17
Averages and spread
CHAPTER
17.1 Mean, mode and median The heights, in metres, of the 11 football players are 1.78 1.83
1.68 1.97
1.88 1.81
1.82 1.88
1.80 1.74
2.05
What is the average height of these 11 football players? In statistics there are three different measures of average, the mean, the mode and the median. To find the mean height, add the eleven heights together and divide by 11 1.78 1.68 1.88 1.82 1.80 2.05 1.83 1.97 1.81 1.88 1.74 The mean height 11 20.24 11 1.84 metres In general, to find the mean of a set of numbers sum of all the numbers mean how many numbers there are x This can be written as x n where x is the mean, x is the sum of all the numbers and n is the number of numbers. The mode of a set of numbers is the number that occurs most often. The mode of the heights (modal height) of the football team is 1.88 metres. To find the median of a set of numbers, list the numbers in order; the median is then the middle number. If there are two middle numbers the median is halfway between the two numbers. In increasing order the heights are 1.68
1.74
1.78
1.80
1.81
1.82
1.83
1.88
1.88
1.97
2.05
The median height is 1.82 metres. In general the median is the 12(n 1)th number. Here, n 11, so 12(11 1) 6th number. The 6th height is 1.82 metres.
Example 1 In one week a salesman makes 9 journeys from his office. Here is the length, in kilometres, of each of the journeys. 120
135
117
10
140
150
50
130
28
a Find the median length of journey. b Find the mean length of journey. Give your answer to an appropriate degree of accuracy. 263
Averages and spread
CHAPTER 17
Solution 1 a 10 28 50 117 120 130 135 140 150
List the numbers in order and select the middle number.
Median length of journey 120 km x Mean =
120 135 117 10 140 150 50 130 28 b 9
n
880 97.77777 … 9
97.77777 … km cannot be measured so is not an appropriate answer.
Mean length of journey 98 km (to the nearest km)
All the given lengths are whole number of kilometres so an appropriate degree of accuracy is to the nearest km.
Some calculators can be used to calculate the mean directly. Check that you know how to calculate the mean using your own calculator.
Example 2 Seven people work in an office. The table shows their earnings last year. a Work out the mean earnings last year. b Find the median earnings last year. c Leaving out Julian’s earnings, work out, for the remaining six office workers, i their mean earnings ii their median earnings.
Name
Earnings last year
Arthur
£12 000
Bob
£ 8 500
Bradley
£30 000
Jim
£11 000
Julian
£73 000
Pamela
£29 500
Tracey
£11 000
Solution 2
12 000 8500 30 000 11 000 73 000 29 500 11 000 a Mean earnings 7 175 000 £25 000 7 b List the seven earnings in order £8500
£11 000
£11 000
£12 000
£29 500
£30 000
£73 000
Median earnings £12 000 c
12 000 8500 30 000 11 000 29 500 11 000 i Mean earnings 6 102 000 £17 000 6 median ii List the six earnings in order £8500
£11 000
£11 000 £12 000
£29 500
£30 000
Find the number which is halfway between 11 000 and 12 000 11 000 12 000 23 000 11 500 2 2 So the median earnings £11 500
264
11 500 is the mean of the middle two numbers.
17.1 Mean, mode and median
CHAPTER 17
Look at the means for the two sets of earnings in part a and part c i. One extreme value (£73 000) has a big effect on the mean but little effect on the median. The median can be used for average earnings to avoid this effect.
Example 3 The mean height of a group of eight girls is 1.56 m. When another girl joins the group the mean height is 1.55 m. Work out the height of this girl.
Solution 3
x Mean height so n
Mean height n x sum of all the heights 1.56 8 12.48
Find the sum of the eight heights.
1.55 9 13.95
Find the sum of the nine heights.
13.95 12.48 1.47
Find the difference between these totals.
Height of the ninth girl 1.47 m.
Exercise 17A 1 Here is a list of five numbers. 2 6 3 7 2 a Write down the mode. b Find the median.
c Work out the mean.
2 Here is a list of six numbers. 2 5 3 7 2 11 a Find the median. b Work out the mean. 3 The list shows the number of cars sold at a garage in the last ten days. 3 2 7 8 4 9 7 5 7 3 a Write down the mode. b Find the median. c Work out the mean number of cars sold per day.
