Download - Describing Data: Summary Measures
© 1999 Prentice-Hall, Inc. Chap. 3 - 1
Measures of Central Location Mean, Median, Mode
Measures of Variation Range, Variance and Standard Deviation
Measures of Association Covariance and Correlation
Describing Data: Summary Measures
© 1999 Prentice-Hall, Inc. Chap. 3 - 2
•It is the Arithmetic Average of data values:
•The Most Common Measure of Central Tendency
•Affected by Extreme Values (Outliers)
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
Sample Mean
Mean
nxxx n2i
n
xn
1ii
x
© 1999 Prentice-Hall, Inc. Chap. 3 - 3
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
•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.•Not Affected by Extreme Values
Median
© 1999 Prentice-Hall, Inc. Chap. 3 - 4
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
•A Measure of Central Tendency•Value that Occurs Most Often•Not Affected by Extreme Values•There May Not be a Mode•There May be Several Modes•Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6
No Mode
Mode
© 1999 Prentice-Hall, Inc. Chap. 3 - 5
Variation
Variance Standard Deviation Coefficient of Variation
PopulationVariance
Sample
Variance
PopulationStandardDeviationSample
Standard
Deviation
Range
100%
X
SCV
Measures Of Variability
© 1999 Prentice-Hall, Inc. Chap. 3 - 6
• Measure of Variation
• Difference Between Largest & Smallest Observations:
Range =
• Ignores How Data Are Distributed:
The Range
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
SmallestrgestLa xx
© 1999 Prentice-Hall, Inc. Chap. 3 - 7
•Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
Variance
For the Population: use N in the denominator.
For the Sample : use n - 1 in the denominator.
N
(Xi 2
2
1
22
n
XXs i
© 1999 Prentice-Hall, Inc. Chap. 3 - 8
•Most Important Measure of Variation
•Shows Variation About the Mean:
•For the Population:
•For the Sample:
Standard Deviation
For the Population: use N in the denominator.
For the Sample : use n - 1 in the denominator.
N
Xi
2
1
2
n
XXs i
© 1999 Prentice-Hall, Inc. Chap. 3 - 9
Sample Standard Deviation
For the Sample : use n - 1 in the denominator.
Data: 10 12 14 15 17 18 18 24
s =
n = 8 Mean =16
18
1624161816171615161416121610 2222222
)()()()()()()(
Sample Standard Deviation= 4.2426
s
:X i
1
2
n
XX i
© 1999 Prentice-Hall, Inc. Chap. 3 - 10
Comparing Standard Deviations
Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 s = 4.57
Data C
© 1999 Prentice-Hall, Inc. Chap. 3 - 11
Coefficient of Variation
•Measure of Relative Variation
•Always a %
•Shows Variation Relative to Mean
•Used to Compare 2 or More Groups
•Formula ( for Sample):
100%
X
SCV
© 1999 Prentice-Hall, Inc. Chap. 3 - 12
Comparing Coefficient of Variation
Stock A: Average Price last year = $50
Standard Deviation = $5
Stock B: Average Price last year = $100
Standard Deviation = $5
100%
X
SCV
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
© 1999 Prentice-Hall, Inc. Chap. 3 - 13
Shape
• Describes How Data Are Distributed
• Measures of Shape: Symmetric or skewed
Right-SkewedLeft-Skewed SymmetricMean = Median = ModeMean Median Mode Median MeanMode
