Statistics 3. Continuous data Collecting continuous data (measured data, few values the same.) We...

Statistics 3

Continuous data

• Collecting continuous data (measured data, few values the same.)

• We collect data using tally charts when we want to group data into categories

• or stem and leaf plots when we want to have a record of each data value.

Oyster length data tally chartLength Tally Frequency 150 - 160 l 1 160 - 170 0 170 - 180 l 1 180 - 190 0 190 - 200 llll ll 7 200 - 210 llll llll 10 210 - 220 l 1 220 - 230 llll l 6 230 - 240 lll 3 240 - 250 0 250 - 260 0 260 - 270 0 270 - 280 0 280 - 290 0 290 - 300 0 300 - 310 0 310 - 320 l 1

150 - 160 means all data that is 150 mm up to 160 mm (but not including 160 mm)

The best graph to show this data is a histogram.

Histogram

Histogram: Lengths of Oyster

0

2

4

6

8

10

12

150 -160

160 -170

170 -180

180 -190

190 -200

200 -210

210 -220

220 -230

230 -240

240 -250

250 -260

260 -270

270 -280

280 -290

290 -300

300 -310

310 -320

Length (mm)

Frequency

Histogram


0

2

4

6

8

10

12

150 -160

160 -170

170 -180

180 -190

190 -200

200 -210

210 -220

220 -230

230 -240

240 -250

250 -260

260 -270

270 -280

280 -290

290 -300

300 -310

310 -320

Length (mm)

Frequency

• Note that we always put units on the axes.

• There are no gaps between bars as the data is continuous.

Histogram


0

2

4

6

8

10

12

150 -160

160 -170

170 -180

180 -190

190 -200

200 -210

210 -220

220 -230

230 -240

240 -250

250 -260

260 -270

270 -280

280 -290

290 -300

300 -310

310 -320

Length (mm)

Frequency

• The graph shows us that we have some outliers.

• Most of the lengths are between 190 and 240 mm.

• The modal interval is 200mm to 210 mm.

Stem and Leaf Plot15 91617 01819 0 3 4 5 7 7 820 1 3 3 3 5 6 7 7 8 821 722 2 4 7 8 8 923 0 3 42425262728293031 8

It looks a lot like the histogram just on its side.

Stem and Leaf Plot15 91617 01819 0 3 4 5 7 7 820 1 3 3 3 5 6 7 7 8 821 722 2 4 7 8 8 923 0 3 42425262728293031 8

We can get the same information from this plot as we can from the histogram.

Line graphs

• Line graphs are used to show some kind of change or distinction between data.

Example

• We could look at the pre-test and post-test scores of a class to gauge if there has been a change.

Pre-test and post-test results

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12

Marks

Frequency

Pre-testPost-test


0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10 12

Marks

Frequency

Pre-testPost-test

• This gives us a picture of what happened.

• Another way to show this would have been to draw a box and whisker plot.

Note that the LQ and Median were the same value


0 2 4 6 8 10 12

Marks

Pre-testPost-test

Note that the LQ and Median were the same value


0 2 4 6 8 10 12

Marks

Pre-testPost-test

• The box and whisker plots give us a clear indication that there has been a change but they don’t show us that there were 25 marks in the first test and only 17 recorded in the second and hence we cannot be certain of the conclusions that we make.

Paying our bills

• Media often talk about people receiving a percentage increase in their wages.

• Is this a fair way to look at things when everyone pays the same amount for goods?

Example

• Mr Jones gets $381 a week. He spends $108 on food for his family of four.

• Mrs Smith gets $684 a week. She spends $146 on food for her family of four.

• The best way to show this situation is to draw 2 pie graphs.

Comparing

Mr Jones

Rest72%

Food28%

Mrs Smith

Rest79%

Food21%

Although Mrs Smith spends more on food, the percentage of her income is less.

