1. Integrated Math 2 - Santowski
Unit 6, Lesson 5 – Measures of Dispersion – 5 Number Summary & Box Plots
Integrated Math 2 - Santowski

2. REVIEW: Measures of Central Tendency
To review, one way to analyze a data set is to determine a SINGLE NUMBER to help look at the “center” of the data. (that is Mean, Median, Mode)
Measures of central tendency attempt to identify the most representative value in a set of data.
So, now we may now the “typical” value, the “average” value, the “central” value

3. (A) Opening Exercise: Which Class did “better”?
Central Tendencies Results Summary from Class A
Central Tendencies Results Summary from Class B
Mean grade  63%
Median grade  60%
Modal Grade  60%, 80%
Mean grade  63%
Median grade  60%
Modal Grade  60%
Can we determine which class did “better” if all we know are the central tendencies?

4. (A) Opening Exercise: Which Class did “better”?
Central Tendencies Results Summary from Class A
Central Tendencies Results Summary from Class B
Mean grade  63%
Median grade  60%
Modal Grade  60%, 80%
Mean grade  63%
Median grade  60%
Modal Grade  60%
Can we determine which class did “better” if all we know are the central tendencies  WE CANNOT REALLY TELL AS WE DON’T HAVE ENOUGH DETAILS ABOUT THE DATA

5. (A) Opening Exercise: Which Class did “better”?
Actual Results from
Class A
Actual Results from
Class B
Class 1 30
100 40 80 50 60
Class 2 70 80 60 50
Can we now determine which class did “better”? CLEARLY, ONE CLASS’ RESULTS ARE MORE SPREAD OUT/DISPERSED (Class A)

6. (A) Opening Exercise: Which Class did “better”?
What are some other ways of analyzing a data set?
How about knowing how the data is spread out or dispersed or distributed?
So how about knowing the RANGE of the data?

7. (A) Opening Exercise: Which Class did “better”?
Results Summary from Class A
Results Summary from Class B
Mean grade  63%
Median grade  60%
Modal Grade  60%
Range  70%
Mean grade  63%
Median grade  60%
Modal Grade  60%
Range  70%
Can we determine which class did “better” if all we know are the central tendencies and the range as well?

8. (A) Opening Exercise: Which Class did “better”?
Results Summary from Class A
Results Summary from Class B
Mean grade  63%
Median grade  60%
Modal Grade  60%
Range  70%
Mean grade  63%
Median grade  60%
Modal Grade  60%
Range  70%
Can we determine which class did “better” if all we know are the central tendencies and the range as well?  WE STILL HAVE NO IDEA ABOUT IF THE DATA IS EVENLY SPREAD OUT OR CLUSTERED

9. (A) Opening Exercise: Which Class did “better”?
Actual Results from
Class A
Actual Results from
Class B
Class 1 20 60 40 80 90
Class 2 30 70 60
100
Can we now determine (or at least make some judgements) about which class did “better”?

10. (B) Measures of Dispersion
So what else do we need?  we need some way to come up with some other “summary numbers” that allow us to get a picture of the “spread” or “dispersion” or “distribution” or “variance” in the data
One such number is the RANGE (as we have discussed before)

11. (C) The Range
The range is simply the difference between the largest and the smallest observed values in a data set.
Thus, range, including any outliers, is the actual spread of data.
A great deal of information is ignored when computing the range, since only the largest and smallest data values are considered.

12. (D) Quartiles

13. (D) Quartiles
25%
25%
25%
25% Q1 Q2 Q3
> first quartile (designated Q1) = lower quartile = cuts off lowest 25% of data = 25th percentile
> second quartile (designated Q2) = median = cuts off data set in half (50% of data) = 50th percentile
> third quartile (designated Q3) = upper quartile = cuts off highest 25% of data, or lowest 75% = 75th percentile
Note that the second quartile Q2 (the 50th percentile) is the median

14. (D) Quartiles

15. (D) Quartiles
20, 20, 21, 24, 27, 29, 33, 33, 36, 39, 50, 57, 60, 65, 65

16. (D) Quartiles
18, 23, 25, 27, 27, 37, 38, 45, 47, 49, 49, 50, 50, 58, 61, 66, 69

17. (D) Quartiles Golf Club A Golf Club B Lower Quartile 24 27 Median 3347
Upper Quartile 57 54
Which golf club has more younger members?
Which golf club has more older members?
What can you say about the age profile of each golf club in general?

18. Box Plots
A box plot is a way of illustrating key information about a set of data
They are also very useful for comparing the distribution of two sets of data (e.g. boys vs girls)

19. Box Plots To draw a box plot, you need FIVE pieces of information:
The minimum value
The lower quartile
The median
The upper quartile
The maximum value

20. Box Plots

21. Quartiles

22. 1 2 3 4 5 6 7 8 9 10 Maths Test Scores Minimum Value: 3
Lower Quartile: 4
Median: 7
Upper Quartile: 9
Maximum Value: 10
Interquartile Range: 5
The Interquartile Range (IQR) is the UQ – LQ.
It illustrates the spread of the middle 50% of the data.
The larger the IQR, the more spread out, and less consistent the data. 1 2 3 4 5 6 7 8 9 10
Maths Test Scores

23. GIRLS BOYS 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Maths Test Scores
The box plot below shows the data for 23 female students.
GIRLS 1 2 3 4 5 6 7 8 9 10
BOYS 1 2 3 4 5 6 7 8 9 10
Maths Test Scores

24. Starter
The number of spelling mistakes of 7 male students on their homework is shown below
4, 9, 3, 5, 8, 10, 5
Calculate a five figure summary for this data
Draw a box plot for the data
Write a couple of sentences to interpret what your box plot shows you

25. Starter
The five figure summary for the girls is shown in the table below
Use it to draw a box plot for the girls
Write a couple of sentences to compare the number of spelling mistakes made by boys and girls
Minimum value 4
Lower Quartile 5
Median 8
Upper Quartile 9
Maximum value 12

26. Box Plots
Below are the midday temperatures for Ringwood over the past 11 days (in oC).
20, 22, 16, 17, 16, 18, 20, 18, 16, 21, 18
Below are the midday temperatures for Glasgow over the past 11 days (in oC).
17, 20, 20, 17, 16, 14, 13, 19, 21, 17, 18
Draw box plots for the data and write some comments on what they tell you about the temperatures in Ringwood and Glasgow over the past 11 days.