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Mean, Median and Mode: Describe a Data Set

Order data, find the mode, median and mean, see the mean as a fair share, and choose the measure that best describes a data set.

Grades 4–6 · 29 practice questions · Original material
Learning objectiveFind the mode, median and mean of a small data set, including data with zeros and an even number of values, explain the mean as a fair share, and choose the measure that best describes the data.

The idea

A set of numbers can be described with one number. Five days of snowfall in an invented northern Ontario town were 2, 2, 3, 5 and 8 cm. Three different numbers each tell part of the story: the mode is 2 cm (it happened twice), the median is 3 cm (the middle day) and the mean is 4 cm (the fair share).

The mode is the value that appears most often. It is the only one of the three that also works for categories, such as the most popular lunch. A data set can have one mode, two modes or no mode.

The median is the middle value after the data are put in order. With an odd number of values it is the middle one. With an even number of values there are two middle values, and the median is halfway between them. Always order the data first, because the middle of an unordered list means nothing.

The mean is the fair-share value. Imagine each day's snow as a stack of cubes and move cubes from tall stacks to short ones until every stack is equal. The total does not change, so the equal height is total ÷ number of values. For 2, 2, 3, 5 and 8: 20 ÷ 5 = 4. A value of zero still counts in the total and in the number of values.

Which one to use depends on the question. The mean uses every value, so one very large or very small value pulls it. The median is not pulled by an extreme value. The mode answers 'what is most common?'. A mean always lies between the least and greatest values.

Worked examples

Mode, median and mean of the data 2, 2, 3, 5 and 8Left panel: a dot plot of five invented snowfall values on a number line from 0 to 8 centimetres. Two dots are stacked at 2, so the mode is 2. The ordered values are 2, 2, 3, 5, 8 and the middle one, 3, is the median. A triangle under the number line at 4 marks the mean. Right panel: five stacks of cubes with heights 2, 2, 3, 5 and 8 are levelled by moving cubes from tall stacks to short ones until all five stacks have height 4. The total of 20 cubes does not change, and 20 divided by 5 equals 4, so the mean is 4.Three ways to describe 2, 2, 3, 5, 8Original schematic with invented data, drawn to the labelled values (not to scale).1 · Dot plot of the ordered data012345678snowfall (cm)median 3middle of the ordered listmode 2(2 dots)mean 4Order first: 2, 2, 3, 5, 8. Five values, so the medianis the 3rd value. Range: 8 − 2 = 6.2 · The mean is a fair sharebeforeafter levelling2, 2, 3, 5, 84, 4, 4, 4, 4level 4The total stays 20 cubes.20 ÷ 5 = 4 → mean = 4Green cubes were moved in from taller stacks.
Original schematic with invented snowfall data, drawn to the labelled values. Order the data first: the median is the middle value (3), the mode is the value with the most dots (2) and the mean is the level the stacks reach when cubes are moved from tall stacks to short ones until all are equal (20 ÷ 5 = 4). An extreme value such as 8 pulls the mean up but not the median.

Example 1 · Mode, median and mean

Snowfall (cm) on five days in an invented northern Ontario town: 2, 8, 3, 2, 5. Find the mode, median and mean.

  1. Order the data: 2, 2, 3, 5, 8.
  2. Mode: 2 appears twice and the others once, so the mode is 2.
  3. Median: the middle of five ordered values is the 3rd, which is 3.
  4. Mean: 2 + 2 + 3 + 5 + 8 = 20 and 20 ÷ 5 = 4.

Mode 2 cm, median 3 cm, mean 4 cm

Example 2 · Median with an even count

Books borrowed by six students (invented): 12, 4, 6, 4, 8, 2. Find the median and the mean.

  1. Order the data: 2, 4, 4, 6, 8, 12.
  2. Six values have two middle values, the 3rd and 4th: 4 and 6.
  3. The median is halfway between them: (4 + 6) ÷ 2 = 5.
  4. Mean: 2 + 4 + 4 + 6 + 8 + 12 = 36 and 36 ÷ 6 = 6.

Median 5 books, mean 6 books

Example 3 · The mean as a fair share

Four skaters scored 2, 6, 3 and 5 goals in a season (invented). If the goals were shared out equally, how many would each skater have?