4 A rugby team plays 12 games. Here are the number of points they scored. 24 10 23 16 12 8 19 23 16 37 16 27 a Write down the mode. b Work out the mean number of points per game. 5 Here are the lengths, in centimetres, of five used matchsticks. 2.7 2.8 3.0 3.2 2.8 a Work out the mean length of these matchsticks. b The mean length of ten other matchsticks is 3.0 cm. Find the total of the lengths of these ten matchsticks. 6 Five people work in a canteen. The table shows their earnings last year. a Work out the mean earnings last year. b Find the median earnings last year. c Pamela is the canteen manager. How much more were Pamela’s earnings than the mean earnings of the remaining four workers?
Name
Earnings last year
Leanne
£2500
Mike
£3200
Nazia
£5800
Owen
£4100
Pamela
£22 400 265
Averages and spread
CHAPTER 17
7 The mean weight of a group of five boys is 56 kg. a Work out the total weight of these five boys. When a sixth boy joins the group the mean weight is 58 kg. b Work out the weight of the sixth boy. 8 The mean number of runs scored by a cricketer in his last 10 innings is 47.3 Work out the number of runs that the cricketer must score in the next innings for the mean to be exactly 50. 9 Ted had 20 DVDs. The mean playing time for these 20 DVDs was 145 minutes. Ted gave away 4 of his DVDs. The mean playing time of the 16 DVDs left was 152 minutes. Work out the mean playing time of the 4 DVDs that Ted gave away. 10 8 men and 5 women work in an office. The mean weekly wage of the men is £338 The mean weekly wage of the women is £289 Work out the mean weekly wage of all 13 workers.
17.2 Using frequency tables to find averages Example 4 A dice is rolled ten times. The frequency table shows the scores. a Write down the modal score. b Find the median for the ten scores. c Find the mean for the ten scores.
Solution 4 a Modal score 2
Score
Frequency
1
1
2
3
3
2
4
2
5
1
6
1
Modal score is the score with the greatest frequency.
1 2
b Median (10 1) 5.5 The median will be halfway between the 5th and 6th scores.
Median 12 (n 1)th number. From the frequency table the scores in order are 1 2 2 2 3 3 4 4 5 6
5th score is 3, 6th score is 3 Median 3 c
Score (x)
Frequency ( f )
1
1
2
3
3
2
4
2
5
1
6
1 f 10
Mean 3120 3.2
fx 1 6 6 8 5 6 fx 32
sum of the scores Mean number of scores
x is the score; f is the frequency. The total of each score is the score multiplied by its frequency, ( f x) for example, a score of 2 occurs 3 times so the total of 3 lots of 2 is 326 Find fx, the sum of the ten scores, and f, the number of scores.
sum of the ten scores Mean 10
fx In general, for data given in a frequency table, mean f 266
17.2 Using frequency tables to find averages
CHAPTER 17
Example 5 The frequency table shows information about the number of certificates awarded to each student in a class last month. a How many students were in the class? b Work out the total number of certificates awarded. c Work out the mean number of certificates awarded. d Work out the median number of certificates awarded.
Solution 5 a 3 7 3 9 8 30 students in the class.
Find f
b (3 0) (7 1) (3 2) (9 3) (8 4) 0 7 6 27 32 72
Find fx
Number of certificates
Frequency
0
3
1
7
2
3
3
9
4
8
Total number of certificates awarded 72 c 72 30 2.4 Mean number of certificates awarded 2.4
fx Find f
d Median 12 (30 1) 15.5 The median will be halfway between the 15th and 16th numbers.
Median 12 (n 1)th number.
The 15th student received 3 certificates. The 16th student received 3 certificates. The median number of certificates 3
Exercise 17B 1 The table shows the results of rolling a dice 10 times. a Find the median score. b Work out the mean score.
2 The table shows the numbers of cakes sold in a shop to the first 30 customers. a Write down the modal number of cakes sold. b Work out the total number of cakes sold to these 30 customers. c Work out the mean number of cakes sold. d Find the median number of cakes sold.
Number of certificates 0 1 2 3 4
Frequency 3 7 3 9 8
1st to 3rd 4th to 10th 11th to 13th 14th to 22nd 23rd to 30th
Score
Frequency
1
3
2
3
3
1
4
2
5
1
6
0
Number of cakes
Number of customers
0
2
1
9
2
6
3
6
4
5
5
2 267
Averages and spread
CHAPTER 17
3 The table shows the numbers of goals scored by a hockey team in each of 25 matches. a Write down the mode of the number of goals scored. b Work out the mean number of goals scored per match.