Mr Jones

Rest72%

Food28%

Mrs Smith

Rest79%

Food21%

Misleading Graphs

Company profits

1000

1050

1100

1150

1200

1250

1300

1350

1400

2000 2001 2002 2003

Year

Profit ($000s)

Misleading Graphs

Company profits

1000

1050

1100

1150

1200

1250

1300

1350

1400

2000 2001 2002 2003

Year

Profit ($000s)

• The impression we get is that this company is improving quite dramatically.

• Notice where the y-axis begins

Misleading Graphs

Company profits

1000

1050

1100

1150

1200

1250

1300

1350

1400

2000 2001 2002 2003

Year

Profit ($000s)

Misleading Graphs

Company profits

0

200

400

600

800

1000

1200

1400

1600

2000 2001 2002 2003

Year

Profit ($000s)

Misleading Graphs

Company profits

0

200

400

600

800

1000

1200

1400

1600

2000 2001 2002 2003

Year

Profit ($000s)

• Using ‘0’ as a starter value gives us a more honest representation of the data and we now see that the increase is less dramatic.

School Exam Passes

0

20

40

60

80

100

120

A B C D

School

Number of passes

Misleading graphs

School Exam Passes

0

20

40

60

80

100

120

A B C D

School

Number of passes

• The principal of school B says her school is the most successful.

• Do you agree?

Misleading graphs

School Exam Passes

0

20

40

60

80

100

120

A B C D

School

Number of passes

• We need more information to make a judgement.

School data

School Number who sat

Number who

passed

Percentage who passed

A 60 40 67%

B 257 100 39%

C 75 60 80%

D 180 70 39%

Comparing graphs

School Exam Passes

0

20

40

60

80

100

120

A B C D

School

Number of passes

Percentage who Passed

0

10

20

30

40

50

60

70

80

90

A B C D

School


We now think that C is the best school because the pass rate is better.

You now find out that school C will not let weak students sit the exam as it spoils their

percentage.

School Exam Passes

0

20

40

60

80

100

120

A B C D

School

Number of passes

Percentage who Passed

0

10

20

30

40

50

60

70

80

90

A B C D

School


We now think that A is the best school as the results for school C don’t represent the situation correctly.

Exam results

Exam results George Smythe

English M Mathematics E Economics N Geography A French A

• From George’s results we know his best subject is mathematics and his worst subject is Economics.

• True or False?

Comment on features that are misleading.

Sales of shoes Jan-Mar 1990

100

102

104

106

108

110

3 4 5 5.5 6 6.5 7 7.5 8 9 10

Shoe Size

No. of pairs sold



100

102

104

106

108

110

3 4 5 5.5 6 6.5 7 7.5 8 9 10

Shoe Size

No. of pairs sold

• Inappropriate graph. It does not show the data correctly. Data is discrete and yet this shows data as continuous.

• Should have been a bar graph.



100

102

104

106

108

110

3 4 5 5.5 6 6.5 7 7.5 8 9 10

Shoe Size

No. of pairs sold

• Scale on y-axis is not uniform.

• Suppression of ‘0’. I.e. the graph does not start at zero.

• Non-uniform scale on the x-axis.

Comment on the features that are misleading.

Use of poisoned bait

0

2

4

6

8

10

12

1985 1986 1987 1988 1989 1990

Year

Amount used

Comment on the features that are misleading.

Use of poisoned bait

0

2

4

6

8

10

12

1985 1986 1987 1988 1989 1990

Year

Amount used

• No units on the vertical axis.

• As the height increases so does the width (and thickness) and visual impact of area (and volume) gives a distorted impression.

• Not clear where carrots should be read from.

Misleading data

Average weekly wages

Men’s suits

1971 $64 1971 $62 1982 $280 1982 $230

(40 hours work)

(Average price)

Misleading data

Average weekly wages

Men’s suits

1971 $64 1971 $62 1982 $280 1982 $230

(40 hours work)

(Average price)

The Minister of Consumer Affairs says that suits are really cheaper in 1982 than 1971. Explain his reasoning.

Statistics 3. Continuous data Collecting continuous data (measured data, few values the same.) We...

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