  1. Total goals: 2 + 6 + 3 + 5 = 16.
  2. Share among 4 skaters: 16 ÷ 4 = 4.
  3. Check by levelling: the skater with 6 gives 2 away, the skater with 5 gives 1 away, and the skaters with 2 and 3 receive 2 and 1. All four have 4.

Mean = 4 goals each

Example 4 · Choosing a measure

Riders on a school bus on five days (invented): 10, 12, 15, 18, 95. The 95 was a field-trip day. Which value best describes a typical day?

  1. Mean: 150 ÷ 5 = 30. Median: in order 10, 12, 15, 18, 95, the middle is 15.
  2. Four of the five days had 18 or fewer riders, so 30 is not typical. The 95 pulls the mean up.
  3. The median, 15, is not pulled by the extreme day. Report it, and mention the field-trip day.

The median, 15 riders, describes a typical day better.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Finding the median without ordering the data (4, 9, 2, 7, 3 gives median 2).

Why this fails: The median is the middle of the ordered list. In order the values are 2, 3, 4, 7, 9, so the median is 4.

Saying the mean is the middle number, or that the mean and median are always the same.

Why this fails: The mean is the fair share (total ÷ count) and the median is the middle by position. For 2, 3, 3, 4, 13 the mean is 5 but the median is 3.

Giving the largest value as the mode.

Why this fails: The mode is the value that appears most often, not the greatest. In 1, 3, 3, 9 the mode is 3.

Leaving out zeros when finding the mean (0, 4, 0, 6, 2, 0 gives 12 ÷ 3 = 4).

Why this fails: A zero is a real data value. There are 6 days, so the mean is 12 ÷ 6 = 2.

Picking one middle value when the count is even (6 values, middle values 4 and 6, median 4).

Why this fails: With an even count the median is halfway between the two middle values: (4 + 6) ÷ 2 = 5. Averaging the middle of an unordered list is also wrong, so order first.

Foundation

Foundation practice

Find the mode, median and mean of small data sets.

Mode

1. Seven students were asked how many pets they have (invented data): 1, 0, 2, 1, 3, 1, 2. What is the mode?

  1. 3
  2. 2
  3. 1
  4. 0
Median

2. What is the median of 9, 2, 7, 4, 5?

  1. 5
  2. 7
  3. 4
  4. 9
Fair share

3. Three friends have 2, 4 and 6 hockey cards (invented). They put them together and share them equally. How many does each friend get?

  1. 6
  2. 12
  3. 3
  4. 4
Vocabulary

4. A store manager wants the value that appears most often in a list of sizes sold. Which measure should she use?

  1. Mean
  2. Mode
  3. Range
  4. Median
Even count

5. What is the median of 3, 5, 7, 11?

  1. 5
  2. 6
  3. 7
  4. 6.5
Mean

6. Find the mean of 4, 8, 6 and 2.

  1. 20
  2. 4
  3. 8
  4. 5
Mode

7. Find the mode of 2, 5, 5, 7, 9, 9, 10.

  1. 5 and 9
  2. 5
  3. 7
  4. There is no mode

Practice

Practice practice

Order data, handle zeros and even counts, and work with totals.

Mean with zeros

8. Snowfall in cm on six days in an invented northern Ontario town: 0, 4, 0, 6, 2, 0. What is the mean daily snowfall?

  1. 4
  2. 12
  3. 2
  4. 1
Median

9. Minutes of homework one week (invented): 12, 15, 9, 20, 14, 11, 16. What is the median?

  1. 20
  2. 15
  3. 12
  4. 14
Even count

10. Minutes six students take to walk to school (invented): 7, 12, 9, 15, 10, 12. What is the median?

  1. 12
  2. 11
  3. 10
  4. 15
Missing value

11. The mean of four numbers is 7. Three of them are 5, 9 and 6. What is the fourth number?

  1. 8
  2. 7
  3. 20
  4. 10
Mean

12. The mean of 5, 7 and 9 is 7. A fourth student adds another 7 to the data. What is the mean of 5, 7, 9 and 7?

  1. 14
  2. 28
  3. 7
  4. 5.25
Compare

13. Data (invented): 2, 3, 3, 4, 13. Which statement is true?

  1. The mean and the median are both 3.
  2. The mean is 5 and the median is 3, so the mean is greater.
  3. The mean is 3 and the median is 5.
  4. The mean and the median are both 5.
Mode of categories

14. Class votes for the favourite lunch (invented): sandwich 8, soup 5, pasta 9, wrap 4. What is the mode?

  1. pasta
  2. 9
  3. sandwich
  4. wrap
Total from mean

15. The mean of 6 quiz scores is 8. What is the total of the 6 scores?

  1. 14
  2. 42
  3. 8
  4. 48

Think

Think practice

Choose a measure, spot errors and see what changes a mean or median.

