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Comparing Distributions

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How to compare two sets of data using an average and the range, how to write each comparison in the context of the question, and how to do this from lists and stem and leaf diagrams.

Comparing two distributions

A distribution is the whole set of values in some data: how big the values are, and how spread out they are. To compare two distributions, such as the times of two groups of runners, you compare two things: an average and the spread.

  • An average, the median or the mean, tells you about a typical value: which group is higher or lower on the whole.
  • The range tells you about the spread: how close together the values are.

In the exam you are often asked to find a median and a range, from a list or a stem and leaf diagram, and then to compare the distribution with another group's. The comparison is usually worth 2 marks: one for comparing the averages and one for comparing the ranges, each written about the context. Sometimes only one is asked for, such as "Who had the more consistent scores? Give a reason for your answer", for 1 mark.

Comparing averages

Compare the same average for both groups: median with median, or mean with mean. Say which group's average is higher, and what that means in the context of the question.

Think about what a higher value means. Higher marks or more points mean a group did better. Longer times mean a group was slower, so for times a lower average means quicker.

For example: "On average, the dogs at Centre 1 were heavier than the dogs at Centre 2, because their median mass is greater (18 kg compared with 14 kg)."

An average describes a group as a whole, not every value in it. If one class has the higher median mark, some students in the other class may still have scored more than some students in the first class.

Worked example 1: Comparing medians

Here are the times, in seconds, that 7 members of Club P took to swim one length.

48,\ 35,\ 52,\ 41,\ 39,\ 60,\ 44

For the members of Club Q, the median time was 37 seconds.

Compare the times of the two clubs.

  1. Write the times for Club P in order: 35,\ 39,\ 41,\ 44,\ 48,\ 52,\ 60

  2. Your turn. There are 7 times, so the median is the 4th. What is the median time for Club P?

    The 4th time is 44, so the median for Club P is 44 seconds

  3. Club Q's median, 37 seconds, is lower. A lower time means a quicker swim.
  4. Answer: On average, the members of Club Q were quicker, because their median time is lower (37 seconds compared with 44 seconds).

Comparing the spread

The range shows how spread out the values are.

\text{range} = \text{largest value} - \text{smallest value}

A smaller range means the values are closer together: the group is more consistent. A larger range means the values are more spread out, or more varied, so the group is less consistent.

The range says nothing about which group is higher. A group can have the lower average and still be the more consistent one.

One very large or very small value can make the range large on its own. Times of 20, 21, 21, 22 and 40 minutes have a range of 20 minutes, even though four of the times are within 2 minutes of each other.

Worked example 2: Comparing ranges

Maya and Leon each threw a ball 6 times. Here are the distances, in metres.

Maya: 23,\ 27,\ 25,\ 24,\ 26,\ 25

Leon: 18,\ 31,\ 22,\ 29,\ 20,\ 30

Who threw more consistently? Give a reason for your answer.

  1. Maya's longest throw is 27 m and her shortest is 23 m, so her range is 27 - 23 = 4 m

  2. Your turn. Work out the range of Leon's distances.

    Leon's longest throw is 31 m and his shortest is 18 m, so his range is 31 - 18 = 13 m

  3. Maya's range is smaller, so her distances are closer together.
  4. Answer: Maya, because the range of her distances is smaller (4 m compared with 13 m).

Writing a full comparison

For "Compare the distributions", write two comparisons: one about the averages and one about the spread. Each must name the group you mean and say what it means in the context, such as taller, quicker or scored more.

  • Average: "On average, the ... were ... than the ..., because their median (or mean) is higher (or lower)."
  • Spread: "The ... were more consistent than the ..., because their range is smaller."

Writing down the two medians and the two ranges is not a comparison, and neither is giving the difference between them. Use comparing words such as higher, lower, more, less or more consistent. You can quote the figures as well, but they must be correct.

A comparison describes the data you were given. If there are only a few values in each group, be careful about saying it is true for everyone in a whole year group or a whole town.

Worked example 3: A full comparison from two lists

Some plants were given plant food X and some were given plant food Y. Here are the heights, in cm, of 5 plants from each group after four weeks.

Plant food X: 12,\ 15,\ 11,\ 14,\ 13

Plant food Y: 9,\ 18,\ 16,\ 10,\ 17

Compare the heights of the two groups of plants.

