📊 Maths: Averages, Range and Statistics

1. How is the mean of a data set calculated?

1. Add up all the values and divide the total by the number of values
2. Subtract the smallest value from the largest value
3. Pick the value that appears most often
4. Take the middle value once the data is in order

The mean is found by adding all the values together and dividing by how many values there are. (CIMT/MEP, Mathematics Enhancement Programme, Unit 5.2 'Mean, Median, Mode and Range')

2. A delivery driver records the parcels dropped off on five days: 12, 15, 18, 9 and 11. What is the mean number of parcels per day?

1. 13
2. 15
3. 11
4. 18

The total is 12 + 15 + 18 + 9 + 11 = 65, and 65 divided by 5 days gives a mean of 13. (CIMT/MEP Unit 5.2; DfE Functional Skills subject content L1 item 25 'find the mean and range')

3. Six bags of flour weigh 4 kg, 8 kg, 6 kg, 10 kg, 7 kg and 13 kg. What is the mean weight?

1. 8 kg
2. 9 kg
3. 6 kg
4. 48 kg

The total weight is 48 kg and there are 6 bags, so 48 divided by 6 gives a mean of 8 kg. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range')

4. A worker's daily wages over five days were £180, £200, £220, £240 and £160. What was the mean daily wage?

1. £200
2. £220
3. £180
4. £1,000

The total is £1,000 and dividing by 5 days gives a mean daily wage of £200. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range')

5. The mean of five numbers is 20. Four of the numbers are 18, 22, 15 and 19. What is the fifth number?

1. 26
2. 20
3. 18
4. 74

For a mean of 20 over 5 values the total must be 100; the four known values sum to 74, so the fifth is 100 − 74 = 26. (CIMT/MEP Unit 5.2; mean = sum of values ÷ number of values)

6. Midday temperatures over five days were 14°C, 16°C, 15°C, 19°C and 11°C. What was the mean temperature?

1. 15°C
2. 16°C
3. 19°C
4. 8°C

The total is 75°C and dividing by 5 days gives a mean of 15°C. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range')

7. A grouped frequency table records waiting times: 0–10 minutes (4 people), 10–20 minutes (6 people) and 20–30 minutes (10 people). Using the midpoint of each class, what is the estimated mean waiting time?

1. 18 minutes
2. 20 minutes
3. 15 minutes
4. 13.3 minutes

Using midpoints 5, 15 and 25 gives (5×4) + (15×6) + (25×10) = 360, divided by 20 people = 18 minutes. (DfE Functional Skills subject content L2 item 24 'estimate the mean of a grouped frequency distribution'; CIMT/MEP grouped data)

8. When estimating the mean from a grouped frequency table, which value is used to represent each class interval?

1. The midpoint of each class interval
2. The lowest value in each class interval
3. The highest value in each class interval
4. The frequency of each class interval

Because individual values are not known, the midpoint of each class is multiplied by its frequency to estimate the mean. (DfE Functional Skills subject content L2 item 24; CIMT/MEP Mathematics Enhancement Programme, grouped data section)

9. Why is the mean described as an estimate when it is found from a grouped frequency table?

1. The exact individual values are unknown, so midpoints are used to represent each class
2. The frequencies are only rough guesses
3. The mean is always rounded to the nearest whole number
4. Grouped data cannot have a mean at all

In a grouped table the actual values within each class are not recorded, so using the class midpoints gives an estimate rather than an exact mean. (DfE Functional Skills subject content L2 item 24 'estimate the mean of a grouped frequency distribution from discrete data')

10. Five test marks are 7, 9, 10, 6 and 8. What is the mean mark?

1. 8
2. 9
3. 7
4. 40

The total is 40 and dividing by 5 gives a mean of 8. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range')

11. A learner adds five values correctly to get a total of 90 but then divides by 4 instead of 5. What mistake has been made when finding the mean?

1. Dividing by the wrong number of values
2. Adding the values incorrectly
3. Forgetting to put the values in order
4. Using the largest value instead of the total

The mean must be found by dividing the total by the number of values; dividing by 4 instead of 5 uses the wrong count. (CIMT/MEP Unit 5.2; mean = sum of values ÷ number of values)

12. At Level 2, which set of measures should learners use to compare two sets of data?

1. The mean, median, mode and range
2. Only the mean
3. Only the mode
4. The total and the number of values

The Level 2 content statement requires learners to use the mean, median, mode and range to compare two sets of data. (DfE, Functional Skills subject content: mathematics (2018), Level 2 content, item 25 — GOV.UK)

13. When comparing two data sets, what does the range tell you about each set?

1. How spread out or consistent the values are
2. The most common value in the set
3. The total of all the values
4. The middle value of the set

The range measures spread, so it shows how consistent the data is; a smaller range means more consistent data. (CIMT/MEP Unit 5.2 interpretation of range as spread; DfE Functional Skills subject content L2 item 25)

14. When comparing two data sets, what does an average such as the mean tell you?

1. The typical or central value of the set
2. How spread out the values are
3. The difference between the highest and lowest value
4. The number of values in the set

An average summarises the typical or central value, while the range describes the spread of the data. (DfE Functional Skills subject content L2 item 25 'compare 2 sets of data'; CIMT/MEP Unit 5.2)

15. Two batsmen each have a mean of 7 runs. Player A's scores have a range of 4 and Player B's a range of 10. What is the best conclusion?