4 The table shows the numbers of planes landing at a small airport during each hourly period yesterday. a Work out the total number of planes that landed at the airport yesterday. b Work out the mean number of planes landing per hour.
5 Mrs Fox did a survey of the number of books each pupil in her class borrowed from the library last month. The frequency table shows her results. a Work out the number of pupils that Mrs Fox asked. b Ben thinks that the average number of books borrowed in this survey is 6 Explain why Ben cannot be correct. c Find the median number of books borrowed. d Work out the mean number of books borrowed. 6 A wedding photographer recorded the number of weddings he attended each week last year. The table shows his results. a Find the median number of weddings he attended per week. b Find the mean number of weddings he attended per week. c The photographer is paid £250 for each wedding he attends. Work out the total amount the photographer was paid last year.
Number of goals
Frequency
0
9
1
5
2
5
3
4
4
2
Number of planes
Frequency
0
5
1
0
2
1
3
2
4
4
5
12
Number of books borrowed
Frequency
0
4
1
6
2
6
3
8
Number of weddings
Frequency
0
10
1
14
2
4
3
2
4
13
5
5
6
3
7
1
17.3 Range and interquartile range In statistics, range is a measure of how spread out numerical data is. To find the range of a set of numbers, work out the difference between the highest number and the lowest number. The heights, in metres, of 11 football players listed in order are 1.68 1.74 1.78 1.80 1.81 1.82 1.83 1.88 1.88 1.97 2.05 The range of these heights is 2.05 1.68 0.37 metres. The median height is the 6th ( 12 (11 1)th) height, which is 1.82 metres. 268
17.3 Range and interquartile range
CHAPTER 17
The median is the value that is halfway through the data. The lower quartile is the value that is a quarter of the way through the data. The upper quartile is the value that is three-quarters of the way through the data. For n numbers listed in order: ● the median is the 12 (n 1)th number ● the lower quartile is the 14 (n 1)th number ● the upper quartile is the 34 (n 1)th number. It is sometimes useful to know how spread out values are over the middle 50% of data. This is called the interquartile range. interquartile range upper quartile lower quartile For the 11 heights in order, 1.68
1.74
1.78
1.80
1.81
1.82
1.83
1.88
1.88
1.97
2.05
lower quartile is the 14 (11 1) 3rd number 1.78 upper quartile is the 34 (11 1) 9th number 1.88
The The Interquartile range 1.88 1.78 0.1 metres.
Example 6 The table shows information about the ages, in years, of junior members of a tennis club. a Find the range of their ages. b Find the interquartile range of their ages.
Solution 6 a 15 9
Age in years
Frequency
9
30
10
40
11
19
12
38
13
11
14
18
15
13
Range highest number lowest number.
Range 6 years b
Frequency table gives ages in order.
Lower quartile is the 14 (169 1) 4212th number. The 42nd age is 10, the 43rd age is 10
Lower quartile is 14 (n 1)th number n f 169
Lower quartile 10 Upper quartile is the 34 (169 1) 12712th number. The 127th age is 12, the 128th age is 13
Upper quartile is 34 (n 1)th number.
Upper quartile 12.5
Halfway between 12 and 13
12.5 10 2.5
Interquartile range upper quartile lower quartile.
Interquartile range 2.5 years 269
Averages and spread
CHAPTER 17
17.4 Stem and leaf diagrams A stem and leaf diagram is a diagram that shows data in a systematic way. The mode, median and range of a set of data can be found easily from a stem and leaf diagram. The ages, in years, of 11 people are 12
9
20
24
15
17
31
4
15
17
28
Here is a stem and leaf diagram showing these ages. The key is part of the stem and leaf diagram. Stem
Leaves
0
4
9
1
2
5
5
2
0
4
8
3
1
7
In this case, the key makes it clear that the stem shows the tens and the leaves show the units. The data is written so that every number has a tens and units value: 12 09 20 24 15 17 31 04 15 17 28 The data is ordered from 0 | 4 ( age 4) to 3 | 1 ( age 31)
7 Key 1 | 2 means age 12
Example 7 Here is a stem and leaf diagram showing the ages, in years, of 15 office workers. Find a the range of the 15 ages b the median age.
Solution 7 a Range 47 16 31 years b The median age is 29 years.
1
6
7
9
2
1
2
5
5
3
0
4
5
8
4
1
7
9
9 Key 1 | 6 means age 16
The data is ordered from 1 | 6 ( age 16) to 4 | 7 ( age 47) The median is the eighth age which is 2 | 9 (The middle of 15 numbers is the eighth number.)