Choose a measure

16. Riders on a school bus on five days (invented): 10, 12, 15, 18, 95 (a field-trip day). Which value best describes a typical day?

  1. The mean, 30, because it uses every value
  2. The mode, because every value is different
  3. The median, 15, because one very large day pulls the mean up
  4. The range, 85, because it shows the biggest change
Error analysis

17. Sam says the median of 4, 9, 2, 7, 3 is 2 because it is the middle number in the list. What did Sam do wrong?

  1. Sam did not order the numbers first. In order, 2, 3, 4, 7, 9, the median is 4.
  2. Sam should have added the numbers and divided by 5.
  3. Sam should have picked the number that appears most often.
  4. Nothing. The median is the middle number in the list as written.
Reasonable?

18. The mean of 12, 14, 15, 16 and 18 cannot be which of these?

  1. 14
  2. 15
  3. 16
  4. 19
Fair share

19. Four stacks have 2, 3, 5 and 10 cubes. Cubes are moved from taller stacks to shorter ones until all four stacks are the same height. How tall is each stack?

  1. 4
  2. 5
  3. 10
  4. 20
Effect of a new value

20. The numbers 4, 6 and 8 have median 6. A new value, 20, is added. What is the new median?

  1. 8
  2. 20
  3. 7
  4. 6
Target mean

21. Maya scored 70, 80 and 90 on three tests (invented). What does she need on a fourth test so that her mean for the four tests is 82?

  1. 82
  2. 328
  3. 80
  4. 88
Compare sets

22. Set A is 5, 5, 5, 5. Set B is 1, 3, 7, 9. Which statement is true?

  1. Both sets have mean 5, but the values in B are more spread out.
  2. Set A has the greater mean because all its values are equal.
  3. Set B has the greater mean because it has the largest value.
  4. The sets must be the same because the means are the same.

Challenge

Challenge practice

Frequency tables, missing values and building a data set.

Frequency table

23. Goals scored by a school team in 10 games (invented): 0 goals in 3 games, 1 goal in 4 games, 2 goals in 2 games, 3 goals in 1 game. What is the median number of goals?

  1. 1.5
  2. 1
  3. 1.1
  4. 2
Missing value

24. Five numbers in order are 2, 5, 6, ?, 12 and their mean is 7. What is the missing number?

  1. 7
  2. 6
  3. 10
  4. 12
Two-step

25. Five students read a mean of 10 books each (invented). A sixth student who read 16 books joins the group. What is the new mean?

  1. 11
  2. 13
  3. 16
  4. 10
Choose a measure

26. A shirt shop sold 20 shirts (invented): small 3, medium 8, large 6, extra large 3. Which measure helps the owner decide which size to stock the most of?

  1. The mean size, because it uses every sale
  2. The median size, because it is in the middle of the list
  3. The range, because it shows how many sizes sell
  4. The mode, medium, because it was sold the most
Change the data

27. The data 4, 6, 8, 10, 12 have mean 8 and median 8. Which change keeps the median at 8 but makes the mean greater?

  1. Change 8 to 20
  2. Change 12 to 20
  3. Change 4 to 0
  4. Change 10 to 6
Explain

28. Bus riders on five days (invented): 10, 12, 15, 18, 95. Explain why the median describes a typical day better than the mean. Use numbers from the data.

