1. © 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

2. © 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 = 5Mean = 6 Sample Mean Mean n xxx n2i   n x n 1i i    x

3. © 1999 Prentice-Hall, Inc. Chap. 3 - 3 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 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

4. © 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

5. © 1999 Prentice-Hall, Inc. Chap. 3 - 5 Variation VarianceStandard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Measures Of Variability

6. © 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 

7. © 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 (X i    2 2    1 2 2     n XX s i

8. © 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 X i    2    1 2     n XX s i

9. © 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 Sample Standard Deviation= 4.2426 s  1 2     n XX i

10. © 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 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

11. © 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):

12. © 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 Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5%

13. © 1999 Prentice-Hall, Inc. Chap. 3 - 13 Shape Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Right-Skewed Left-SkewedSymmetric Mean =Median =Mode Mean Median Mode Median Mean Mod e