Example 8 Fifteen boys are timed over a 10 metre sprint. Here are their times to the nearest tenth of a second. 2.6 4.0
3.8 2.4
3.0 3.7
4.7 2.7
4.1 5.1
2.1 2.8
3.1 4.7
3.9
a Draw a stem and leaf diagram to show these results. b Use your stem and leaf diagram to find i the range of the times ii the median time iii the interquartile range.
Solution 8 a
270
2
1 4
6
7
8
3
0 1
7
8
9
4
0
7
7
5
1
1
Key 3 | 9 means 3.9 seconds
17.4 Stem and leaf diagrams
b
CHAPTER 17
i 5.1 2.1
Range highest number lowest number.
Range 3.0 seconds ii
Stem and leaf diagram gives times in order.
Median 12 (15 1)th number 8th number
Median 12 (n 1)th number. n total number of boys 15
Median 3.7 seconds
3 | 7 3.7 seconds.
iii Lower quartile is 14 (15 1)th 4th number
Lower quartile is 14 (n 1)th number.
Lower quartile 2.7
2 | 7 2.7 seconds.
Upper quartile is 34 (15 1) 12th number
Upper quartile is 34 (n 1)th number.
Upper quartile 4.1
4 | 1 4.1 seconds.
4.1 2.7 1.4
Interquartile range upper quartile lower quartile.
Interquartile range 1.4 seconds Stem and leaf diagrams can be used to compare two sets of data. In these cases two diagrams are combined to make a back-to-back stem and leaf diagram.
Example 9 David chooses a sample of 30 girls and 30 boys in Year 10 to compare their weights. This back-to-back stem and leaf diagram shows his results. Girls
Boys
9 9 8 7 7 0
4
9 9 9 8 7 7 6 4 4 3 2 1
5
6 7 7 8 9
8 7 6 6 6 5 4 4 3 2
6
0 1 1 2 4 5 6 6 6 6 8 9
2 1
7
0 0 1 2 2 3 4 4 5 6 7
8
0 1
Key 5 | 1 means 51 kg
Compare the weights of the girls and boys by finding the range and the median of each distribution.
Solution 9 For the girls, the data is ordered from 0 | 4 ( 40 kg) to 2 | 7 ( 72 kg) The range 72 40 32 kg. For the boys, the data is ordered from 5 | 6 ( 56 kg) to 8 | 1 ( 81 kg) The range 81 56 25 kg. This shows that the weights of the girls are more spread out. The median is in the middle of the 30 weights for each distribution. This is halfway between the 15th weight and 16th weight. For the girls, the 15th weight is 8 | 5 which is 58 kg the 16th weight is 9 | 5 which is 59 kg The median for the girls 58.5 kg
For the boys, the 15th weight is 6 | 6 which is 66 kg the 16th weight is 6 | 8 which is 68 kg The median for the boys 67 kg
This shows that in general the boys are heavier than the girls. Look back at the stem and leaf diagram. The distribution of the weights of the boys is towards the higher values. 271
Averages and spread
CHAPTER 17
Exercise 17C 1 The weights, in grams, of nine potatoes are 262
234
208
248
239
210
206
227
254
a Find the range of the weights of these potatoes. b Find the interquartile range of the weights of these potatoes. 2 In an experiment some people were asked to estimate the length, in centimetres, of a piece of string. The frequency table shows their estimates. a Find the range of their estimates. b Find the interquartile range of their estimates.
3 The stem and leaf diagram shows the number of minutes taken by each student in a class to complete a puzzle. a Find the number of students in the class. b Write down how many students took 24 minutes to complete the puzzle. c Write down how many students took 10 minutes longer than the quickest student to complete the puzzle. d Find the range of the times. e Find the median time. f Find the interquartile range.
Length (cm)
Frequency
11
1
12
2
13
4
14
18
15
16
16
14
17
20
18
18
19
0
20
2
0
8
9
9
1
0
1
3
3
8
8
9
2
1
4
4
4
6
6
9
3
0
2
3
4
7
7
8
4
1
8
1 2 2 4 Key 4 | 1 means for 41 minutes
4 Here are the number of minutes a sample of 15 patients had to wait before seeing a hospital doctor. a Draw a stem and leaf diagram to show this information. b Use your stem and leaf diagram to find i the range of the times ii the median time. c Find the interquartile range.
