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F04 Skewness by Inspection

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each statement about skewness, write down whether it is True or False.

(i)In a symmetric distribution, the mean, median and mode are all equal (or very close).(1)

(ii)A distribution is positively skewed when the mean is greater than the median.(1)

(iii)Negative skew means the distribution has a long tail on the right-hand (high-value) side.(1)

(iv)If the mode is greater than the mean, the distribution is likely to be positively skewed.(1)

(Total for Question 1 is 4 marks)

The diagram shows three distributions, A, B and C, drawn as smooth frequency curves.

(a)State whether distribution A is symmetric, positively skewed or negatively skewed.(1)

(b)State whether distribution B is symmetric, positively skewed or negatively skewed.(1)

(c)State whether distribution C is symmetric, positively skewed or negatively skewed.(1)

(Total for Question 2 is 3 marks)

In a spelling test out of 20, a class of pupils obtained a mean score of 14.2, a median score of 15 and a modal score of 16.

(a)State whether the distribution of scores is symmetric, positively skewed or negatively skewed.(1)

(b)Justify your answer to part (a) by referring to the mean, median and mode.(2)

(Total for Question 3 is 3 marks)

The number of pets owned by each of 9 pupils in a class is: 1, 2, 2, 3, 3, 3, 4, 4, 5.

(a)Work out the mean number of pets.(2)

(b)Write down the median number of pets.(1)

(c)Write down the modal number of pets.(1)

(d)State, with a reason, whether this distribution is symmetric, positively skewed or negatively skewed.(1)

(Total for Question 4 is 5 marks)

The times, in minutes, taken by 15 runners to complete a fun run gave a lower quartile of 28 minutes, a median of 33 minutes and an upper quartile of 45 minutes.

(a)Work out median - lower quartile.(1)

(b)Work out upper quartile - median.(1)

(c)Compare your answers to parts (a) and (b) and state, giving a reason, whether the distribution of times is symmetric, positively skewed or negatively skewed.(2)

(Total for Question 5 is 4 marks)

A holiday company recorded the number of nights that 200 customers stayed at a resort. The distribution of the lengths of stay is negatively skewed. Describe what this tells you about the shape of the distribution, and about how the lengths of stay are spread either side of the median.

(Total for Question 6 is 2 marks)

The histogram shows the waiting times, in minutes, of 60 customers in a bank queue.

(a)State whether the distribution of waiting times is symmetric, positively skewed or negatively skewed.(1)

(b)Give a reason for your answer, based on the shape of the histogram.(2)

(Total for Question 7 is 3 marks)

50 students each completed a puzzle. The frequency polygon shows the time taken, in minutes, by each student.

(a)State whether the distribution of completion times is symmetric, positively skewed or negatively skewed.(1)

(b)Give a reason for your answer, based on the shape of the frequency polygon.(2)

(Total for Question 8 is 3 marks)

The box plot shows the masses, in kg, of 40 parcels handled by a courier.

(a)Write down the median mass.(1)

(b)Work out the interquartile range.(1)

(c)By comparing median - lower quartile with upper quartile - median, state, giving a reason, whether the distribution of masses is symmetric, positively skewed or negatively skewed.(2)

(Total for Question 9 is 4 marks)

The box plots show the commuting times, in minutes, of workers travelling to Town A and to Town B, drawn on the same scale.

(a)Write down the median commuting time for Town A.(1)

(b)Write down the median commuting time for Town B.(1)

(c)Compare the skewness of the two distributions, referring to the shape of each box plot.(2)

(Total for Question 10 is 4 marks)

A survey recorded the annual income of every adult living on one street. Most of the adults earn a modest income, but the street also contains one company director with a very high income.

(a)State whether you would expect the distribution of incomes to be symmetric, positively skewed or negatively skewed.(1)

(b)Explain your answer, referring to the mean and the median.(2)

(Total for Question 11 is 3 marks)

The selling prices of houses on a small road are: £180,000, £185,000, £190,000, £195,000, £200,000, £205,000 and £595,000 (one much larger detached house).

