Statistics for Managers using Microsoft Excel 3 rd

Statistics for Managers using Microsoft Excel 3 rd Edition Chapter 3 Numerical Descriptive Measures © 2002 Prentice-Hall, Inc. Chap 3 -1

Chapter Topics n Measures of central tendency n Mean, median, mode, geometric mean, midrange n Quartile n Measure of variation n n Range, interquartile range, variance and standard deviation, coefficient of variation Shape n Symmetric, skewed, using box-and-whisker plots © 2002 Prentice-Hall, Inc. 2

Chapter Topics n n (continued) Coefficient of correlation Pitfalls in numerical descriptive measures and ethical considerations © 2002 Prentice-Hall, Inc. 3

Summary Measures Central Tendency Mean Median Mode Quartile Range Variance Geometric Mean © 2002 Prentice-Hall, Inc. Variation Coefficient of Variation Standard Deviation 4

Measures of Central Tendency Average Median Mode Geometric Mean © 2002 Prentice-Hall, Inc. 5

Mean (Arithmetic Mean) n Mean (arithmetic mean) of data values n Sample mean n Population mean Sample Size Population Size © 2002 Prentice-Hall, Inc. 6

Mean (Arithmetic Mean) (continued) n n 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 © 2002 Prentice-Hall, Inc. 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 7

Median n n Robust measure of central tendency Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 Median = 5 n 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 In an ordered array, the median is the “middle” number n n If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers © 2002 Prentice-Hall, Inc. 8

Mode n n n A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 © 2002 Prentice-Hall, Inc. 0 1 2 3 4 5 6 No Mode 9

Geometric Mean n n Useful in the measure of rate of change of a variable over time Geometric mean rate of return n Measures the status of an investment over time © 2002 Prentice-Hall, Inc. 10

Example An investment of $100, 000 declined to $50, 000 at the end of year one and rebounded to $100, 000 at end of year two: © 2002 Prentice-Hall, Inc. 11

Quartiles n Split Ordered Data into 4 Quarters 25% n 25% 25% Position of i-th Quartile Data in Ordered Array: 11 12 13 16 16 17 18 21 22 n n and Are Measures of Noncentral Location = Median, A Measure of Central Tendency © 2002 Prentice-Hall, Inc. 12

Measures of Variation Variance Range Population Variance Sample Variance Interquartile Range © 2002 Prentice-Hall, Inc. Standard Deviation Coefficient of Variation Population Standard Deviation Sample Standard Deviation 13

Range n Measure of variation Difference between the largest and the smallest observations: n Ignores the way in which data are distributed n Range = 12 - 7 = 5 7 8 © 2002 Prentice-Hall, Inc. 9 10 11 12 7 8 9 10 11 12 14

Interquartile Range n n Measure of variation Also known as midspread n n Spread in the middle 50% Difference between the first and third quartiles Data in Ordered Array: 11 12 13 16 16 17 n 17 18 21 Not affected by extreme values © 2002 Prentice-Hall, Inc. 15

Variance n n Important measure of variation Shows variation about the mean n Sample variance: n Population variance: © 2002 Prentice-Hall, Inc. 16

Standard Deviation n Most important measure of variation Shows variation about the mean Has the same units as the original data n Sample standard deviation: n Population standard deviation: © 2002 Prentice-Hall, Inc. 17

Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Data B 11 12 13 14 15 16 17 18 19 20 21 Mean = 15. 5 s =. 9258 20 21 Mean = 15. 5 s = 4. 57 Data C 11 12 © 2002 Prentice-Hall, Inc. 13 14 15 16 17 18 19 Mean = 15. 5 s = 3. 338 18

Coefficient of Variation n Measures relative variation n Always in percentage (%) n Shows variation relative to mean n Is used to compare two or more sets of data measured in different units n © 2002 Prentice-Hall, Inc. 19

Comparing Coefficient of Variation n Stock A: n n n Stock B: n n n Average price last year = $50 Standard deviation = $5 Average price last year = $100 Standard deviation = $5 Coefficient of variation: n Stock A: n Stock B: © 2002 Prentice-Hall, Inc. 20

Shape of a Distribution n Describes how data is distributed n Measures of shape n Symmetric or skewed Left-Skewed Symmetric Mean < Median < Mode Mean = Median =Mode © 2002 Prentice-Hall, Inc. Right-Skewed Mode < Median < Mean 21

Exploratory Data Analysis n Box-and-whisker plot n Graphical display of data using 5 -number summary Median( X smallest 4 © 2002 Prentice-Hall, Inc. 6 8 ) Xlargest 10 12 22

Distribution Shape and Box-and-Whisker Plot Left-Skewed © 2002 Prentice-Hall, Inc. Symmetric Right-Skewed 23

Coefficient of Correlation n Measures the strength of the linear relationship between two quantitative variables © 2002 Prentice-Hall, Inc. 24

Features of Correlation Coefficient n Unit free n Ranges between – 1 and 1 n n n The closer to – 1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship © 2002 Prentice-Hall, Inc. 25

Scatter Plots of Data with Various Correlation Coefficients Y Y r = -1 X Y r = -. 6 Y © 2002 Prentice-Hall, Inc. X r=0 X Y r =. 6 X r=1 X 26

Pitfalls in Numerical Descriptive Measures n Data analysis is objective n n Should report the summary measures that best meet the assumptions about the data set Data interpretation is subjective n Should be done in fair, neutral and clear manner © 2002 Prentice-Hall, Inc. 27

Ethical Considerations Numerical descriptive measures: n n n Should document both good and bad results Should be presented in a fair, objective and neutral manner Should not use inappropriate summary measures to distort facts © 2002 Prentice-Hall, Inc. 28

Chapter Summary n Described measures of central tendency n Mean, median, mode, geometric mean, midrange n Discussed quartile n Described measure of variation n n Range, interquartile range, variance and standard deviation, coefficient of variation Illustrated shape of distribution n Symmetric, skewed, box-and-whisker plots © 2002 Prentice-Hall, Inc. 29

Chapter Summary n n (continued) Discussed correlation coefficient Addressed pitfalls in numerical descriptive measures and ethical considerations © 2002 Prentice-Hall, Inc. 30