49 10 25 39 14
23 28 45 35 48
34 28 20 15 10
5 Tony records the number of emails he receives each day. Here are his results for the last 20 days. 178 171 189 147
189 153 166 158
147 171 165 148
147 164 155 152
166 158 152 172
a Draw a stem and leaf diagram to show this information. b Use your stem and leaf diagram to find the median number of emails. c Find the interquartile range. 272
17.5 Estimating the mean of grouped data
CHAPTER 17
6 The table gives the number of days each of 25 people had been on holiday in Spain. 0 30 21
7 9 14
14 10 25
38 21 31
26 25 30
16 22 7
13 7 7
21 15
35 7
a Draw a stem and leaf diagram to show this information. b Use your stem and leaf diagram to find the median number of days. c Find the interquartile range. 7 Nicki weighs 20 parcels. Here are her results in kilograms. 0.7 3.3
1.6 2.6
2.3 1.6
3.4 1.1
2.8 2.7
1.7 2.7
1.5 1.8
1.1 2.0
1.4 0.9
0.8 3.0
a Draw a stem and leaf diagram to show this information. b Use your stem and leaf diagram to find the range of the weights. c Use your stem and leaf diagram to find the median weight. 8 The back-to-back stem and leaf diagram shows the percentage marks of a group of boys and girls in a science test. Girls Boys 9 7 6 1
3
9 8 7 5 5 0
4
0 2 8
9 9 8 8 7 7 3 3 2 1 1
5
2 3 3 8
6 5 5 4 3 2 0
6
0 1 1 2 3 5 5 6 6 7 8 9
2 1
7
0 0 1 2 2 3 4 4
8
0 1 6
Key 5 | 2 means 52%
a Find the range of percentage marks for
i the boys
ii the girls.
b Find the median of percentage marks for
i the boys
ii the girls.
c Compare and comment on the marks for the boys and girls.
17.5 Estimating the mean of grouped data This table appeared in the Burwich Guardian. It shows some information about the number of road accidents in Burwich each day last September. There were 7 days on which there were fewer than 5 accidents. It is impossible to tell from the table the exact number of accidents on each day so an exact value for the average number of accidents per day cannot be found.
Number of road accidents
Number of days
0 to 4
7
5 to 9
14
10 to 14
8
15 to 19
1
It is possible to find an estimate for the mean number of accidents per day. We use the middle of 04 each class interval. For example, we assume that there were 2 accidents on each of the 2 7 days. This gives a total of 7 2 14 accidents. 273
Averages and spread
CHAPTER 17
Example 10 The table shows some information about the number of road accidents in Burmage last April. Number of road accidents
Number of days
0 to 4
7
5 to 9
14
10 to 14
8
15 to 19
1
Work out an estimate for the mean number of accidents per day in Burmage last April.
Solution 10 Number of road accidents
Number of days ( f )
0 to 4
7
5 to 9
9
10 to 14
12
15 to 19
2
Middle of class interval (x) 04 2 2 59 7 2 10 14 12 2 15 19 17 2
Totals of the numbers of accidents ( fx) 7 2 14 9 7 63 12 12 144 2 17 34
The ‘number of days’ is the frequency, f. Let the middle of class interval be x. The total number of accidents in April is fx 14 63 144 34 255 The total number of days in April is f 7 9 12 2 30 fx 255 Estimated mean 8.5 accidents per day. 30 f
Example 11 The table shows some information about the annual earnings of 140 employees of a company. Annual earnings (£P)
Frequency
10 000 P 10 000
30
10 000 P 20 000
42
20 000 P 30 000
28
30 000 P 40 000
20
40 000 P 50 000
18
50 000 P 60 000
2
10 000 P 20 000 means earnings above £10 000 up to and including £20 000 (See Section 15.1) a Work out an estimate for the mean earnings. b Find the class interval that contains the median earnings. 274
17.5 Estimating the mean of grouped data
CHAPTER 17
Solution 11 a
Annual earnings (£P)
Frequency (f)
Middle of class interval (x)
Totals of earnings ( fx)
10 000 P 10 000
30
5 5 000
5 000 30 150 000
10 000 P 20 000
42
15 000
15 000 42 630 000
20 000 P 30 000
28
25 000
25 000 28 700 000
30 000 P 40 000
20
35 000
35 000 20 700 000
40 000 P 50 000
18
45 000
45 000 18 810 000
50 000 P 60 000
2
55 000
55 000 2 110 000
f 140
fx 3 100 000
fx 3 100 000 Estimated mean £22 143 (to the nearest £). 140 f b 12(140 1) 70.5 The median is in the middle of the 70th and the 71st earnings. Annual earnings (£P)
Frequency
10 000 P 10 000
30
1st to the 30th earnings
10 000 P 20 000
42
31st to 72nd earnings
Both the 70th and the 71st earnings lie in the class interval 10 000 P 20 000 So the median lies in the class interval 10 000 P 20 000
Exercise 17D 1 Twenty people took part in a competition. The points scored are grouped in the frequency table. a Find the class interval which contains the median. b Work out an estimate for the mean number of points scored.
