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Statistics 2. Quantitative (Numerical) (measurements and counts). Qualitative (categorical) (define groups). Continuous. Discrete. Categorical (no idea of order). Ordinal (fall in natural order). We are only going to consider quantitative variables in this AS. Variables. - PowerPoint PPT Presentation
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Statistics 2
Variables
DiscreteContinuous
Quantitative(Numerical)
(measurements and counts)
Qualitative(categorical)
(define groups)
Ordinal(fall in natural order)
Categorical(no idea of order)
We are only going to consider quantitative variables in this AS
Quantitative
Discrete• Many repeated
values• Age groups• Marks
Continuous• Few repeated
values• Height• Length• Weight
Qualitative
Categorical• Gender• Religious
denomination• Blood types• Sport’s numbers
(e.g. He wears the number ‘8’ jersey)
Ordinal• Grades• Places in a race
(e.g. 1st, 2nd, 3rd)
Collecting data
• Tally charts • Stem and leaf plots
How we collect the data usually depends on what question we wish to
answer.
Tally chart
• If we were asking people what they had for breakfast we might set up a table like this…
Tally chart
Breakfast Tally Frequency
Toast
Cereal
Eggs
Porridge
Rice
No breakfast
Tally Chart
• We use a tally chart when data fits easily into categories.
Stem and leaf plot
• A stem and leaf plot sorts data that has few values the same.
Example
• The number of punnets of strawberries picked by Carol over a 17-day period. (This example is in your text book)
• 65 73 86 90 99 106 45 92 94 102 107 107 99 83 101 91
Example
• Set up a ‘stem’ based on the fact that the numbers picked are between 40 and 110
Example
Stem
4
5
6
…
10
Example
• The first number is 65 and the next is 73.
• They are recorded like this
Example
Stem Leaf
4
5
6 5
7 3
…
Example
Stem Leaf
4 5
5
6 5
7 3
8 6 3
9 0 9 2 4 7 9 1
10 6 2 7 7 1
Sort the data in order
Stem Leaf
4 5
5
6 5
7 3
8 3 6
9 0 1 2 4 7 9 9
10 1 2 6 7 7
Lowest and highest values
Stem Leaf
4 5 = 45
5
6 5
7 3
8 3 6
9 0 1 2 4 7 9 9
10 1 2 6 7 7 = 107
Median and quartiles
Stem Leaf
4 5 = 45
5
6 5
7 3
8 3 6 = 84.5
9 0 1 2 4 7 9 9
10 1 2 6 7 7 = 107
Median and quartiles
Stem Leaf
4 5
5
6 5
7 3
8 3 6
9 0 1 2 4 7 9 9
10 1 2 6 7 7
• 5- number summary• Lowest = 45• LQ = 84.5• Median = 94• UQ = 101.5• Highest = 107
Stem Leaf
4 5
5
6 5
7 3
8 3 6
9 0 1 2 4 7 9 9
10 1 2 6 7 7
Median and quartiles
Pictures that tell a story
• Drawing a picture of our data.
• Our data is discrete and hence a bar graph is an appropriate way of showing our ‘picture’.
A bar graph
A bar graph
• We use a bar graph (spaces between bars) because we are dealing with discrete data (counted data, many repeated values)
Bar graph
• A bar graph gives us a picture of the data and we can easily see many features of our data.
Bar graph
• Lowest = 3 letters• Highest = 8 letters• Mode = 5 letters• The graph is
approximately symmetrical and uni-modal (has only one mode)
Bar graph
• To find out how many were surveyed, you add the frequencies together.
Pie graph
• Each category makes up a certain percentage of the ‘pie’.
• A pie graph does not tell us how many were in the data set.
• You must be careful when comparing data from 2 pie graphs.
Pie graph
Letters Frequency Angle of pie
3 2 360÷35x2=21
4 5 360÷35 x 5=51
5 14 144
6 7 72
7 5 51
8 2 21
Pie graph
Pie Graph
• This also is an appropriate graph as it shows the relative numbers in each category.
• It does not give us a lot of specific information like how many were surveyed or how many had 8 letters in their name.
Box and Whisker plot
• The box and whisker plot is a picture of the 5-number summary and it shows us where the cut-off is for every quarter of the data.
• Again, the box and whisker plot does not tell us how many were in the sample just how the quarters were distributed.
Box and Whisker plot
Box and Whisker plot
• This gives us a lot of information.
• The lowest and highest values.
• The median, upper and lower quartiles.
• We also get a sense of how the data is distributed.
Box and Whisker Plot
• Box and whisker plots can also be used to compare two sets of data.
Back to strawberry picking!
• Who would you employ?
Strawberry picking
Comparing
Carol Dilip
Mean 90.4 90.1
Median 94 99
Mode 99 95
Comparing
Carol Dilip
Mean 90.4 90.1
Median 94 99
Mode 99 95
• Carol has the higher mean.
• Dilip has the higher median.
• Carol has the higher mode.
Central tendency
• Which central tendency is more useful in measuring the punnets picked overall?
Comparing
Carol Dilip
Range 62 108
Interquartile range
17 7.5
Lowest 45 0
Highest 107 108
Comparing
Carol Dilip
Range 62 108
Interquartile range
17 7.5
Lowest 45 0
Highest 107 108
• Carol has the lower range.
• Dilip has the lower interquartile range.
• Carol’s lowest value is higher than Dilip’s.
• Dilip’s highest value is higher than Carol’s.
Spread
• Which picker is more reliable?
Back to the data
Comparing using a picture
Box and whisker
Box and whisker
• Overall they both picked roughly the same number of punnets.
• Carol 1537• Dilip 1532
Box and whisker
• The long tails on the box and whisker plots suggest outliers (extreme values).
• 45 is a likely outlier for Carol and suggests she worked a half day.
• 0 suggests that Dilip did not work on one of the days which would have pulled his mean value down.
• 49 is also an outlier for Dilip suggesting he also worked half a day.
Box and whisker
• Dilip is more reliable as his spread as shown by the interquartile range is smaller.
• (This is presuming he doesn’t just take days off when he wants to.)
What not to do!!!
No! No! No!- this is not a good idea!
No! No! No!- this is not a good idea!
• Axes need to be labelled.
• Colour distorts the graph.
• Lines also distort the graph- take a look at these.
Are the lines parallel?
Are these lines parallel?
Are these lines parallel?
Are the lines parallel?
• This kind of graph gives us very little information.
Negatively skewed (unimodal)
Positively skewed
Symmetric
Uniform
Groupings (bimodal)
Outlier
Bi-variate data
• Looking for relationships between two variables.
Example
• Is there a relationship between the amount of study a person does and their test result?
Consider data on ‘hours of study’ vs ‘ test score’
Hours Score Hours Score Hours Score
18 59 14 54 17 59
16 67 17 72 16 76
22 74 14 63 14 59
27 90 19 72 29 89
15 62 20 58 30 93
28 89 10 47 30 96
18 71 28 85 23 82
19 60 25 75 26 35
22 84 18 63 22 78
30 98 19 61
Relationship
• There is a positive linear relationship between the amount of study and the test score. This means that as the hours of study increases, we expect an increase in test score.