Build a data set

29. Write five whole numbers with mean 6, median 5 and mode 4. Show how you know each measure is right.

Answer key

  1. 1. 1
    The mode is the value that appears most often. The value 1 appears 3 times, 2 appears twice, and 0 and 3 appear once each, so the mode is 1. The answer 3 is the largest value, but the mode is about how often, not how big. The answer 2 is the second most common. The answer 0 is the smallest value.
  2. 2. 5
    Put the numbers in order first: 2, 4, 5, 7, 9. The middle number is 5. The answer 7 is the middle of the unordered list, which does not mean anything. The answer 4 is the value just before the middle. The answer 9 is the largest value.
  3. 3. 4
    The total is 2 + 4 + 6 = 12 cards. Shared equally among 3 friends, that is 12 ÷ 3 = 4 each, which is the mean. The answer 6 is the largest pile, 12 is the total before sharing, and 3 is the number of friends rather than the number of cards.
  4. 4. Mode
    The mode is the value that appears most often. The mean is the fair-share value (total ÷ count). The median is the middle value of the ordered list. The range is the greatest value minus the least value, so it says how spread out the data are.
  5. 5. 6
    With 4 values there are two middle values, 5 and 7. The median is halfway between them: (5 + 7) ÷ 2 = 6. The answers 5 and 7 each pick only one of the two middle values. The answer 6.5 is the mean (26 ÷ 4), a different measure.
  6. 6. 5
    Add the values: 4 + 8 + 6 + 2 = 20. Divide by how many values there are: 20 ÷ 4 = 5. The answer 20 is the total and forgets to divide. The answer 4 divides by 5 instead of 4. The answer 8 is the largest value.
  7. 7. 5 and 9
    5 and 9 each appear twice and every other value appears once, so there are two modes. The answer 5 alone ignores the tie. The answer 7 is the median, not the mode. A data set has no mode only when no value is repeated.
  8. 8. 2
    A day with no snow is still a day. Total: 0 + 4 + 0 + 6 + 2 + 0 = 12 cm over 6 days, and 12 ÷ 6 = 2 cm. The answer 4 leaves out the three zero days (12 ÷ 3). The answer 12 is the total. The answer 1 is the median, found by averaging the middle values 0 and 2.
  9. 9. 14
    In order the values are 9, 11, 12, 14, 15, 16, 20. The middle of 7 values is the 4th one, 14. The answer 20 is the 4th value in the original unordered list. The answers 15 and 12 are the second and third values of the original list, not the middle of the ordered list.
  10. 10. 11
    In order: 7, 9, 10, 12, 12, 15. The two middle values are 10 and 12, and halfway between them is 11. The answer 12 comes from averaging the middle of the unordered list (9 and 15), and it also happens to be the mode. The answer 10 uses only the lower middle value. The answer 15 is the largest value.
  11. 11. 8
    A mean of 7 for 4 numbers means the total is 4 × 7 = 28. The three known numbers add to 20, so the fourth is 28 − 20 = 8. The answer 7 is just the mean. The answer 20 is the total of the three known numbers. The answer 10 comes from a subtraction slip.
  12. 12. 7
    The new total is 28 and there are 4 values, so the mean is 28 ÷ 4 = 7. A value equal to the mean does not move the fair-share level. The answer 14 adds 7 + 7. The answer 28 is the total. The answer 5.25 divides the old total, 21, by 4.
  13. 13. The mean is 5 and the median is 3, so the mean is greater.
    Mean: 25 ÷ 5 = 5. Median: in order the middle value is 3. The large value 13 pulls the mean up but leaves the middle of the ordered list at 3. The other statements either swap the two measures or assume the mean and median are always equal.
  14. 14. pasta
    The mode is the category chosen most often, and pasta has the most votes (9). The number 9 is how many votes pasta got, not the mode itself. Sandwich is only the first one listed. Wrap has the fewest votes. Categories like lunches have a mode but no mean or median.
  15. 15. 48
    Mean = total ÷ count, so total = mean × count = 8 × 6 = 48. The answer 14 adds 8 + 6. The answer 42 multiplies 7 × 6. The answer 8 is the mean itself.
  16. 16. The median, 15, because one very large day pulls the mean up
    The mean is 150 ÷ 5 = 30, but four of the five days had 18 riders or fewer, so 30 is not typical. The median, 15, is the middle of the ordered list and is not pulled by the 95. There is no mode because no value repeats. The range of 95 − 10 = 85 describes spread, not a typical value.
  17. 17. Sam did not order the numbers first. In order, 2, 3, 4, 7, 9, the median is 4.