2 The table shows information about the number of minutes that 125 students spent doing homework yesterday. a Find the class interval which contains the median. b Work out an estimate for the mean number of minutes that the students spent doing homework yesterday.
Points scored
Number of people
1 to 5
1
6 to 10
2
11 to 15
2
16 to 20
6
21 to 25
7
26 to 30
2
Number of minutes (t)
Frequency (f)
0 t 20
10
20 t 40
20
40 t 60
30
60 t 80
35
80 t 100
25
100 t 120
5 275
Averages and spread
CHAPTER 17
3 The table shows some information about the lifetimes, in hours, of 50 light bulbs. a Find the class interval which contains the median. b Work out an estimate for the mean number of hours.
4 The table shows some information about the number of text messages Simon received each day in December. a Find the class interval which contains the median. b Work out an estimate for the mean number of messages received. Give your answer correct to one decimal place.
5 Jack grows onions. The table shows some information about the weights (w) of some onions. a Work out an estimate for the mean weight of these onions. Give your answer correct to the nearest gram. b Find the class interval that contains the median.
Number of hours (t)
Number of light bulbs
0 t 50
2
50 t 100
3
100 t 150
6
150 t 200
9
200 t 250
19
250 t 300
11
Number of text messages
Frequency (f)
0 to 4
3
5 to 9
11
10 to 14
12
15 to 19
3
20 to 24
0
25 to 29
2
Weight (w grams)
Frequency ( f )
0 w 40
10
40 w 60
16
60 w 80
25
80 w 100
28
100 w 120
17
120 w 150
8
17.6 Moving averages The table shows the number of units of gas used in a village, in each season, over a three-year period. Season
Year 1
Year 2
Year 3
Winter (W)
5900
5500
5100
Spring (Sp)
2400
2000
1600
Summer (Su)
1600
1200
1200
Autumn (A)
5700
5300
4900
This information can be plotted on a graph. The graph shows the changes in the number of units of gas used over the three-year period. It is called a time series graph. A time series is a set of readings taken over a period of time. In this time series graph there are high points, showing the number of units of gas used in Winter, and low points, showing the number of units of gas used in Summer. These are seasonal variations in the gas used. 276
6000 5000 4000 Units of 3000 gas 2000 1000
W Sp Su A W Sp Su A W Sp Su A Year 1 Year 2 Year 3
17.6 Moving averages
CHAPTER 17
Has the amount of gas used in the village increased or decreased over this three-year period? This cannot be answered easily from the graph. The question can be answered using the following calculations: The mean of the first four values (W, Sp, Su and A of Year 1) 5900
2400
1600
5700
5500
2000
1200
5300
5100
1600
1200
4900
1600
1200
4900
1600
1200
4900
5900 2400 1600 5700 15 600 3900 units of gas. 4 4 The mean of the four values (Sp, Su and A of Year 1 and W of Year 2) 5900
2400
1600
5700
5500
2000
1200
5300
5100
2400 1600 5700 5500 15 200 3800 units of gas. 4 4 The mean of the 4 values (Su and A of Year 1 and W and Sp of Year 2) 5900
2400
1600
5700
5500
2000
1200
5300
5100
is then found, and so on. These are called moving averages. Since 4 consecutive values are used in each case, they are called 4-point moving averages. The moving averages are then plotted on the time series graph, and joined. For example the point representing the 1st moving average is plotted halfway between Spring and Summer of Year 1 at a height of 3900.
6000 5000 3900 4000 Units of 3000 gas
It is clear from this graph that the number of units used in the village over the three-year period decreases. This is called the trend of the graph.
2000 1000 middle point W Sp Su A W Sp Su A W Sp Su A Year 1 Year 2 Year 3
Example 12 The table shows the attendances at a cinema during each day of one week.