  1. Plant food X in order: 11,\ 12,\ 13,\ 14,\ 15. The median is the 3rd value, so it is 13 cm
  2. The range for plant food X is 15 - 11 = 4 cm
  3. Plant food Y in order: 9,\ 10,\ 16,\ 17,\ 18

  4. Your turn. What is the median height for plant food Y?

    The 3rd value is 16, so the median for plant food Y is 16 cm

  5. The range for plant food Y is 18 - 9 = 9 cm
  6. Answer: On average, the plants given plant food Y were taller, because their median height is greater (16 cm compared with 13 cm). The heights of the plants given plant food X were more consistent, because their range is smaller (4 cm compared with 9 cm).

Stem and leaf diagrams

In a stem and leaf diagram the stem is usually the tens digit and each leaf is a units digit. The key tells you how to read it: 4 \mid 7 represents 47. The leaves are in order, so the values are already sorted from smallest to largest.

Count the leaves to find how many values there are. The median is the middle one: with 15 values it is the 8th, with 7 values before it and 7 after it. The range is the last value subtract the first. Always write the whole value, such as 47, not just the leaf 7

Worked example 4: Comparing with a stem and leaf diagram

The stem and leaf diagram shows the masses, in grams, of 13 eggs from Hill Farm.

For the eggs from Brook Farm, the median mass was 58 g and the range was 35 g.

Compare the masses of the eggs from the two farms.

  1. The key says 5 \mid 2 represents 52 g. There are 13 leaves, so there are 13 eggs, already in order.
  2. With 13 values the median is the 7th, with 6 values on each side of it.

  3. Your turn. Count along the leaves to the 7th value. What is the median mass for Hill Farm?

    Counting along: 52,\ 54,\ 57,\ 60,\ 61,\ 61,\ 63. The 7th value is 63, so the median is 63 g

  4. The range is the last value subtract the first: 81 - 52 = 29 g
  5. Answer: On average, the eggs from Hill Farm were heavier, because their median is greater (63 g compared with 58 g). The masses of the eggs from Hill Farm were more consistent, because their range is smaller (29 g compared with 35 g).

Worked example 5: Working out the range from the least and greatest values

The stem and leaf diagram shows the masses, in kg, of 17 suitcases on Flight 2.

For the suitcases on Flight 1, the lightest mass was 9 kg, the median was 18 kg and the heaviest was 31 kg.

Compare the masses of the suitcases on the two flights.

  1. The key says 1 \mid 2 represents 12 kg. There are 17 leaves, so there are 17 suitcases on Flight 2, already in order.
  2. With 17 values the median is the 9th. The first row has 6 values, then 20,\ 21,\ 23, so the median for Flight 2 is 23 kg
  3. The range for Flight 2 is the last value subtract the first: 39 - 12 = 27 kg
  4. For Flight 1 you are given the lightest and heaviest masses, not the range, so you need to work it out.

  5. Your turn. Work out the range of the masses for Flight 1.

    The range for Flight 1 is 31 - 9 = 22 kg

  6. Answer: On average, the suitcases on Flight 2 were heavier, because their median is greater (23 kg compared with 18 kg). The masses of the suitcases on Flight 1 were more consistent, because their range is smaller (22 kg compared with 27 kg).

Worked example 6: Comparing when there is an even number of values

Here are the distances, in km, that 10 members of Cycle Club A rode last Saturday.

34,\ 28,\ 41,\ 25,\ 37,\ 30,\ 45,\ 29,\ 36,\ 33

For the members of Cycle Club B, the median distance was 38 km and the range was 14 km.

Compare the distances ridden by the two clubs.

  1. Write the distances for Club A in order: 25,\ 28,\ 29,\ 30,\ 33,\ 34,\ 36,\ 37,\ 41,\ 45
  2. There are 10 distances, an even number, so there is no single middle value. The median is halfway between the 5th and 6th values, 33 and 34

  3. Your turn. Work out the median distance for Club A.

    The median is (33 + 34) \div 2 = 33.5 km

  4. The range for Club A is 45 - 25 = 20 km
  5. Answer: On average, the members of Club B rode further, because their median distance is greater (38 km compared with 33.5 km). The distances for Club B were more consistent, because their range is smaller (14 km compared with 20 km).

Common mistakes

  • Writing down the values without comparing them. Say which group is higher and which is more consistent, using words such as higher, lower or more consistent.
  • Giving only the difference between the medians. "The difference is 7" is not a comparison: say which group is higher, and what that means.
  • Comparing only the averages. A full comparison needs the spread as well: compare the ranges too.
  • Saying a higher median time means quicker. For times, a lower average means quicker.
  • Thinking a larger range means more consistent. A smaller range means the values are closer together, so more consistent.
  • Writing only a leaf from a stem and leaf diagram. Use the key to write the whole value, such as 57, not 7

Practice questions

More on this topic: Comparing Distributions worksheet with full solutions