Statistics 1
Exploring data
Chapter assessment
1. George records the time he spends per day surfing the internet for the first three
weeks of May. The times, given to the nearest minute, are as follows:
0
24
0
26
61
73
13
16
21
5
10
17
18
26
16
12
15
42
35
0
32
(i) Illustrate the data using a sorted stem and leaf diagram with eight stems.
Comment briefly on the shape of the distribution.
[3]
(ii) Find the mode, median and mean, commenting on their relative usefulness as
measures of central tendency for this data set.
[5]
(iii) Calculate the standard deviation and hence find any outliers.
[4]
(iv) George’s Dad claims that he is spending too much time on the internet. He
tells George to reduce his usage so that the mean daily time for May is 20%
less than the current mean.
What is the maximum total time George can spend surfing the internet for the
remaining 10 days of May?
[3]
2. Over a period of time, a teacher recorded the number of time, x, each of the 20
students in the mathematics class was absent. The distribution was as follows.
Number of times
absent, x
0
1
2
3
4
5
6
7
8
9
10
11 or
more
Number of students, f
4
6
3
2
0
2
0
1
1
0
1
0
 f  20,  fx  53,  fx
2
 299
(i) Illustrate the data using a suitable diagram.
[2]
(ii) State the mode and find the median for the data set.
[2]
(iii) Calculate the mean and the standard deviation of the data set.
[3]
During this period of time, there were 30 mathematics lessons. The teacher needs
to analyse the distribution of the number of times each student was present during
the 30-lesson session.
(iv) Without creating a new frequency distribution, deduce values for the mean
and standard deviation of the numbers of times students were present.
Describe the shape of the new distribution.
[3]
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There are 12 boys and 8 girls in the class. The mean of the numbers of times boys
were absent was 3, and the standard deviation was also 3.
(v) Show that the mean of the numbers of times girls were absent is 2.125.
[2]
(vi) Find the standard deviation of the numbers of times girls were absent.
[3]
3. A bookshop has a selection of 60 different mathematics books. The number of
chapters contained in these books are summarised in the table.
Number of chapters
Number of books
3–5
6–8
9 – 11
12 – 16
17 – 21
22 – 30
4
6
12
14
14
10
(i) Calculate estimates of the mean and standard deviation of the number of
chapters per book.
[5]
(ii) Why is it not possible to obtain exact values for the mean and standard
deviation from the data in the table?
[1]
In fact, the exact values of the mean and standard deviation of the number of
chapters per book are 14.7 and 6.1. Use these exact values for the remainder of the
question.
(iii) For each of the two statements below, state with reasons whether it is
definitely true, definitely false or possibly true.
Statement 1
Statement 2
There is at least one outlier below the mean.
There is at least one outlier above the mean.
[3]
[3]
The number of pages, p, in a book is modelled by
p  20 x  15
where x is the number of chapters in the book.
(iv) Calculate the mean and standard deviation of the number of pages, as given
by this model.
[3]
4. The first paragraph of the children’s book Stig of the Dump contains 107 words.
The number of letters per word is summarised in the following table.
Word length (x)
1
2
3
4
5
6
7
8
9
10
Frequency (f)
2
15
30
23
14
12
6
1
3
1
 f  107,  fx  443,  fx
2
 2183 .
(i) Illustrate the distribution of word lengths by a suitable diagram.
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(ii) State the mode, median and mid-range of the data. What feature of the
distribution accounts for the different values of these quantities?
[4]
(iii) Calculate the mean and standard deviation of the data.
Hence identify the outliers and discuss briefly whether or not they should be
excluded from the sample.
[6]
A passage of an adult fiction book was analysed in a similar way. The mean
number of letters per work was 5.07 and the standard deviation was 2.62.
(iv) Compare the word lengths in the two passages of writing, commenting briefly
on the differences.
[3]
Total 60 marks
Exploring data
Solutions to Chapter assessment
1. (i)
0
10
20
30
40
50
60
70
0
0
1
2
2
0 0 5
2 3 5 6 6 7 8
4 6 6
5
Key: 20 | 6 means
26 minutes
1
3
The distribution has a positive skew.
(ii) Mode = 0
Median is 11th item = 17
Mean 
462
 22
21
The mode is not very useful as it is not representative of the data. Most of
the values appear only once.
The mean is skewed by two unusually large values.
The median is the most representative of the data.
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(iii) S xx   x 2  nx 2  17220  21  22 2  7056
Standard deviation 
S xx