Number of people
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
170
380
530
560
500
1250
1160
a Find the set of 3-point moving averages for this information. b Comment on the trend of the number of attendances throughout the week. 277
Averages and spread
CHAPTER 17
Solution 12 170 380 530 1080 a 360 2 3 380 530 560 1470 490 3 3 530 560 500 1590 530 3 3 560 500 1250 2310 770 3 3 2910 500 1250 1160 970 3 3 b Attendances increase throughout the week.
Find the mean of Mon, Tues and Wed attendances. Find the mean of Tues, Wed and Thurs attendances. Find the mean of Wed, Thurs and Fri attendances. Find the mean of Thurs, Fri and Sat attendances. Find the mean of Fri, Sat and Sun attendances. Compare 360, 490, 530, 770, 970
Exercise 17E 1 The table shows the number of tyres fitted each month by a local garage. Month Number of tyres fitted
May
June
July
Aug
Sept
Oct
Nov
Dec
32
27
34
19
24
16
12
23
Work out the first two 4-month moving averages for this data. 2 The table shows the number of books sold in a shop each month from January to June.
Jan
Feb
Mar
Apr
May
Jun
149
156
154
161
159
169
Work out the 3-month moving averages for this information. 3 A shop sells televisions. The table shows the number of televisions sold in every quarter in a two-year period. a Calculate the set of 4-point moving averages for this data. b What do your moving averages in part a tell you about the trend in the sale of televisions?
Year 1
Year 2
Months
Number of televisions sold
Jan–Mar
46
Apr–Jun
23
Jul–Sep
36
Oct–Dec
71
Jan–Mar
54
Apr–Jun
31
Jul–Sep
44
Oct–Dec
71
4 The table shows the total charges, in £, Season for gas used by Mr Smith, in each season, Winter (W) over a three-year period. Spring (Sp) a Plot this information on a time series graph. Summer (Su) b Work out the 4-point moving averages Autumn (A) for this period. c On your time series graph plot the moving averages. d Comment on the seasonal gas charges during this period. 278
Year 1
Year 2
Year 3
218
228
242
163
166
138
68
43
38
179
139
122
Chapter 17 review questions
CHAPTER 17
Chapter summary You should now know that:
for a list of numbers, sum of all the numbers ● the mean how many numbers there are ● the mode is the number which occurs most often ● the median is the middle number when the numbers are written in order ● the range is the difference between the highest and lowest numbers
when data is arranged in order, ● the median is the value that is halfway through the data, ● the lower quartile is the value that is a quarter of the way through the data ● the upper quartile is the value that is three-quarters of the way through the data
interquartile range upper quartile lower quartile and represents the middle 50% of the data. You should also be able to:
find the mean, mode and median of a list of numbers
find the mean, mode and median of data given in a frequency table
find the range and interquartile range of data given in a list or in a frequency table
draw single and back-to-back stem and leaf diagrams and use them to find the median, range and interquartile range
use average and range to compare distributions
estimate the mean of grouped data by using the middle value of each class interval
draw time series graphs from given data
find moving averages and use these to describe the trend for the data over a given time period.
Chapter 17 review questions
1 a Kuldip recorded the numbers of people getting off his tram at 10 stops. Here are his results for Monday. 3 7
5 4
4 9
7 8
4 12
For these 10 numbers, work out iii the range iii the median iii the mean. b On Tuesday, the mean number of people getting off his tram at the same ten stops is 8.6 How many more people got off the tram on Tuesday at these ten stops? 279
Averages and spread
CHAPTER 17
2 Andy did a survey of the number of cups of coffee some pupils in his school had drunk yesterday. The frequency table shows his results. a Work out the number of pupils that Andy asked.
Number of cups of coffee
Frequency
2
1
3
3
4
5
5
8
6
5
Andy thinks that the average number of drinks pupils in his survey had drunk is 7. b Explain why Andy cannot be correct.
(1387 June 2003)
3 Rosie had 10 boxes of drawing pins. Number of She counted the number of drawing pins drawing pins in each box. 29 The table gives information about her results. 30 a Write down the modal number of 31 drawing pins in a box. 32 b Work out the range of the number of drawing pins in a box. c Work out the mean number of drawing pins in a box.
Frequency 2 5 2 1
(1387 June 2003)
4 Amy had 30 CDs. The mean playing time of these 30 CDs was 42 minutes. Amy sold 5 of her CDs. The mean playing time of the 25 CDs left was 42.8 minutes. Calculate the mean playing time of the 5 CDs that Amy sold.