1. Player A is more consistent because the smaller range shows less spread
2. Player B is more consistent because the larger range shows more runs
3. Both players are equally consistent because the means are equal
4. Player B is more consistent because a larger range is always better

With equal means, the smaller range (Player A) shows the scores are closer together, so Player A is more consistent. (DfE Functional Skills subject content L2 item 25; CIMT/MEP Unit 5.2 (smaller range = more consistent))

16. Shop A's daily takings are £200, £210, £190, £205 and £195. Shop B's are £150, £300, £100, £250 and £200. Both have a mean of £200. Which shop has the more consistent daily takings?

1. Shop A, because its range of £20 is smaller than Shop B's range of £200
2. Shop B, because its range is larger
3. Both are equally consistent because the means match
4. Shop B, because it had a day of £300

Both means are £200, but Shop A's range (£20) is far smaller than Shop B's (£200), so Shop A's takings are more consistent. (DfE Functional Skills subject content L2 item 25; Range = highest − lowest (CIMT/MEP Unit 5.2))

17. Class 1 scored 60, 62, 58, 61 and 59. Class 2 scored 40, 80, 50, 70 and 60. Both classes have a mean of 60. What does comparing the ranges show?

1. Class 1's scores are more consistent because its range of 4 is smaller than Class 2's range of 40
2. Class 2 performed better because it had the highest single score
3. Class 1 performed better because every score was above 50
4. The classes cannot be compared because the means are equal

Equal means make the range the key comparison; Class 1's range (4) is much smaller than Class 2's (40), so Class 1's results are more consistent. (DfE Functional Skills subject content L2 item 25 'compare 2 sets of data'; CIMT/MEP Unit 5.2)

18. When comparing two data sets, a smaller range indicates that the data is:

1. More consistent
2. Larger on average
3. More spread out
4. Always higher

A smaller range means the values are closer together, so the data is more consistent. (CIMT/MEP Unit 5.2; DfE Functional Skills subject content L2 item 25)

19. Bus A is late by 2, 3, 2, 4 and 4 minutes over a week. Bus B is late by 0, 1, 8, 2 and 4 minutes. Both have a mean lateness of 3 minutes. Which statement is correct?

1. Bus A is more reliable because its range of 2 is smaller than Bus B's range of 8
2. Bus B is more reliable because it was once on time
3. Both buses are equally reliable because the means are equal
4. Bus B is more reliable because its range is larger

The means are identical, so the range decides reliability; Bus A's range (2) is smaller than Bus B's (8), making Bus A more reliable. (DfE Functional Skills subject content L2 item 25; CIMT/MEP Unit 5.2 (smaller range = more consistent))

20. To compare the typical performance of two football teams over a season, which pair of measures gives the most complete picture?

1. An average to show the typical result and the range to show the consistency
2. The highest score of each team only
3. The number of matches played by each team
4. The mode of each team and nothing else

A full comparison needs an average (typical value) together with the range (spread), as a single measure alone can be misleading. (DfE Functional Skills subject content L2 item 25 'use the mean, median, mode and range to compare 2 sets of data')

21. Which average is most affected by an outlier (an extreme value very different from the rest)?

1. The mean
2. The median
3. The mode
4. The range

The mean uses every value in its calculation, so a single extreme value pulls it up or down. (Pass Functional Skills, 'Mean, Median, Mode and Range', Level 2 (passfunctionalskills.co.uk))

22. Why is the median often a better central measure than the mean when a data set contains extreme values?

1. The median depends only on the middle value(s), so it resists outliers
2. The median always equals the mode
3. The median uses every value, so it is more accurate
4. The median is the largest value in the set

Because the median is just the middle value of the ordered data, extreme values do not pull it, making it resistant to outliers. (vedantu.com / Bhanzu, 'Median: Definition, Formula & Examples'; Pass Functional Skills L2)

23. Five salaries in a small firm are £16,000, £17,000, £18,000, £19,000 and £200,000. Which average best represents a typical salary?

1. The median, because the £200,000 outlier inflates the mean
2. The mean, because it uses every value
3. The range, because it shows the spread
4. The highest value, because it is the most important

The mean here is £54,000, which no ordinary worker earns; the median of £18,000 better represents a typical salary because it is not distorted by the outlier. (Pass Functional Skills L2 (mean sensitive to outliers); vedantu.com (median resistant to extreme values))

24. A job advert claims 'average pay £54,000' but most staff earn around £18,000 because one director earns £200,000. Why is this average misleading?

