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BUSINESS MATHEMATICS
&
STATISTICS
LECTURE 27
Measures of Dispersion and Skew ness
Part 2
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
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
Summary Measures
Summary Measures
Central Tendency
Mean
Quartile
Variation
Range
Coefficient
of Variation
Mode
Median
Variance
Geometric Mean
Standard Deviation
Measures of Central Tendency
Central Tendency
Mean
Median
Mode
n
X 
X
i 1
i
Geometric Mean
n
N

X
i 1
N
i
X G  ( X1  X 2   X n )
1n
The Mean (Arithmetic Average)
The Arithmetic Average of data values:
X1  X 2      X n
X   Xi / n 
n
i 1
n
Sample Mean
Sample Size
X1  X 2      X N
   Xi / N 
N
i 1
N
Population Mean
Population Size
The Mean
The Most Common Measure of Central Tendency
Affected by Extreme Values (Outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
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
The Median
(continued)
Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10
Median = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
The Mode
A Measure of Central Tendency
Value that Occurs Most Often
Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Mode = 8
The Mode
There May Not be a Mode
There May be Several Modes
Used for Either Numerical or Categorical Data
0 1 2
3 4 5
No Mode
6
0 1
2 3
4 5
Two Modes
6
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
Midrange
A Measure of Central Tendency
Average of Smallest and Largest Observation:
Midrange 
X largest  X smallest
2
Midrange
Affected by Extreme Value
0 1 2 3 4 5 6 7 8 9 10
Midrange = 5
0 1 2 3 4 5 6 7 8 9 10
Midrange = 3
Quartiles
Not a measure of central tendency
Split ordered data into 4 quarters
25%
25%
Q1
25%
Q2
25%
Q3
Position of i-th quartile:
Qi 
i(n+1)
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1= 1•(9 + 1) = 2.50
4
Q1 =12.5
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
= 8 + 0.75(9 - 8)= 8 + 0.75 x 1= 8.75
Q3= 3(2.75) = 7.75th value
= 8th value + 0.25(8th value –7th value)
= 17 + 0.75 (20 –17) = 19.25
Q.D =(19.25 –8.75)/2 = 5.25
Exploratory Data Analysis
Box-and-whisker:
Graphical display of data using 5-number
X smallest
4
summary
Median(Q2)
Q1
Q3
Xlargest
6
12
8
10
Distribution Shape &
Box-and-whisker Plots
Left-Skewed
Q1
Q3
Median
Symmetric
Q1
Q3
Median
Right-Skewed
Q1
Q3
Median
Five-Number Summary
(Symmetrical Distribution)
Smallest
Q1
Median
Q3
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
Five-Number Summary
(Nonsymmetrical Distribution)
Xsmallest Q1
Median
Q3
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
Determining the 5-Number Summary
Example
Example
For the sample representing annual costs (in Rs.000) for
attending 10 Conferences, the ordered array is:
Ordered:
13.0 14.3 14.9 15.2 15.2 15.4 15.6 16.4
17.0 23.1
State the 5-number summaries:
Xsmallest = 13.0
Q1
= 14.9
Median = 15.3
Q3
= 16.4
Xlargest = 23.1
Data appear right-skewed
Median < Midhinge < Midrange
Box-and-Whisker Plot
Graphical display of data using
5-number summary
X smallest Q 1 M edian Q 3
4
6
8
10
X largest
12
Shape &
Box-and-Whisker Plot
Left-Skewed
Q1 Median Q3
Symmetric
Q1
Median Q3
Right-Skewed
Q1 Median Q3
Summary Measures
Summary Measures
Central Tendency
Variation
Range
Mean
Mode
Median
Midrange
Interquartile
Range
Midhinge
Variance
Standard
Deviation
Coefficient of
Variation
Measures of Variation
Variation
Variance
Range
Population
Variance
Sample
Variance
Interquartile Range
Standard Deviation
Population
Standard
Deviation
Sample
Standard
Deviation
Coefficient of
Variation
S
CV= 
X

 100%

The Range
Measure of Variation Difference Between Largest & Smallest
Observations:
Range =
x La rgest  x Smallest
Ignores How Data Are Distributed:
Range = 12 - 7 = 5
7
8
9
10
Range = 12 - 7 = 5
11
12
7
8
9
10
11
12
Interquartile Range
Measure of Variation
Also Known as Midspread:
Spread in the Middle 50%
Difference Between Third & First Quartiles: Interquartile Range
=
Data in Ordered Array: 11 12 13 16 16 17
Q3  Q1
= 17.5 - 12.5 = 5
Not Affected by Extreme Values
17 18 21
BUSINESS MATHEMATICS
&
STATISTICS