n 1
7056
 18.78 minutes (2 d.p.)
20
Outliers are more than 2 standard deviations from the mean
i.e. greater than 22  2  18.78  59.57
or less than 22  2  18.78  15.57
The outliers are therefore 61 and 73.
(iv) Target mean  0.8  22  17.6
Maximum total for the whole of May  17.6  31  545.6
Maximum total time available for the last 10 days of May
 545.6  462  83.6 minutes
2. (i)
frequency
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9 10
Number of absences
(ii) Mode = 1
Median is halfway between 10th and 11th values which are 1 and 2
Median = 1.5
(iii) Mean 
 fx
n

53
 2.65 days
20
S xx  fx 2  nx 2  299  20  2.65 2  158.55
S xx
158.55

 2.89 days (3 s.f.)
Standard deviation 
n 1
19
(iv) Number of times present = p, so p  30  x
p  30  x  30  2.65  27.35
Standard deviation = 2.89 (3 s.f.) as before.
The new distribution is negatively skewed.
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(v) Total number of absences in class = 53
Total number of times boys were absent  3  12  36
Total number of times girls were absent  53  36  17
17
Mean number of times girls were absent 
 2.215 .
8
S xx
3
(vi) For boys:
11
 S xx  99
99   fx 2  12  3 2
fx  299
fx  299  207  92
S xx  fx  nx  92  8  2.125
2
Total value of
2
For girls:
2
Standard deviation 
3. (i)
Number of chapters
Mid-interval value x
x²
Number of books, f
fx
fx²
Mean 
  fx 2  207
 fx
n

2
2
 55.875
S xx
55.875

 2.83 days (3 s.f.)
n 1
7
3–5
4
16
4
16
64
6–8 9–11
7
10
49 100
6
12
42 120
294 1200
12–16
14
196
14
196
2744
17–21
19
361
14
266
5054
22–30
26
676
10
260
6760
Total
60
900
16116
900
 15 chapters
60
S xx  fx 2  nx 2  16116  60  15 2  2616
S xx
2616

 6.66 chapters (3 s.f.)
Standard deviation 
n 1
59
(ii) The raw data is not available, so each piece of data in a particular interval
is taken to be the mid-interval value.
(iii) Outliers are more than two standard deviations from the mean, i.e. below
2.5 or above 26.9.
Statement 1 is definitely false, since the lowest class interval is 3 – 5.
Statement 2 is possibly true. There may be one or more data items in the
22 – 30 class interval which are above 26.9, but it is not possible to be
certain, since all the items in this class could be less than 26.9.
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(iv) p  20 x  15
p  20 x  15
 p  20 x  15  20  14.7  15  309
 s p  20s x  20  6.1  122
The mean number of pages is 309 and the standard deviation is 122.
4. (i)
Frequency
30
25
20
15
10
5
1
2
3
4
5
6
7
8
9
10
Number of letters
(ii) Mode = 3
Median is 54th word, which is 4
Midrange 
1  10
 5.5
2
The distribution is positively skewed, which is why the mode and median
are smaller than the mid-range.
(iii) Mean 
fx
n

443
 4.14 letters (3 s.f.)
107
S xx  fx 2  nx 2  2183  107  4.142  348.897
348.897
S xx

 1.81 letters (3 s.f.)
Standard deviation 
n 1
106
Outliers are more than 2 standard deviations from the mean.
In this case outliers are less than 0.5 or greater than 7.8, i.e, words of 8
letters or more. So the words of 8, 9 or 10 letters are outliers.
These should not be excluded from the sample since they are valid items of
data. There is no reason to exclude them.
(iv)The adult fiction book has a greater mean word length and also a greater
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Statistics 1
standard deviation, so the adult fiction book has in general longer words
and a wider variety of word lengths. It would be expected that an adult
book would have a greater proportion of long words.
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