1. The mean has been pulled up by one very high salary, so it does not reflect typical pay
2. The mode has been used instead of the mean
3. The range has been used instead of an average
4. The median is too high to be useful

A single very high salary inflates the mean, so quoting it makes typical pay look far higher than it really is. (Pass Functional Skills, 'Mean, Median, Mode and Range', Level 2 (mean is sensitive to outliers))

25. House prices on a street are £150,000, £160,000, £155,000, £165,000 and £800,000. An estate agent advertises the 'average' price as £286,000. Which average did they use and is it fair?

1. The mean, and it is misleading because the £800,000 house inflates it
2. The median, and it is fair because it ignores extremes
3. The mode, and it is fair because it is the most common price
4. The range, and it is misleading because it is too small

The £286,000 figure is the mean, pulled up by the £800,000 outlier; the median of £160,000 would give a fairer picture of a typical house. (Pass Functional Skills L2 (mean sensitive to outliers); CIMT/MEP Unit 5.2)

26. A report wants to show the typical value of a data set that contains one extreme outlier, while still using all the data. What is the best approach?

1. Use the median, as it is resistant to the outlier
2. Use the mean, as it is pulled towards the outlier
3. Use the range, as it ignores the middle values
4. Remove all values except the outlier

The median resists the effect of an extreme value, so it gives a more representative typical figure than the outlier-sensitive mean. (vedantu.com / Bhanzu, 'Median: Definition, Formula & Examples'; Pass Functional Skills L2)

27. Which statement about the mean and the median is correct?

1. The mean is sensitive to extreme values, whereas the median is resistant to them
2. The median is sensitive to extreme values, whereas the mean is resistant to them
3. Both the mean and the median ignore extreme values completely
4. Neither the mean nor the median is affected by the data values

Because the mean uses every value it is affected by outliers, while the median depends only on the middle value(s) and so resists them. (Pass Functional Skills L2 (mean sensitive to outliers); vedantu.com (median resistant to extreme values))

28. What is the mode of a set of data?

1. The value that appears most frequently
2. The middle value when the data is in order
3. The total of all values divided by how many there are
4. The difference between the highest and lowest value

The mode is the value that occurs most often in a data set. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range' (cimt.org.uk))

29. How is the median of a data set found?

1. It is the middle value once the values are arranged in order of size
2. It is the value that appears most often
3. It is the largest value in the set
4. It is the sum of the values divided by the number of values

The median is the middle value of a data set after the values have been put in order. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range' (cimt.org.uk))

30. Find the mode of these test marks: 7, 4, 9, 4, 6, 4, 8.

1. 4
2. 7
3. 6
4. 9

The value 4 appears three times, more often than any other value, so it is the mode. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range' (cimt.org.uk))

31. Find the median of these five numbers: 3, 8, 2, 9, 5.

1. 5
2. 8
3. 2
4. 9

In order the values are 2, 3, 5, 8, 9; the middle value is 5. (CIMT/MEP Unit 5.2 'Mean, Median, Mode and Range' (cimt.org.uk))

32. Find the median of these six values: 12, 5, 9, 14, 7, 10.

1. 9.5
2. 9
3. 10
4. 11

In order: 5, 7, 9, 10, 12, 14; the two middle values are 9 and 10, so the median is (9 + 10) ÷ 2 = 9.5. (CIMT/MEP Unit 5.2; Pass Functional Skills 'Mean, Median, Mode and Range' (passfunctionalskills.co.uk))

33. When a data set has an even number of values, how is the median calculated?

1. By taking the mean of the two middle values
2. By choosing the larger of the two middle values
3. By choosing the smaller of the two middle values
4. By adding the two middle values together

With an even number of values there are two middle values, and the median is their mean (their sum divided by two). (CIMT/MEP Unit 5.2; Pass Functional Skills 'Mean, Median, Mode and Range' (passfunctionalskills.co.uk))

34. Which statement about the mode is correct?

1. A data set can have more than one mode, or no mode at all
2. Every data set has exactly one mode
3. The mode must always be the largest value
4. The mode is found by dividing the total by the number of values

If two or more values are equally most common there can be several modes, and if no value repeats there is no mode. (Pass Functional Skills 'Mean, Median, Mode and Range', Level 2 (passfunctionalskills.co.uk))

35. State the mode of these shoe sizes: 6, 8, 6, 9, 8, 7, 8, 6.

1. 6 and 8
2. 8 only
3. 6 only
4. 7

Both 6 and 8 appear three times each, more than any other value, so the data set has two modes: 6 and 8. (Pass Functional Skills 'Mean, Median, Mode and Range', Level 2 (passfunctionalskills.co.uk))