Statistics 4

This is an interesting bar graph. What does it tell us?

This is an interesting bar graph. What does it tell us?

• We get an idea of how the temperature is changing in December.

• We also can see the daily ranges of temperature.

State whether the following data is discrete or continuous.

• Number of books on library shelves• Lengths of library shelves• Time that the bells ring at a railway crossing• Weights of bales of wool• Number of bales of wool on trucks• Magnitude of earthquakes• Shoe size• Lengths of feet of year 9 boys

State whether the following data is discrete or continuous.

• Number of books on library shelves Discrete• Lengths of library shelves Continuous• Time that the bells ring at a railway crossing

Continuous• Weights of bales of wool Continuous• Number of bales of wool on trucks Discrete• Magnitude of earthquakes Continuous• Shoe size Discrete• Lengths of feet of year 9 boys Continuous

State whether the following data is discrete or continuous.

• Number of pinetrees in plantations• Diameters of pinetrees in plantations• Weights of logs of wood• Number of times a telephone rings before it is picked

up• Time a telephone rings before it is picked up• Lengths of songs in an album• Number of songs in an album

State whether the following data is discrete or continuous.

• Number of pinetrees in plantations Discrete• Diameters of pinetrees in plantations Continuous• Weights of logs of wood Continuous• Number of times a telephone rings before it is picked

up Discrete• Time a telephone rings before it is picked up

Continuous• Lengths of songs in an album Continuous• Number of songs in an album Discrete

Looking at examples of Data and Graphs

Example 1. Oil Consumption

• The following chart was published by USA Today in June 1994

Oil Consumption

Oil Consumption

• For what period of time is the fastest increase (either actual or projected) in world oil consumption shown? What is the rate of increase during this period?

Oil Consumption

If we check out the percentage increases:

• 68.4/67.6 = 1.011= 1.1%

• 75.6/68.4 = 1.105 = 10.5%

• 81.3/75.6 = 1.075 = 7.5%

• ..

• Greatest increase was between 2000 and 2005.

Oil Consumption

Are you sure?

Take another look at the x-axis scale.

Oil Consumption

Are you sure?

Take another look at the x-axis scale.

• The scales were not even and hence our perspective is distorted.

• The later figures represent 5 year periods.

•

The greatest increase between 2000 and 2005 is represented by a yearly increase according to the equation

68.4 x (increase)5 = 75.6Increase = 1.02 i.e. 2% per

year This is still the greatest

increase.

Notice one more detail

Notice one more detail

• We haven’t reached 2010 yet!!!

Stock Market

• The Dow Jones Industrial Average (DJIA) is the best-known index measuring the rise and fall of the U.S. stock market. It is a weighted composite of the stock prices of 30 major companies.

• The next chart, taken from a daily newspaper, shows the movement of the DJIA over a three-month period. Each bar represents one weekday (Monday through Friday), and dates are shown for every Monday.– The Boston Globe, June 21, 1994, p. 40.

Stock Market

Do you think this is the most appropriate graph?

Example 3

• Maru works in a video store. For 60 consecutive days, he counted the number of videos that were hired. These were the figures he got:

Example 3

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36

Data is discrete and hence a tally chart is the best way to organise this data.

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36

Organised data

No. of videos Tally Frequency

30 llll llll 9

31 llll llll ll 12

32 llll llll 10

33 llll ll 7

34 llll llll lll 13

35 llll 5

36 llll 4

Appropriate graphVideos hired

0

2

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12

14

30 31 32 33 34 35 36

Number of videos

Frequency

Appropriate graph

Videos hired

0

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30 31 32 33 34 35 36

Number of videos

Frequency

• Answer the questions on the sheet.

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• In how days did Maru hire 34 videos?

• 13 days

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• In how days did Maru hire more than 34 videos?

• 9 days

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• In how days did Maru hire less than 34 videos?

• 38 days

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• What was the greatest number of videos hired?

• 36

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• What was the least number of videos hired?

• 30

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• Find the range (greatest minus least) of videos hired.

• 36 - 30 = 6

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• For this data find

• The median• The mean• The mode

Appropriate graph

Videos hired

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30 31 32 33 34 35 36

Number of videos

Frequency

• For this data find• The median• =average of 30th and 31st data = 32

• The mean • Put it in the calculator to get 32.57

• The mode• The highest bar gives 34

Complete question 2 for practice.

Example 4

• For the following sets of data:• Pose a statistical question

• Calculate the mean, median and mode for the data. (Central tendencies)

• Calculate the range and interquartile range. (Measures of spread)

• State what two graphs, in order of preference, you would draw to help answer your question.

