Statistics

​Univariate Data and Bivariate Data Revision

​Key Concepts...

​

• Measures of centre

• Measures of spread

• Plotting a comparative box plot

• Plotting a histogram

• Grouped Data

• Pearson's Correlation

• Least Squares Regression Line

​Measures of Centre (Raw Data)

​The measures of centre we are concerned with are the mean and median. We need an understanding of the mode but we also need to recognise its vast limitations i.e. multiple modes or no mode is an option.

​Consider the following raw data which shows the shoe sizes of Class 10A:

​12	​ 10	​ 8	​ 7	​ 5	​ 3	​ 11	​ 11	​ 12	​ 10
​ 7	​ 6	​ 4	​ 12	​ 10	​ 9	​ 10	​ 10	​ 9	​ 12

​Therefore, the mean shoe size is 8.9

​Organising the shoe sizes in ascending order, our median is 10. The CAS can help, our key code is: Menu - 4 - 1 - 1

Fill in the Blank

The median is the best measure of centre is it is not affected by ________

Multiple Choice

Consider the following data set:

12, 13, 15, 18, 20, 22, 22, 23, 25, 29, 35, 40

• Mean = 22.8

  Median = 22

• Mean = 22

  Median = 22.8

• Mean = 24

  Median = 22.5

• Mean = 274

  Median = 22

Multiple Choice

Find the mean: 12, 15, 18, 20, 24, 29, 30, 32

• 20

• 22.5

• 24.4

• 32

Multiple Choice

Find the median: 12, 15, 18, 20, 24, 29, 30, 32

• 15

• 20

• 22

• 24

​Measures of Spread (Raw Data)

​The measures of range we are concerned with are the range, interquartile range and the standard deviation. The range is the maximum - minimum, the IQR is the Upper Quartile - Lower Quartile and the standard deviation represents how close the values are to the mean.

​Consider the following raw data which shows the shoe sizes of Class 10A:

​12	​ 10	​ 8	​ 7	​ 5	​ 3	​ 11	​ 11	​ 12	​ 10
​ 7	​ 6	​ 4	​ 12	​ 10	​ 9	​ 10	​ 10	​ 9	​ 12

​Range = 12 - 3

Therefore, the range is 9.​

​IQR = 11 - 7

Therefore, the IQR is 4.​

We'd expect the​ standard deviation to be close the the IQR.

In this example, the SD is 2.68.​

Fill in the Blank

What percentage of the population lie between the LQ and UQ?

Multiple Choice

Consider the following data set:

2, 3, 3, 4, 5, 7, 7, 8, 10, 12, 14, 15, 16, 16, 20

What is the Range?

• 8

• 11

• 18

• 20

Multiple Choice

Consider the following data set:

2, 3, 3, 4, 5, 7, 7, 8, 10, 12, 14, 15, 16, 16, 20

What is the IQR?

• 8

• 11

• 18

• 20

Multiple Choice

Consider the following data set:

2, 3, 3, 4, 5, 7, 7, 8, 10, 12, 14, 15, 16, 16, 20

If the LQ was 4 and the UQ was 15.

What percentage of the data was over 15?

• 0%

• 25%

• 50%

• 75%

Multiple Choice

Consider the following data set:

2, 3, 3, 4, 5, 7, 7, 8, 10, 12, 14, 15, 16, 16, 20

If the LQ was 4 and the UQ was 15.

What percentage of the data was less than 15?

• 0%

• 25%

• 50%

• 75%

Multiple Choice

Consider the following data set:

2, 3, 3, 4, 5, 7, 7, 8, 10, 12, 14, 15, 16, 16, 20

If the Median was 8.

What percentage of the data was less than 8?

• 0%

• 25%

• 50%

• 75%

Multiple Choice

Minimum = 10

Lower Quartile = 12

Median = 15

Upper Quartile = 20

Maximum = 30

What percentage of the results lie between 12 and 20?

• 0%

• 25%

• 50%

• 75%

Multiple Choice

Minimum = 10

Lower Quartile = 12

Median = 15

Upper Quartile = 20

Maximum = 30

What percentage of the results lie between 15 and 20?

• 0%

• 25%

• 50%

• 75%

​Outliers

​Outliers represent extreme values. Our CAS can determine outliers for us and it represents an outlier with an enclosed circle. We need to be able to find the upper and lower fences associated with outliers.

​Consider the following raw data which shows the shoe sizes of Class 10A:

​12	​ 10	​ 8	​ 7	​ 5	​ 3	​ 11	​ 11	​ 12	​ 10
​ 7	​ 6	​ 4	​ 12	​ 10	​ 9	​ 10	​ 10	​ 9	​ 12

​Anything less than 1 or greater than 17 would be considered outliers.

Multiple Choice

Find the outlier:

12, 15, 18, 20, 100

• 12

• 15

• 20

• 100

Multiple Choice

The upper fence for the following data is: 10, 14, 16, 18, 20, 23, 40, 60, 100, 120, 300

• -110

• 16

• 226

• 180

Multiple Choice

The outlier(s) for the following data set is: 3, 4, 6, 8, 10, 13, 23, 45, 65, 150, 300

• 3

• 3 and 4

• 150 and 300

• 300

​Box Plots

​Consider the following raw data which shows the shoe sizes of Class 10A:

​12	​ 10	​ 8	​ 7	​ 5	​ 3	​ 11	​ 11	​ 12	​ 10
​ 7	​ 6	​ 4	​ 12	​ 10	​ 9	​ 10	​ 10	​ 9	​ 12

Multiple Choice

What is the median of this data set?

