For a positively skewed distribution, write down whether the mean is usually greater than, or less than, the median.
(Total for Question 1 is 1 mark)
2
Write down whether this statement is True or False. In a symmetric distribution, the mean, median and mode are all equal.
(Total for Question 2 is 1 mark)
3
A data set has a long tail stretching towards the higher values, with most of the data bunched at the lower end. Write down whether this distribution is positively skewed or negatively skewed.
(Total for Question 3 is 1 mark)
4
A data set has a long tail stretching towards the lower values, with most of the data bunched at the higher end. Write down whether this distribution is positively skewed or negatively skewed.
(Total for Question 4 is 1 mark)
5
A class test out of 30 has a mean score of 19.5, a median score of 22, and a modal score of 24. State whether the distribution of scores is positively skewed or negatively skewed, giving a reason.
(Total for Question 5 is 2 marks)
6
The times taken by 9 runners to finish a fun run, in minutes, are: 30, 31, 31, 32, 32, 32, 33, 33, 52. Work out the mean and the median of these times.
(Total for Question 6 is 2 marks)
7
Using your answers to Question 6, state whether the distribution of race times is positively skewed or negatively skewed. Give a reason based on the mean and median.
(Total for Question 7 is 2 marks)
8
A holiday company records the number of nights that 150 customers stayed at a resort. The distribution of the lengths of stay is negatively skewed. Write down what this tells you about the relationship between the mean and the median number of nights stayed, and explain why.
(Total for Question 8 is 2 marks)
9
A survey of 40 employees' annual leave days used has a lower quartile of 8, a median of 12 and an upper quartile of 15. Compare Q3 - Q2 with Q2 - Q1 to determine whether the distribution is positively skewed or negatively skewed.
(Total for Question 9 is 2 marks)
10
A different survey has a lower quartile of 5, a median of 9, and an upper quartile of 20. Compare Q3 - Q2 with Q2 - Q1 to determine whether this distribution is positively skewed or negatively skewed.
(Total for Question 10 is 2 marks)
11
The masses, in kg, of 11 parcels are: 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6. Write down the mode, and state whether the mode is greater than, less than, or equal to the median.
(Total for Question 11 is 2 marks)
12
A small business records the annual bonus, in pounds, paid to each of its 8 staff: 500, 520, 510, 505, 515, 495, 505, 3000. Work out the mean bonus, giving your answer to the nearest pound.
(Total for Question 12 is 2 marks)
13
The selling prices of 7 houses on a small road are: 180,000 pounds, 185,000 pounds, 190,000 pounds, 195,000 pounds, 200,000 pounds, 205,000 pounds, 595,000 pounds (one much larger detached house). Work out the median price and the mean price, and state which average better represents a 'typical' house price on this road, giving a reason.
(Total for Question 13 is 3 marks)
14
Priya, a Year 11 student, wants to investigate the statement: 'Most staff at my part-time job have a short commute, but a few travel a long way.' She collects the commuting time, in minutes, of all 12 staff. State whether you would expect the resulting distribution of commuting times to be positively skewed or negatively skewed, give a reason referring to the mean and median, and suggest an appropriate average to use to summarise a 'typical' commuting time for this group, giving a reason.
(Total for Question 14 is 3 marks)
15
Two classes sat the same maths test, out of 80 marks. Class A: lower quartile = 40, median = 55, upper quartile = 62. Class B: lower quartile = 45, median = 55, upper quartile = 65. Both classes have the same median score. Compare Q3 - Q2 with Q2 - Q1 for each class to determine the skew of each distribution, and comment on which class has a more symmetric spread of results.
(Total for Question 15 is 3 marks)
16
A data set has a mean of 48, a median of 42, and a lower quartile of 38. A student claims: 'Since the mean is greater than the median, this data set must be negatively skewed.' Comment on whether the student is correct, explaining your reasoning.
(Total for Question 16 is 3 marks)
17
The ages, in years, of 11 people at a family gathering are: 4, 6, 7, 8, 9, 9, 10, 11, 12, 13, 64. Work out the median age, work out the mean age to 1 decimal place, and explain why the mean is not a good measure of the 'typical' age at this gathering, suggesting a more appropriate average.
