1. BUSINESS MATHEMATICS
&
STATISTICS

2. Measures of Dispersion and Skew ness
LECTURE 27
Measures of Dispersion and Skew ness
Part 2

3. Measures of Central Tendency, Variation and Shape for a Sample
Mean, Median, Mode, Midrange, Quartiles, Midhinge
Range, Interquartile Range, Variance, Standard Deviation, Coefficient of Variation
Right-skewed, Left-skewed, Symmetrical

4. Measures of Central Tendency, Variation and Shape
Exploratory Data Analysis
Five-Number Summary
Box-and-Whisker Plot
Proper Descriptive Summarization
Exploring Ethical Issues
Coefficient of Correlation

5. Coefficient of Variation
Summary Measures
Summary Measures
Central Tendency
Variation
Quartile
Mean
Mode
Coefficient of Variation
Median
Range
Variance
Standard Deviation
Geometric Mean

6. Measures of Central Tendency
Mean
Median
Mode
Geometric Mean

7. The Mean (Arithmetic Average)
The Arithmetic Average of data values:
Sample Mean
Sample Size
Population Size
Population Mean

8. The Mean The Most Common Measure of Central Tendency
Affected by Extreme Values (Outliers)
Mean = 5
Mean = 6

9. The Median Important Measure of Central Tendency
In an ordered array, the median is the
“middle” number.
If n is odd, the median is the middle number
If n is even, the median is the average of the 2 middle numbers

10. The Median (continued)
Not Affected by Extreme Values
Median = 5
Median = 5

11. The Mode A Measure of Central Tendency Value that Occurs Most Often
Not Affected by Extreme Values
Mode = 8

12. The Mode There May Not be a Mode There May be Several Modes
Used for Either Numerical or Categorical Data
No Mode
Two Modes

13. R = Largest – Smallest Value
DISPERSION OF DATA
Range
R = Largest – Smallest Value
Example
Find range:
31, 26, 15, 43, 19, 27, 22, 12, 36, 33, 30, 24, 20
Largest value = 43
Smallest = 12
Range = 43 – 12 = 31

14. Midrange Midrange A Measure of Central Tendency
Average of Smallest and Largest Observation:
Midrange

15. Affected by Extreme Value
Midrange
Affected by Extreme Value
Midrange = 5
Midrange = 3

16. Quartiles
Not a measure of central tendency
Split ordered data into 4 quarters
Position of i-th quartile:
25%
25%
25%
25% Q1 Q2 Q3
i(n+1) Q  i 4
Data in Ordered Array:
1•(9 + 1)
Position of Q1 =
= 2.50 Q1
=12.5 4

17. DISPERSION OF DATA Quartile Deviation Q.D = (Q3 – Q1)/2 Example
Find Q.D
14, 10, 17, 5, 9, 20, 8, 24, 22, 13
Q1 = (n+1)/4th value = (10+1)/4 = 2.75th
= (9 - 8)= x 1= 8.75
Q3= 3(2.75) = 7.75th value
= 8th value (8th value –7th value)
= (20 –17) = 19.25
Q.D =(19.25 –8.75)/2 = 5.25

18. Exploratory Data Analysis
Box-and-whisker:
Graphical display of data using 5-number summary
Median(Q2) X X Q Q
largest
smallest 1 3 4 6 8 10 12

19. Distribution Shape & Box-and-whisker Plots
Left-Skewed
Symmetric
Right-Skewed Q Q Q Q Q Q 1 1 3 3 1 3
Median
Median
Median

20. Five-Number Summary (Symmetrical Distribution)
Smallest Q1 Median Q Xlargest
Data perfectly symmetrical if:
Distance from Q1 to Median = Distance from Median to Q3
Distance from Xsmallest to Q1 = Distance from Q3 to Xlargest
Median = Midhinge = Midrange

21. Five-Number Summary (Nonsymmetrical Distribution)
Xsmallest Q1 Median Q Xlargest
Right-skewed distribution
Median < Midhinge < Midrange
Distance from Xlargest to Q3 greatly exceeds distance from Q1 to Xsmallest
Left-skewed distribution
Median > Midhinge > Midrange
Distance from Q1 to Xsmallest greatly exceeds distance from Xlargest to Q3

22. Determining the 5-Number Summary Example
For the sample representing annual costs (in Rs.000) for attending 10 Conferences, the ordered array is:
Ordered:
State the 5-number summaries:
Xsmallest = 13.0
Q = 14.9
Median = 15.3
Q = 16.4
Xlargest = 23.1
Data appear right-skewed
Median < Midhinge < Midrange

23. Graphical display of data using 5-number summary
Box-and-Whisker Plot
Graphical display of data using 5-number summary

24. Shape & Box-and-Whisker Plot

25. Coefficient of Variation
Summary Measures
Summary Measures
Central Tendency
Variation
Range
Standard Deviation
Mean
Mode
Interquartile Range
Midrange
Median
Coefficient of Variation
Midhinge
Variance

26. Measures of Variation Variation Variance Standard Deviation
Coefficient of Variation
Range
Population
Variance
Population
Standard
Deviation
Sample
Variance
Sample
Standard
Deviation
Interquartile Range

27. Ignores How Data Are Distributed:
The Range
Measure of Variation Difference Between Largest & Smallest Observations:
Range =
Ignores How Data Are Distributed:
Range = = 5
Range = = 5

28. Interquartile Range = 17.5 - 12.5 = 5 Measure of Variation
Also Known as Midspread:
Spread in the Middle 50%
Difference Between Third & First Quartiles: Interquartile Range =
Not Affected by Extreme Values
Data in Ordered Array:
= = 5

29. BUSINESS MATHEMATICS
&
STATISTICS