These eight questions cover the four pages in choosing the right summary of a data set. Write both the working and the reason on paper first.
Questions
Question 1. Two groups each have a mean of 50. Group A has a standard deviation of 4 and Group B has a standard deviation of 15. What does this tell you (2 marks)?
Answer
The centres are equal, but the marks in Group B are far more spread out. Group A is more consistent, with values close to 50.
Question 2. The data 8, 9, 10, 11, 12 has mean 10. The value 12 is changed to 42. Find the new mean and median (3 marks).
Answer
The new data is 8, 9, 10, 11, 42. The sum is 80, so the mean is 16. The median is the middle value, 10.
Question 3. In Question 2, which summary better describes a typical value? Give a reason (2 marks).
Answer
The median (10), because four of the five values are between 8 and 11, and the mean of 16 is pulled up by the single value of 42.
Question 4. The mean of six scores is 70. Five scores are 65, 80, 72, 60 and 75. Find the sixth (3 marks).
Answer
Total = 70 × 6 = 420. Known sum = 65 + 80 + 72 + 60 + 75 = 352. Missing = 420 − 352 = 68.
Check: 420 ÷ 6 = 70, and 68 is a possible score near the others.
Question 5. The mean of four marks, each out of 50, is 45. Three marks are 40, 42 and 44. Find the fourth and comment (3 marks).
Answer
Total = 45 × 4 = 180. Known sum = 126. Missing = 54.
A mark of 54 is impossible when the maximum is 50, so the given mean cannot be correct for these marks.
Question 6. A class of 20 has a mean of 60. A new student joins, and the mean of 21 becomes 61. Find the new student’s mark (3 marks).
Answer
Old total = 60 × 20 = 1 200. New total = 61 × 21 = 1 281. New mark = 1 281 − 1 200 = 81.
Question 7. Test M (out of 50) has a mean of 35 and a standard deviation of 5. Test N (out of 100) has a mean of 68 and a standard deviation of 8. Which has the higher mean and the smaller spread, after rescaling to 100 (4 marks)?
Answer
Multiply Test M by 2: mean 70 and standard deviation 10. So Test M has the higher mean (70 against 68), and Test N has the smaller spread (8 against 10).
Question 8. A report says “the mean mark of Class R is 60, and the median mark of Class S is 62, so S did better.” Explain why this comparison is not fair (2 marks).
Answer
It compares a mean with a median, which describe the centre in different ways. To compare fairly, use the mean for both classes or the median for both.
If you got these wrong
Question 1 uses comparing two classes with equal means but different spread. Questions 2 and 3 use explaining how an outlier changes the mean and median differently.
Questions 4 to 6 use reconstructing a missing observation from a mean. Questions 7 and 8 use deciding whether two data summaries are genuinely comparable.
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