    The median is the middle of the ordered list, so Sam must write 2, 3, 4, 7, 9 first. Then the middle value is 4. Adding and dividing by 5 finds the mean. The most common value is the mode (here every value appears once). The position in the list as written has no meaning.
  18. 18. 19
    A mean always lies between the least and greatest values, here 12 and 18. The actual mean is 75 ÷ 5 = 15. A mean of 19 is greater than every value, which is impossible, because the fair share cannot be more than the largest pile. 14, 15 and 16 are all inside the range, so they are possible.
  19. 19. 5
    Moving cubes does not change the total, 2 + 3 + 5 + 10 = 20, so four equal stacks have 20 ÷ 4 = 5 cubes each. The answer 4 is the median of the four stacks. The answer 10 is the tallest stack, and 20 is the total.
  20. 20. 7
    In order the four values are 4, 6, 8, 20. The middle two are 6 and 8, so the median is (6 + 8) ÷ 2 = 7. The answer 8 picks only one of the two middle values. The answer 20 is the new value, which is at the end, not the middle. The answer 6 assumes that an extreme value cannot change the median; it moves it a little.
  21. 21. 88
    A mean of 82 over 4 tests means a total of 4 × 82 = 328. Her first three tests add to 240, so she needs 328 − 240 = 88. The answer 82 is the target mean, not the score. The answer 328 is the total needed for all four tests. The answer 80 is the median of her first three scores.
  22. 22. Both sets have mean 5, but the values in B are more spread out.
    Both totals are 20 and both have 4 values, so both means are 5. The values in B range from 1 to 9 but those in A do not vary at all. Having the same mean does not make the data sets the same. A single measure hides the spread, so it helps to look at the range or the picture too.
  23. 23. 1
    Write the 10 results in order: 0, 0, 0, 1, 1, 1, 1, 2, 2, 3. The 5th and 6th values are both 1, so the median is 1. The answer 1.5 takes the median of the four different values 0, 1, 2, 3 and ignores how often each happened. The answer 1.1 is the mean (11 ÷ 10). The answer 2 is the third value in the table.
  24. 24. 10
    A mean of 7 for 5 numbers means a total of 35. The known numbers add to 2 + 5 + 6 + 12 = 25, so the missing number is 35 − 25 = 10, which fits between 6 and 12. The answer 7 is the mean itself. The answer 6 is the median of the list. The answer 12 repeats the largest value.
  25. 25. 11
    The first five read 5 × 10 = 50 books. With the sixth student the total is 50 + 16 = 66 books for 6 students, and 66 ÷ 6 = 11. The answer 13 averages 10 and 16 and ignores that five students are in the first group. The answer 16 is the new value. The answer 10 forgets the new student changes the mean.
  26. 26. The mode, medium, because it was sold the most
    Stocking decisions are about what sells most often, and medium was sold 8 times, more than any other size. The sizes are categories, so a mean size has no meaning. A middle size does not show which is most popular. The range only describes how spread out numbers are.
  27. 27. Change 12 to 20
    Changing 12 to 20 gives 4, 6, 8, 10, 20: the middle value is still 8 and the mean is 48 ÷ 5 = 9.6. Changing 8 to 20 gives 4, 6, 10, 12, 20, and the median becomes 10. Changing 4 to 0 lowers the mean and leaves the median at 8. Changing 10 to 6 makes the ordered list 4, 6, 6, 8, 12, so the median drops to 6.
  28. 28. The mean is (10 + 12 + 15 + 18 + 95) ÷ 5 = 150 ÷ 5 = 30, but four of the five days had 18 riders or fewer, so no ordinary day looks like 30. The one extreme day, 95, pulls the mean up. The median is the middle of the ordered list, 15, and it is not pulled by the extreme day, so it is closer to a typical day.
    Self-review with this: did you compute both measures (mean 30, median 15)? Did you say that one extreme value pulls the mean but not the median? Did you use numbers from the data? Check: 10 + 12 + 15 + 18 + 95 = 150.
  29. 29. One answer: 4, 4, 5, 8, 9. Mean: (4 + 4 + 5 + 8 + 9) ÷ 5 = 30 ÷ 5 = 6. Median: the middle value of the ordered list is 5. Mode: 4 appears twice and no other value repeats. Other answers work too, for example 4, 4, 5, 7, 10.
    Self-review: a mean of 6 needs a total of 30. A median of 5 means the middle value in order is 5. A mode of 4 means 4 appears at least twice and more often than any other value, so start 4, 4, 5. The last two numbers must be at least 5 and add to 30 − 13 = 17, for example 8 and 9. Check your own set against all three tests.

Related practice

Bar graphs: intervals, scale and evidenceBar-graph word problems: total, difference and evidenceDivide a 2-digit number by 1 digitEstimate sums and differences