(a)Work out the mean house price.(2)

(b)Write down the median house price.(1)

(c)Explain why the median gives a better representation of a typical house price on this road than the mean.(2)

(Total for Question 12 is 5 marks)

Priya, a Year 11 student, wants to investigate the statement: 'Most students at my school have a short journey to school, but a few travel a long way.' She plans to collect journey times, in minutes, from a sample of students.

(a)Suggest an appropriate sampling method Priya could use to choose her sample of students, and give one reason for your choice.(2)

(b)Priya collects the following journey times, in minutes, from 11 students: 5, 6, 6, 7, 8, 9, 10, 12, 15, 20, 56. Find the median journey time.(1)

(c)Calculate the mean journey time.(2)

(d)Compare the mean and the median, and use this comparison to explain whether Priya's data supports the statement that journey times are positively skewed.(2)

(Total for Question 13 is 7 marks)

The ages, in years, of 11 people at a family gathering are: 3, 5, 6, 7, 8, 8, 9, 10, 11, 12, 61.

(a)Write down the modal age.(1)

(b)Write down the median age.(1)

(c)Work out the mean age, giving your answer correct to 1 decimal place.(2)

(d)Use your answers to parts (a) to (c) to state, giving a reason, whether the distribution of ages is symmetric, positively skewed or negatively skewed.(1)

(Total for Question 14 is 5 marks)

The table shows the number of books read last month by each of 30 pupils in a class.

Books read (x)	0	1	2	3	4	5	6
Frequency (f)	3	5	8	7	4	2	1

(a)Write down the modal number of books read.(1)

(b)Find the median number of books read.(1)

(c)Calculate the mean number of books read, giving your answer correct to 2 decimal places.(3)

(d)State, giving a reason based on your answers to parts (a) to (c), whether the distribution is symmetric, positively skewed or negatively skewed.(1)

(Total for Question 15 is 6 marks)

Two classes sat the same maths test, out of 60 marks. The table shows summary statistics for each class.

Class	Lower quartile	Median	Upper quartile
9A	28	40	44
9B	30	34	52

(a)For Class 9A, work out median - lower quartile and upper quartile - median, and use these to state the type of skew shown.(2)

(b)For Class 9B, work out median - lower quartile and upper quartile - median, and use these to state the type of skew shown.(2)

(Total for Question 16 is 4 marks)

Two data sets of delivery times have the same median value, but Data Set P is positively skewed and Data Set Q is negatively skewed. Explain how this is possible, and describe one difference you would expect to see between their box plots.

(Total for Question 17 is 3 marks)

A data set has a mean of 52, a median of 60 and an upper quartile of 65. A student claims: 'Since the mean is less than the median, this data must be negatively skewed.'

(a)State whether the student's claim about the type of skew is correct, based on the mean and median alone.(1)

(b)The lower quartile of the data is 38. Show that the quartile spacing (median - lower quartile, compared with upper quartile - median) agrees with the student's conclusion.(2)

(c)Give one reason why it is good practice to check for skewness using more than one method (for example, both the mean/median comparison and the quartile spacing).(1)

(Total for Question 18 is 4 marks)

Mark scheme · F04 Skewness by Inspection

Question 1

(i) B1 True cao
(i) Answer: True
(ii) B1 True cao
(ii) Answer: True
(iii) B1 False cao
(iii) Answer: False
(iv) B1 False cao
(iv) Answer: False

Question 2

(a) B1 Symmetric cao
(a) Answer: Symmetric
(b) B1 Positively skewed cao
(b) Answer: Positively skewed
(c) B1 Negatively skewed cao
(c) Answer: Negatively skewed

Question 3

(a) B1 Negatively skewed cao
(a) Answer: Negatively skewed
(b) B1 identifies mean < median < mode (14.2 < 15 < 16), oe
(b) B1 correct interpretation: most pupils scored towards the top of the range, with a small tail of lower scores pulling the mean down (allow equivalent wording about spread either side of the median)
(b) Answer: Since 14.2 < 15 < 16, most scores are bunched at the higher end, with a tail of lower scores pulling the mean down below the median and mode.

