matches with 2 or 3 goals. ✓
Watch out. Mention both an average and a spread.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Think about what an average tells us about individual values.
- What does a mean tell us? It tells us about the group as a whole, not about every individual.Class A did better on average.
- Give an example. Class A could be marks 40, 64, 88 (mean 64) and class B could be 60, 61, 62 (mean 61): the best student is in A, but the weakest is also in A.Some students in B scored higher than some in A.
Check. In the example the mean of A is (40 + 64 + 88) ÷ 3 = 64 and of B is 61. ✓
Watch out. Averages compare groups, not individuals.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. One sentence about the average, one about the spread.
- Compare the averages. The larger mean is the higher average.15 > 12, so B has the higher average.
- Compare the spreads. The smaller range is more consistent.5 < 14, so A is more consistent.
- Write the sentences. B has the higher average (15 compared with 12). A is more consistent (range 5 compared with 14).
Check. Each sentence has numbers. ✓
Watch out. Use both an average and a spread.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Archer 1. An average tells us what is typical.6 + 7 + 7 + 8 + 8 + 8 + 9 = 53; 53 ÷ 7 = 7.6
- Work out the mean for Archer 2. 3 + 6 + 8 + 8 + 9 + 10 + 10 = 54; 54 ÷ 7 = 7.7
- Work out the range for Archer 1. A spread tells us how consistent the data is.9 − 6 = 3
- Work out the range for Archer 2. 10 − 3 = 7
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Archer 2 has the higher score (mean 7.7 compared with 7.6).
Archer 1 is more consistent (smaller range: 3 compared with 7), so Archer 2 is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The median is the average and the IQR is the spread: compare each.
- Compare the medians. 52 is more than 50.Group 1 has the higher median.
- Compare the IQRs. The smaller IQR means the middle half of the data is closer together.IQR 4 < 9, so group 1 is more consistent.
- Write in context. Group 1 has the higher median (52 compared with 50) and is more consistent (IQR 4 compared with 9).
Check. Both sentences contain numbers. ✓
Watch out. The IQR is not affected by outliers.
Method. Check what the range tells us.
- What does the range measure? The range measures how spread out the data is, not what is typical.Spread, not average.
- Can we tell the mean from it? No. Two groups with the same mean can have very different ranges, and a big range does not mean a big mean.Class X: 30 to 65 has mean 47.5, class Y: 50 to 70 has mean 60.
Check. In the example, class X has the larger range but the lower mean. ✓
Watch out. Compare averages with averages and spreads with spreads.
Method. Use total = mean × number of values before and after the swap.
- Total height of team A before. mean × 5.180 × 5 = 900
- Total height of team A after. 178 × 5 = 890
- Take away the player who left. He is 190 cm.900 − 190 = 710
- Find the new player’s height. The new total is 890.890 − 710 = 180 cm
Check. 710 + 180 = 890 and 890 ÷ 5 = 178. ✓
Watch out. Work with totals, then divide.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the median) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the median for Player A. An average tells us what is typical.In order: 4, 5, 6, 7, 8; median = 6
- Work out the median for Player B. In order: 3, 4, 5, 9, 10; median = 5
- Work out the range for Player A. A spread tells us how consistent the data is.8 − 4 = 4
- Work out the range for Player B. 10 − 3 = 7
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Player A has the higher points (median 6 compared with 5).
Player A is more consistent (smaller range: 4 compared with 7), so Player B is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the mean) and a measure of spread (the range) for each group, then write sentences in context.
- Work out the mean for Phone A. An average tells us what is typical.14 + 15 + 13 + 14 + 16 = 72; 72 ÷ 5 = 14.4
- Work out the mean for Phone B. 10 + 18 + 12 + 19 + 16 = 75; 75 ÷ 5 = 15
- Work out the range for Phone A. A spread tells us how consistent the data is.16 − 13 = 3
- Work out the range for Phone B. 19 − 10 = 9
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Phone B has the higher battery life (mean 15 compared with 14.4).
Phone A is more consistent (smaller range: 3 compared with 9), so Phone B is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. One sentence about the average, one about the spread.
- Compare the means. The larger mean is the higher average.30 > 24, so team X is higher on average.
- Compare the ranges. The smaller range is more consistent.6 < 10, so team X is more consistent.
- Write the sentences. Team X scored more on average (mean 30 compared with 24) and was more consistent (range 6 compared with 10).
Check. Each sentence uses numbers and the context. ✓
Watch out. Do not just say “X is better”.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Make P a set of numbers that are close together but high, and Q a set that are spread out but with a lower total.
- Choose set P. Numbers close together, around 12.10, 11, 12, 12, 13
- Choose set Q. Numbers spread out with a low total.2, 4, 6, 8, 20
- Work out the means. Total ÷ 5.P: 58 ÷ 5 = 11.6
Q: 40 ÷ 5 = 8 - Work out the ranges. Largest − smallest.P: 13 − 10 = 3
Q: 20 − 2 = 18 - Compare. P has the higher mean (11.6 > 8) and the smaller range (3 < 18).
Check. Both conditions are satisfied. ✓
Watch out. Many different answers are possible. Always show the mean and the range of each set.