Download - Statistics 4
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
32
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34
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36
Data is discrete and hence a tally chart is the best way to organise this data.
32
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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
4
6
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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
0
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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
0
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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
0
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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
0
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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
0
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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
0
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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
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55
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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)