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.