Question 4

(a) M1 1+2+2+3+3+3+4+4+5 (=27), oe divided by 9
(a) A1 3 cao
(a) Answer: 3
(b) B1 3 cao
(b) Answer: 3
(c) B1 3 cao
(c) Answer: 3
(d) B1 Symmetric, since mean = median = mode (all equal to 3), oe
(d) Answer: Symmetric

Question 5

(a) B1 5 cao
(a) Answer: 5
(b) B1 12 cao
(b) Answer: 12
(c) M1 compares median - LQ (5) with UQ - median (12), oe
(c) A1 Positively skewed, since the upper quartile is further from the median than the lower quartile is (a larger gap on the high-value side / a tail towards the higher times) cao
(c) Answer: Positively skewed

Question 6

B1 describes the shape: most customers stayed for a long time (values bunched at the higher end), with a small tail of customers who stayed for a much shorter time
B1 correctly links this to spread either side of the median: the lengths of stay below the median are more spread out (a longer tail) than those above the median, which are bunched closely together
Answer: Most customers stayed for a long time (values bunched at the higher end), with a small tail of customers who stayed for a much shorter time; the lengths of stay below the median are more spread out than those above it, which are bunched closely together.

Question 7

(a) B1 Positively skewed cao
(a) Answer: Positively skewed
(b) B1 the tallest bar (the peak) is towards the lower end (10-15 minutes) of the range
(b) B1 the bars decrease gradually across the higher classes (15-20, 20-25, 25-30), giving a longer tail towards the higher waiting times
(b) Answer: The peak is at 10-15 minutes, near the low end of the range, and the bars decrease gradually through the higher classes, giving a tail stretching towards the higher waiting times.

Question 8

(a) B1 Negatively skewed cao
(a) Answer: Negatively skewed
(b) B1 the peak (highest frequency) is towards the higher end of the times (the 16-20 minutes class)
(b) B1 frequencies rise gradually across the lower time classes, giving a longer tail towards the lower (shorter) times
(b) Answer: The peak is at 16-20 minutes, near the top of the range, and frequencies build up gradually from the lower classes, giving a tail stretching towards the shorter times.

Question 9

(a) B1 9 cao
(a) Answer: 9
(b) B1 12 cao
(b) Answer: 12
(c) M1 median - LQ (=3) compared with UQ - median (=9), oe
(c) A1 Positively skewed - the upper quartile is much further from the median than the lower quartile is, showing the higher masses are more spread out cao
(c) Answer: Positively skewed

Question 10

(a) B1 30 cao
(a) Answer: 30
(b) B1 18 cao
(b) Answer: 18
(c) B1 Town A's box plot is symmetric (the median is equally spaced between the LQ and UQ / the two box halves are equal)
(c) B1 Town B's box plot is positively skewed (the median is much closer to the LQ than the UQ, so the upper commuting times are more spread out - a long tail towards the higher times)
(c) Answer: Town A is symmetric; Town B is positively skewed, with a tail towards the longer commuting times.

Question 11

(a) B1 Positively skewed cao
(a) Answer: Positively skewed
(b) B1 the one very high income acts as an extreme value which pulls the mean up but has little effect on the median
(b) B1 so the mean is greater than the median, with a long tail towards the higher incomes (most incomes bunched at the lower/modest end)
(b) Answer: The director's very high income pulls the mean up but barely affects the median, so mean > median, with a long tail towards the higher incomes.

Question 12

(a) M1 180+185+190+195+200+205+595 (=1750, in £000s), oe divided by 7
(a) A1 £250,000 cao
(a) Answer: £250,000
(b) B1 £195,000 cao
(b) Answer: £195,000
(c) B1 identifies the £595,000 house as an extreme value (outlier) which increases the mean
(c) B1 states the median is not affected by this outlier and so better represents the typical price, since the mean (£250,000) is higher than 6 of the 7 prices
(c) Answer: The £595,000 house is an outlier that pulls the mean up to £250,000, which is higher than six of the seven prices; the median (£195,000) is unaffected by it and better reflects a typical price on the road.

