tabs-figs 05 1. - macmillanihe.com resources (by author)/o... · l. oakshott, essential...

L. Oakshott, Essential Quantitative Methods, 5th edn Palgrave Macmillan, 2012.

Figure 5.1Prices and wages:monthly percentagechanges comparedwith previous year

0

3

4

6

5

1

Average weekly earningsexcluding bonuses

2

Mon

thly

per

cent

age

chan

ge

J F M A M J J A S O N D J F M A M J J A S O N2009 2010

Source: ONS.

All items RPI

Table 5.1 Comparison of mean and median

Mean Median

One of the actual data items? Not usually Usually

Equal contribution by all data items? Yes No

Influenced by extreme values (outliers)? Yes No

Easy to calculate? Yes No

Table 5.2 Results of a travel to work survey

Mode of travel Frequency

Car 9

Bus 4

Cycle 3

Walk 2

Train 2

Table 5.3 Ungrouped frequency table

No. of sales Frequency

2 3

3 7

4 9

5 6

6 5

7 2

8 1

Table 5.4 Calculation of the mean

No. of salesx

Frequencyf f x

2 3 6

3 7 21

4 9 36

5 6 30

6 5 30

7 2 14

8 1 8

Total 33 145

Table 5.5 Calculation of the median

x f Cumulative f

2 3 3

3 7 10

4 9 19

5 6 25

6 5 30

7 2 32

8 1 33


Table 5.6 Grouped frequency table

Interval Frequency

30 to under 35 mm 4

35 to under 40 mm 7

40 to under 45 mm 14



55 to under 60 mm 9

60 to under 65 mm 4

65 to under 70 mm 1

70 to under 75 mm 1

75 to under 80 mm 0

80 to under 85 mm 1


Table 5.7 Calculation of the mean and median for a frequency distribution

IntervalMid point (x) Frequency

f fxCumulativefrequency

% cumulativefrequency

30 to under 35 mm 32.5 4 130.0 4 5.00

35 to under 40 mm 37.5 7 262.5 11 13.75

40 to under 45 mm 42.5 14 595.0 25 31.25

45 to under 50 mm 47.5 23 1092.5 48 60.00

50 to under 55 mm 52.5 16 840.0 64 80.00

55 to under 60 mm 57.5 9 517.5 73 91.25

60 to under 65 mm 62.5 4 250.0 77 96.25

65 to under 70 mm 67.5 1 67.5 78 97.50

70 to under 75 mm 72.5 1 72.5 79 98.75

75 to under 80 mm 77.5 0 0.0 79 98.75

80 to under 85 mm 82.5 1 82.5 80 100.00

Total 80 3910.0

Figure 5.2Ogive

0

20

40

60

80

100

% c

umul

ativ

e fr

eque

ncy

10

30

50

70

90

35.025.0 45.0Length of bolt (mm)

55.0 65.0 75.0 85.0 95.0

Median

Table 5.8 A stem and leaf plot of the length of bolt data

Frequency Stem & Leaf4.00 3 . 00037.00 3 . 5556667

14.00 4 . 0112222233444423.00 4 . 55555666666677779999999916.00 5 . 00111222223334449.00 5 . 5566667894.00 6 . 01143.00 Extremes (>=68)

3|3 represents 33 mm

The middle two values


Figure 5.3Graphical method forfinding the mode

0.0

5.0

10.0

15.0

20.0

25.0

Freq

uenc

y

30.0 40.0Length of bolt (mm)50.0 60.0 70.0 80.0 90.0

Modal class

Modal value

Table 5.9 Grouped frequency table with open classes

Class interval Frequency

less than 96 g 5

96 to under 98 g 10

98 to under 99 g 15

99 to under 100 g 27

100 to under 102 g 11

more than 102 g 2

Figure 5.4Median weight for jarsof coffee

0

20

40

60

80

100

% c

umul

ativ

e fr

eque

ncy

10

30

50

70

90

9695 98Weight (gm)

100 102 10497 99 101 103


Figure 5.5Graphical method forfinding theinterquartile range

0

20

40

60

80

100

% c

umul

ativ

e fr

eque

ncy

10

30

50

70

90

30 40Length of bolt (mm)

