Maths: Statistics & Probability

The scores of five students are 12, 15, 9, 15 and 19. What is the mean score?

• A15
• B14✓
• C10
• D13

Answer: The mean is the total divided by the number of values: (12 + 15 + 9 + 15 + 19) ÷ 5 = 70 ÷ 5 = 14. 15 is the mode and also the median; 10 is the range (19 − 9).

For the same five scores — 12, 15, 9, 15 and 19 — what is the median?

• A12
• B14
• C15✓
• D9

Answer: Put the values in order: 9, 12, 15, 15, 19. The median is the middle value, which is 15. 14 is the mean, not the median.

A bag contains 4 red, 3 blue and 3 green counters. One counter is picked at random. What is the probability that it is blue?

• A1/3
• B3/10✓
• C7/10
• D3/7

Answer: There are 4 + 3 + 3 = 10 counters and 3 of them are blue, so the probability is 3/10. 7/10 is the probability of not picking blue, and 3/7 ignores the green counters.

A fair six-sided dice is rolled twice. What is the probability of getting a 6 both times?

• A1/12
• B1/36✓
• C1/6
• D2/6

Answer: The rolls are independent, so the probabilities multiply: 1/6 × 1/6 = 1/36. Adding the probabilities (2/6) or halving (1/12) are the usual errors.

A survey of 40 people has a mean score of 12.5. A 41st person scores 33. What is the new mean score?

• A12.5
• B14
• C13✓
• D22.75

Answer: The original total is 40 × 12.5 = 500. Adding 33 gives 533 over 41 values: 533 ÷ 41 = 13. Leaving the mean unchanged ignores the new value, and 22.75 averages the old mean with 33.

A scatter graph of hours of revision against test score shows the points rising from bottom-left to top-right. What does this show?

• AThat revision causes a higher score in every case
• BNo correlation between the two variables
• CPositive correlation between revision time and score✓
• DNegative correlation between revision time and score

Answer: Points rising from bottom-left to top-right show that as one variable increases the other tends to increase: a positive correlation. Correlation on its own does not prove that revision causes the higher score for every student.

Which average is most affected by one extremely high value in a data set?

• AThe range is an average and is unaffected
• BThe median
• CThe mean✓
• DThe mode

Answer: The mean uses every value, so a single very large value pulls it up. The median depends only on the middle position and the mode on the most frequent value, so neither shifts much; the range is a measure of spread, not an average.

The probability that it rains on a given day is 0.3. What is the probability that it does not rain?

• A0.3
• B0.7✓
• C0.6
• D1.3

Answer: The probabilities of an event and its complement add to 1, so P(no rain) = 1 − 0.3 = 0.7.

A spinner has four equal sections coloured red, blue, green and yellow. It is spun twice. What is the probability of red then blue?

• A1/2
• B1/16✓
• C1/8
• D1/4

Answer: Each spin is independent with probability 1/4 for any colour, so P(red then blue) = 1/4 × 1/4 = 1/16. Adding the probabilities would give 1/2, which is wrong for a sequence of two events.

A survey asks 'Don't you agree that the new car park is excellent?' Why is this a poor question?

• AIt is too short to gather useful data
• BIt can only be asked of people who drive and use the car park regularly
• CIt has too many possible answers to analyse in a survey of this kind of size
• DIt is a leading question that pushes respondents towards agreeing✓

Answer: Questions that suggest the expected answer produce biased results; a fair version would ask respondents to rate the car park on a neutral scale. Length and the number of answers are not the problem here.

A frequency table shows how many books 30 students read last month: 2 books (5 students), 4 books (12), 6 books (8), 8 books (5). Estimate the mean number of books, to one decimal place.

• A4.9✓
• B6.0
• C4.0
• D5.0

Answer: Multiply each value by its frequency and add: (2 × 5) + (4 × 12) + (6 × 8) + (8 × 5) = 10 + 48 + 48 + 40 = 146. Divide by the 30 students: 146 ÷ 30 = 4.87, which rounds to 4.9. Averaging the four values (5.0) ignores the frequencies.

A pie chart shows how 120 people travel to work. The 'bus' sector has an angle of 90°. How many people travel by bus?

• A40
• B90
• C30✓
• D25

Answer: A full circle of 360° represents 120 people, so 90° represents 90/360 = 1/4 of them: 120 ÷ 4 = 30.