• 8

• 12

• 15

• 19

Multiple Choice

Find the Minimum

• 98
• 85
• 68
• 93

Multiple Choice

What is the interquartile range?

• A. 4
• B. 6
• C. 2
• D. 8

Multiple Choice

Which group has a greater median shopping time?

• Group A

• Group B

• They're the same.

• You can't tell.

Multiple Choice

Which group has a smaller interquartile range?

• Group A

• Group B

• They're the same.

• You can't tell.

Multiple Choice

What do points B and D represent on the box plot?

• Mean and Mode

• Mean and Median

• Least and Greatest Value

• First and Third Quartiles

​Grouped Data

​Grouped data is used to group larger categories and intervals. We need to be able to find the estimated mean, the median and modal class, the range and the cumulative frequency.

​ Height (h)	​ Frequency	​ Midpoint	​ fx	​ cf
121 - 130​	​ 8	​ 125	​ 1000	8​
​131 - 140	​ 10	135 ​	​1350	18​
​ 141 - 150	​ 7	​ 145	​ 1015	​ 25
​ Total	​ 25	​	​ 3365	​

​Therefore the estimated mean will be 134.6.

​The median value will be in the 13th position, therefore the category will be 131 - 140.

​The modal class is 131 - 140.

The range will be 30.​

​

Multiple Choice

The sum of the frequency column is 34, the sum of the fx column is 156. Find the estimated mean.

• 0.22

• 5,304

• 122

• 4.6

Multiple Choice

The midpoint for the following grouped data is: $$10<b\le18$$  

• 14.5

• 15

• 13

• 14

Multiple Choice

Find the mean of the given data sets.

• 31

• 28

• 23

• 30

Multiple Choice

Which is the modal class?

• 4 - 6

• 7 - 9

• 10 - 12

• 16 - 18

Multiple Select

For this table of data, what is the modal class?

• 0-<10

• 30-<40

• 20-<30

• 40-<50

Multiple Select

For this grouped data, which is the median class?

• 20-<30

• 30-<40

• 40-<50

• 0-<10

Multiple Select

For this grouped data, what is the mean?

• 20

• 25

• 27

• 30

Fill in the Blank

This is an example of which type of skew?

Fill in the Blank

This is an example of which type of skew?

Fill in the Blank

This is an example of which type of skew?

​Bivariate Data

​Bivariate Data compares two sets of data i.e. Height Vs. Weight. The Explanatory Variable (EV) is the variable which is not affected. It is positioned on the x-axis. The Response Variable (RV) is affected by the EV, it is positioned on the y-axis.

​

Pearson's Correlation is represented by the letter r. It gives the correlation between the two variables. The value should be between -1 and +1, when describing the data we focus on the strength and direction.

​

The least squares regression equation provides an accurate line of best fit. We usually substitute x and y for the variables in the question. We utilise the equation for predictions and forecasting.​

​

Fill in the Blank

An r-value of -0.798 would be described as ...

Fill in the Blank

If Pearson's Correlation was 0.26, we would describe the correlation as... ?

Multiple Choice

The study is to analyse the effects of sun screen on catching skin cancer. Which is the EV?

• Sun Screen

• Skin Cancer

• Not enough information is provided

• Medicare

Multiple Choice

If Pearson's Correlation is 0.7456, that could be described as?

• No Correlation

• Weak, Positive, Linear Correlation

• Strong, Positive, Linear Correlation

• Moderate, Positive, Linear Correlation

Multiple Choice

The least squares equation was: $$No.\ of\ Accidents=100-4.5\times No.\ of\ Lessons$$  

Which value is the RV?

• No. of Accidents

• No. of Lessons

• 10.2

• We can't possibly know

Multiple Choice

The least squares equation was: $$No.\ of\ Accidents=100-4.5\times No.\ of\ Lessons$$  

A student having 20 lessons would be expected to have how many accidents?

• 45

• 30

• 10

• 100

Multiple Choice

The least squares equation was: $$No.\ of\ Accidents=100-4.5\times No.\ of\ Lessons$$  

A student has 5 accidents, how many lessons do we estimate they have had?

• 4.5

• 20

• 21

• 35

Multiple Choice

Which value for Pearson's Correlation would we expect to see when comparing the number of hot pies sold and the temperature.

• 0.65

• 0.2

• -0.1

• -0.8

Multiple Choice

Describe the type of linear correlation between the two variables

• Strong positive

• Weak positive

• Strong Negative

• No correlation

Multiple Choice

Which sentence describes the relationship shown on this scatter plot?

• As the temperature decreases, the visitors increase.

• As the temperature increases, the number of visitors increases.

• As the visitors increase, the temperature decreases.

• No correlation

Draw

Draw a line of best fit on the scatter plot below:

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