describing data: summary measures
DESCRIPTION
Describing Data: Summary Measures. Measures of Central Location Mean, Median, Mode Measures of Variation Range, Variance and Standard Deviation Measures of Association Covariance and Correlation. Mean. It is the Arithmetic Average of data values: - PowerPoint PPT PresentationTRANSCRIPT
© 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