Question 13

(a) B1 valid sampling method named (e.g. simple random sample, systematic sample, stratified sample by year group/tutor group)
(a) B1 valid reason linked to representativeness or avoiding bias, matched to the method given
(a) Answer: e.g. A simple random sample, using a random number generator on the full list of students, so that every student has an equal chance of being chosen and the sample is not biased.
(b) B1 9 cao
(b) Answer: 9
(c) M1 5+6+6+7+8+9+10+12+15+20+56 (=154), oe divided by 11
(c) A1 14 cao
(c) Answer: 14
(d) B1 mean (14) is greater than the median (9)
(d) B1 so the data is positively skewed, which supports the statement: most journey times are short, with one or two much longer journeys stretching a tail to the right and pulling the mean up
(d) Answer: Mean (14) > median (9), so the data is positively skewed - this supports the statement, since most journeys are short but one long journey (56 minutes) pulls the mean up.

Question 14

(a) B1 8 cao
(a) Answer: 8
(b) B1 8 cao
(b) Answer: 8
(c) M1 3+5+6+7+8+8+9+10+11+12+61 (=140), oe divided by 11
(c) A1 12.7 cao (awrt 12.7)
(c) Answer: 12.7
(d) B1 Positively skewed - mean (12.7) is greater than the median and mode (both 8), since the one much older grandparent's age (61) is an extreme value pulling the mean upward
(d) Answer: Positively skewed

Question 15

(a) B1 2 cao
(a) Answer: 2
(b) B1 2 cao
(b) Answer: 2
(c) M1 attempt to find sum of f x x (allow one slip)
(c) M1 their sum(fx) divided by 30 (dep on previous method mark)
(c) A1 2.47 cao (awrt 2.47)
(c) Answer: 2.47
(d) B1 Positively skewed - mean (2.47) is greater than the median and mode (both 2), so there is a tail of higher values (pupils who read more books) pulling the mean up
(d) Answer: Positively skewed

Question 16

(a) M1 12 and 4 (median-LQ=12, UQ-median=4), oe
(a) A1 Negative skew (the larger gap is below the median than above it) cao
(a) Answer: Negative skew
(b) M1 4 and 18 (median-LQ=4, UQ-median=18), oe
(b) A1 Positive skew (the larger gap is above the median than below it) cao
(b) Answer: Positive skew

Question 17

B1 the median only measures the middle position of the data, so it says nothing about how the rest of the data is spread out; two data sets can share a median but have different spread above/below it
B1 for P (positive skew), the box plot would show a bigger gap between the median and the upper quartile than between the lower quartile and the median (upper half more spread out / tail towards higher times)
B1 for Q (negative skew), the box plot would show the opposite: a bigger gap between the lower quartile and the median than between the median and the upper quartile (lower half more spread out / tail towards lower times)
Answer: The median only fixes the middle position, not the spread, so two data sets can share it while differing in shape: P's box plot would have a bigger gap between median and UQ (tail to the right), while Q's box plot would have a bigger gap between LQ and median (tail to the left).

Question 18

(a) B1 Yes - mean (52) < median (60) is consistent with negative skew cao
(a) Answer: Yes, negative skew
(b) M1 60-38=22 and 65-60=5, oe
(b) A1 22 > 5, confirming negative skew (the lower half of the data is more spread out), which agrees with the student's conclusion
(b) Answer: Confirmed: negative skew (22 > 5), which agrees with the student.
(c) B1 e.g. using two methods checks that they agree, which reduces the risk of being misled by a single unusual value or measure (oe)
(c) Answer: Checking two methods lets you confirm they agree, reducing the risk that one unusual value or measure alone gives a misleading conclusion.

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