The following sets of maths marks are those obtained by a group of 7 formers in 2 different exams. The top row gives the mark in the term 3 school exam. The bottom row gives

the mark in the Bursary exam.

School 70 44 43 65 53 65 48 59 61 86 69 70 61 45

Bursary 65 52 53 63 53 62 51 58 58 59 56 60 56 55

Question:Is there a relationship between the school exam result and the

bursary result?

School 70 44 43 65 53 65 48 59 61 86 69 70 61 45

Bursary 65 52 53 63 53 62 51 58 58 59 56 60 56 55

Relationship questions don’t usually need us to calculate statistics as they have little meaning

when we are finding a relationship.They are useful for comparing the results.

Statistics

School Bursary Mean 59.93 Mean 57.21 Median 61 Median 57 Mode 61 Mode 53 Range 43 Range 14 Minimum 43 Minimum 51 Maximum 86 Maximum 65 Sum 839 Sum 801 Count 14 Count 14 LQ 49.25 LQ 53.5 UQ 68 UQ 59.75 IQR 18.75 IQR 6.25

Comments

School Bursary Mean 59.93 Mean 57.21 Median 61 Median 57 Mode 61 Mode 53 Range 43 Range 14 Minimum 43 Minimum 51 Maximum 86 Maximum 65 Sum 839 Sum 801 Count 14 Count 14 LQ 49.25 LQ 53.5 UQ 68 UQ 59.75 IQR 18.75 IQR 6.25

• Central tendencies:• Both the mean and

median were higher in the school exam than in the bursary exam but the values were very close.

• The mode has little meaning

Comments

School Bursary Mean 59.93 Mean 57.21 Median 61 Median 57 Mode 61 Mode 53 Range 43 Range 14 Minimum 43 Minimum 51 Maximum 86 Maximum 65 Sum 839 Sum 801 Count 14 Count 14 LQ 49.25 LQ 53.5 UQ 68 UQ 59.75 IQR 18.75 IQR 6.25

• The range of bursary results was much smaller than the school results. This could be that those who had preformed very poorly had done some study.

Comments

School Bursary Mean 59.93 Mean 57.21 Median 61 Median 57 Mode 61 Mode 53 Range 43 Range 14 Minimum 43 Minimum 51 Maximum 86 Maximum 65 Sum 839 Sum 801 Count 14 Count 14 LQ 49.25 LQ 53.5 UQ 68 UQ 59.75 IQR 18.75 IQR 6.25

• The interquartile range for bursary results was also smaller for the bursary exam.

Comparing results

36 40 44 48 52 56 60 64 68 72 76 80 84 88

Bursary Marks

School Marks

Relationship graph

Scatter plot showing Bursary and school marks

y = 0.2459x + 42.476

R2 = 0.5006

45

50

55

60

65

70

35 45 55 65 75 85 95

School marks

Bursary Mark

School marks is considered the predictor variable

and so it is put on the x-axis

Appropriate graph

Scatter plot showing Bursary and school marks

y = 0.2459x + 42.476

R2 = 0.5006

45

50

55

60

65

70

35 45 55 65 75 85 95

School marks

Bursary Mark

Appropriate graph

Scatter plot showing Bursary and school marks

y = 0.2459x + 42.476

R2 = 0.5006

45

50

55

60

65

70

35 45 55 65 75 85 95

School marks

Bursary Mark

• We would conclude that there is a positive linear relationship between school marks and bursary marks I.e. as the school mark increases we generally expect an increase in bursary mark.

• Note this doesn’t apply in all cases- it is a general observation.

Example 5

• Measurements of the wrist width and forearm length of a group of young men provided the following data where both measurements are in mm.

Question

• We can’t really compare wrist and forearm data - it more of a relationship situation.

• An appropriate question would be

• “Is there a relationship between the wrist measurement and length of the forearm?”

Data

Wrist 68 57 60 55 53 51 40 40 58 65

Fore

arm

275 280 275 280 280 249 280 265 273 268

Neither is a distinct predictor variable so we could use either on the x-axis.

Appropriate graph

Scatter plot comparing Wrist vs Forearm

y = 0.1231x + 265.77

R2 = 0.0136

260

265

270

275

280

285

38 42 46 50 54 58 62 66 70

Wrist (cm)

Forearm (cm)

Appropriate graph

• Even though we can put a straight line on the graph, the R2 value is so low that we cannot conclude that there is any relationship between wrist and forearm measurements.

Scatter plot comparing Wrist vs Forearm

y = 0.1231x + 265.77

R2 = 0.0136

260

265

270

275

280

285

38 42 46 50 54 58 62 66 70

Wrist (cm)

Forearm (cm)