(Total for Question 17 is 4 marks)
18
The table shows the number of books read last month by each of 28 pupils in a class. Books read (x): 0, 1, 2, 3, 4, 5, 6 Frequency: 2, 3, 5, 8, 6, 3, 1 Work out the mean number of books read, to 2 decimal places, work out the median number of books read, and use your two answers to state whether the distribution is positively skewed or negatively skewed, giving a reason.
(Total for Question 18 is 4 marks)
Mark scheme · F04D Skewness by Inspection: Fluency and Exam Drill
Question 1
B1 greater than cao
Answer: Greater than
Question 2
B1 True cao
Answer: True
Question 3
B1 positively skewed cao
Answer: Positively skewed
Question 4
B1 negatively skewed cao
Answer: Negatively skewed
Question 5
M1 compares the mean with the median or mode, e.g. recognises mean < median < mode oe
A1 negatively skewed cao
Answer: Negatively skewed, since the mean (19.5) is less than the median (22), which is less than the mode (24).
Question 6
M1 (30+31+31+32+32+32+33+33+52) / 9 oe
A1 mean = 34 and median = 32, both cao
Answer: Mean = 34, median = 32.
Question 7
B1 positively skewed cao
B1 reason: the mean (34) is greater than the median (32), consistent with a tail stretching towards higher values oe
Answer: Positively skewed, since the mean (34) is greater than the median (32).
Question 8
B1 the mean is less than the median cao
B1 explains this is because the long tail of a negative skew pulls the mean below the median oe
Answer: The mean is less than the median, because the long tail of a negatively skewed distribution pulls the mean down below the median.
Question 9
M1 Q3 - Q2 = 3 and Q2 - Q1 = 4 oe
A1 negatively skewed cao
Answer: Negatively skewed, since Q2 - Q1 (4) is greater than Q3 - Q2 (3).
Question 10
M1 Q3 - Q2 = 11 and Q2 - Q1 = 4 oe
A1 positively skewed cao
Answer: Positively skewed, since Q3 - Q2 (11) is greater than Q2 - Q1 (4).
Question 11
B1 mode = 4 cao
B1 equal to the median (both 4) cao
Answer: Mode = 4; the mode is equal to the median (both 4).
Question 12
M1 (500+520+510+505+515+495+505+3000) / 8 oe
A1 819 (awrt) cao
Answer: £819 (awrt)
Question 13
B1 median = 195,000 cao
B1 mean = 250,000 cao
B1 median is more representative, since the mean is pulled up a long way by the one large detached house oe
Answer: Median = 195,000 pounds, mean = 250,000 pounds; the median better represents a typical house price, since the mean is pulled up by the one much larger house.
Question 14
B1 positively skewed cao
B1 reason: the long tail from the few staff with a long commute pulls the mean above the median oe
B1 the median is more appropriate, since it is not affected by the few unusually long commutes oe
Answer: Positively skewed, because the few staff with a long commute create a tail towards higher values, pulling the mean above the median; the median is a more appropriate typical average, since it is not affected by these few long commutes.
Question 15
M1 Class A: Q3 - Q2 = 7 and Q2 - Q1 = 15 (or equivalent working for either class)
A1 Class A is negatively skewed, since Q2 - Q1 (15) is greater than Q3 - Q2 (7)
B1 Class B has a symmetric (evenly spread) interquartile range, since Q3 - Q2 = Q2 - Q1 = 10
Answer: Class A: Q3 - Q2 = 7, Q2 - Q1 = 15, so negatively skewed. Class B: Q3 - Q2 = 10, Q2 - Q1 = 10, so symmetric. Class B has the more symmetric spread of results.
Question 16
B1 the student is incorrect cao
B1 explains that a mean greater than the median is generally a sign of positive skew (a tail towards higher values), not negative oe
B1 states that negative skew would instead show the mean being less than the median oe
Answer: The student is incorrect. A mean greater than the median generally indicates positive skew, since it shows a tail stretching towards higher values; negative skew would instead show the mean being less than the median.
Question 17
B1 median = 9 cao
M1 (4+6+7+8+9+9+10+11+12+13+64) / 11 oe
A1 mean = 13.9 (awrt) cao
B1 the median is more appropriate, since 64 is an outlier that pulls the mean well above the age of most people at the gathering oe
Answer: Median = 9, mean = 13.9 (awrt). The median is more appropriate, since the one 64-year-old is an outlier that pulls the mean a long way above the age of most people there.