50 60 70 80 90Q3Q1

Table 5.10 Calculation of the variance for Mike’s sales

Sales Difference Difference squared

x x � 6 ðx � 6Þ2

3 �3 9

2 �4 16

1 �5 25

32 26 676

2 �4 16

1 �5 25

1 �5 25

Total 792

Table 5.11 Calculation of the variance using the modified formula

x x2

3 9

2 4

1 1

32 1024

2 4

1 1

1 1

42 1044

Table 5.12 Calculation of the variance of Janet’s data

x x2

0 0

1 1

4 16

12 144

10 100

7 49

8 64

6 36

Totals 48 410


Figure 5.6Standard deviationremains unchanged

40 8

Sales

12 16 20 24

Janet’s sales

Janet’s sales plus 3

Figure 5.7Standard deviationdoubles

40 8

Sales

12 16 20 24

Janet’s sales

Janet’s sales times 2

Table 5.13 Calculation of standard deviation for a frequency distribution

Interval x f fx x2 fx2

30 to under 35 mm 32.5 4 130.0 1 056.25 4 255.00

35 to under 40 mm 37.5 7 262.5 1 406.25 9 843.75

40 to under 45 mm 42.5 14 595.0 1 806.25 25 827.50

45 to under 50 mm 47.5 23 1092.5 2 256.25 51 893.25

50 to under 55 mm 52.5 16 840.0 2 756.25 44 100.00

55 to under 60 mm 57.5 9 517.5 3 306.25 29 756.25

60 to under 65 mm 62.5 4 250.0 3 906.25 15 625.00

65 to under 70 mm 67.5 1 67.5 4 556.25 4 556.25

70 to under 75 mm 72.5 1 72.5 5 256.25 5 256.25

75 to under 80 mm 77.5 0 0.0 6 006.25 0.00

80 to under 85 mm 82.5 1 82.5 6 806.25 6 806.25

Total 80 3910.0 39 118.75 197 350.00

Figure 5.8Box and whisker plot 4020 60 80 100

Median

Lowerextreme

Upperextreme

4430 85

48.3 54

Q1 Q2 Q3

Lowerquartile

Upperquartile


Table 5.14 Data from the 2001 census on travel to work

Description Count Percentage

Less than 2 km 185 781 25.13

2 km to less than 5 km 168 878 22.84

5 km to less than 10 km 151 639 20.51

10 km to less than 20 km 127 642 17.27

20 km to less than 40 km 65 983 8.92

40 km and over 39 385 5.33

Total 739 308 100.00

Source: 2001 Census Area Statistics. (Census output is Crown copyright and is reproduced with the permission of the Controller ofHMSO and the Queen’s printer for Scotland.)

Figure 5.9Excel worksheet fordistance to work

Figure 5.10Box and whiskerpolot for distanceto work 0

2

50

5 14


Figure 5.11The completedworksheet

Figure 5.12The Excel worksheetwith the formulaeshown


Figure 5.13Dialog box to createsummary statisticsusing SPSS

Click on Statistics

Figure 5.14Summary statisticsfrom SPSS


Figure 5.15The Explore menu

Figure 5.16Creating a simpleboxplot in SPSS

Click on Plots and then Dependents together

Figure 5.17Boxplots for the salesdata Sales in September

Median

Outliers

Sales in August

1

70000

60000

50000

40000

30000

20000

10000

0

8


Figure 5.18Alternative data layout

Figure 5.19Using the chartbuilder for boxplots Simple boxplot

Drag the simple boxplot into the chart preview area,Sales into the Y-axis and Month into the X-axis

Table 5.15 Number of rejects from a production process

Number of rejects Frequency

0 12

1 45

2 36

3 30

4 20

5 5

6 0


Figure 5.20Histogram of hoursworked from the2001 census

0

400000

300000

200000

100000Fr

eque

ncy

0 20 40 60 80 100Hours worked weekly

Table 5.16 Data for Question 9

Days off work Number of employees

Less than 2 days 45

2 to 5 days 89

6 to 9 days 40

10 to 13 days 25

14 to 21 days 5

22 to 29 days 2


Class interval (mm) Frequency

Machine A Machine B

20 to under 22 5 2

22 to under 24 12 5

24 to under 26 26 20

26 to under 28 11 25

28 to under 30 3 8

30 to under 32 0 2


Wages (£s) No. of employees

20 to under 30 2

30 to under 40 4

40 to under 50 6

50 to under 60 12

60 to under 70 11

70 to under 80 9

80 to under 90 1

45



Gross weekly wages (£s) No. of employees

90 to under 110 14

110 to under 140 54

140 to under 160 16

160 to under 200 12

200 to under 240 2

98


Raw material % increase in price Mean expenditure per month (£)

Lead 10 3000

Copper 4 1000

Zinc 0 650

Adhesives 46 350


Value of sales (£) Number of sales in this range

5 to less than 10 50






Table 5.22 Data for the assignment

Weight rangeNumber of loaves

in this range

780 g to below 790 g 34

790 g to below 795 g 80

795 g to below 800 g 111

800 g to below 805 g 162

805 g to below 810 g 161

810 g to below 815 g 120

815 g to below 820 g 70

820 g to below 830 g 27

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