(1387 June 2004)
5 Mary recorded the heights, in centimetres, of the girls in her class. She put the heights in order. 132 167
144 170
150 172
152 177
160 181
162 182
162 182
167
Find a the lower quartile b the upper quartile. 6 Mrs Chowdery gives her class a maths test. Here are the test marks for the girls. 7 5 8 5 a Work out the mode. b Work out the median.
2
8
7
4
7
10
3
7
4
3
6
The median mark for the boys was 7 and the range of the marks for the boys was 4. The range of the girls’ marks was 8. c By comparing the results explain whether the boys or the girls did better in the test. (1385 June 1998)
7 Here are the weights, in kilograms, of 15 parcels. 1.1 1.5
1.7 2.6
2.0 3.5
1.0 2.1
1.1 0.7
0.5 1.2
3.3 0.6
Draw a stem and leaf diagram to show this information. 280
2.0 (1388 March 2004)
Chapter 17 review questions
CHAPTER 17
8 a Ahmed recorded the number of lorries entering a motorway service station every three minutes between 9 am and 10 am yesterday. The stem and leaf diagram shows this information. 0 1 2
Number of lorries 1 3 3 5 7 0 0 2 3 4 6 0 1 1 1 3 5
7 6
Key 1 | 3 means 13 lorries
9
i Find the median number of lorries. ii Work out the range of the number of lorries. Uzma recorded the number of cars entering the motorway service station in each of the same three-minute periods. The list below shows this information. 13 21 17 25 32 19 17 30 31 24 24 14 32 18 19 27 20 30 37 21 b Draw a stem and leaf diagram to show this information. 9 Mark recorded the number of e-mails he received each day for 21 days. The stem and leaf diagram shows this information. 0 1 2
Number of e-mails 4 5 5 6 7 7 0 1 2 3 3 4 0 1 3 6
8 6
9 7
8
Key 2 | 6 means 26 e-mails
a Find the median number of e-mails that Mark received in the 21 days. b Work out the range of the number of e-mails Mark received in the 21 days. c Find the interquartile range of the number of e-mails Mark received in the 21 days. (1388 March 2005)
10 The table shows information about the number of hours that 120 children used a computer last week. Work out an estimate for the mean number of hours that the children used a computer. Give your answer correct to two decimal places.
Number of hours (h)
Frequency
0h2
10
2h4
15
4h6
30
6h8
35
8 h 10
25
10 h 12
5 (1388 June 2005)
11 Charles found out the length of reign of each of 41 kings. He used the information to complete the frequency table. Length of reign (L years)
Number of kings
0 L 10
14
10 L 20
13
20 L 30
8
30 L 40
4
40 L 50
2
Use the copy of the table on the resource sheet. a Write down the class interval that contains the median. b Calculate an estimate for the mean length of reign.
(1387 November 2003)
281
Averages and spread
CHAPTER 17
12 The table shows the number of orders received each month by a small company. Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Number of orders received
23
31
15
11
19
16
20
13
Work out the first two 4-month moving averages for this data. 13 Paul and Carol open a new shop Month in the High Street. Monthly takings (£) The table shows the monthly takings in each of the first four months.
(1388 June 2003)
Jan
Feb
March
April
9375
8907
9255
9420
Work out the 3-point moving averages for this information 14 The table shows information about the quarterly gas bill, in £s, for Samira’s house, over a period of two years.
(1388 November 2005)
Quarter Year
1
2
3
4
1
£200
£162
£80
£130
2
£216
£166
£96
£142
The data has been plotted as a time series.
Quarterly gas bills
a The first three 4-point moving averages are £143, £147 and £148. i Work out the last two 4-point moving averages. ii Plot all five of the moving averages on the copy of the graph on the resource sheet. b What do the moving averages show about Quarterly the trend of the quarterly gas bills? gas bill (1389 June 2005)
(£)
240 220 200 180 160 140 120 100 80 60 40 20 0
1
2 3 4 1 2 3 4 Year 1 Year 2 Years and quarters
15 A youth club has 60 members. 40 of the members are boys. 20 of the members are girls. The mean number of videos watched last week by all 60 members was 2.8 The mean number of videos watched last week by the 40 boys was 3.3 a Calculate the mean number of videos watched last week by the 20 girls. Ibrahim has two lists of numbers. The mean of the numbers in the first list is p. The mean of the numbers in the second list is q. Ibrahim combines the two lists into one new list of numbers.
pq Ibrahim says ‘The mean of the new list of numbers is equal to .’ 2 One of two conditions must be satisfied for Ibrahim to be correct. b Write down each of these two conditions. 282
(1387 November 2004)
