Business Statistics: A Decision-Making Approach 6th Edition Chapter Describing Data Using Numerical Measures Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-1 Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean, median, and mode for a set of data Compute the range, variance, and standard deviation and know what these values mean Construct and interpret a box and whiskers plot Compute and explain the coefficient of variation and z scores Use numerical measures along with graphs, charts, and tables to describe data Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-2 Chapter Topics Measures of Center and Location Other measures of Location Mean, median, mode, geometric mean, midrange Weighted mean, percentiles, quartiles Measures of Variation Range, interquartile range, variance and standard deviation, coefficient of variation Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-3 Summary Measures Describing Data Numerically Center and Location Other Measures of Location Mean Median Mode Variation Range Percentiles Interquartile Range Quartiles Weighted Mean Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Variance Standard Deviation Coefficient of Variation Chap 3-4 Measures of Center and Location Overview Center and Location Mean Median Mode Weighted Mean n x i x i1 n XW i i1 N A Decision-Making Approach, 6e © 2010 PrenticeBusiness Statistics: Hall, Inc i i i N x wx w wx w W i i i Chap 3-5 Mean (Arithmetic Average) The Mean is the arithmetic average of data values Sample mean x n = Sample Size n x i x1 x x n n i1 n Population mean N = Population Size N x i x1 x x N N N i1 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-6 Mean (Arithmetic Average) (continued ) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 10 Mean = 15 3 5 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc 10 Mean = 10 20 4 5 Chap 3-7 Median Not affected by extreme values 10 10 Median = Median = In an ordered array, the median is the “middle” number 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 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-8 Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes 10 11 12 13 14 Mode = Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc No Mode Chap 3-9 Weighted Mean Used when values are grouped by frequency or relative importance Example: Sample of 26 Repair Projects Days to Complete Frequency 12 8 Weighted Mean Days to Complete: XW wx w Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc i i i (4 5) (12 6) (8 7) (2 8) 12 164 6.31 days 26 Chap 3-10 Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 3.338 20 21 Mean = 15.5 s = 9258 20 21 Mean = 15.5 s = 4.57 Data B 11 12 13 14 15 16 17 18 19 Data C 11 12 13 14 15 16 17 18 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc 19 Chap 3-31 Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units Population σ CV 100% μ Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Sample s 100% CV x Chap 3-32 Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 s $5 CVA 100% 100% 10% $50 x Stock B: Average price last year = $100 Standard deviation = $5 s CVB x $5 100% 100% 5% $100 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Both stocks have the same standard deviation, but stock B is less variable relative to its price Chap 3-33 The Empirical Rule If the data distribution is bell-shaped, then the interval: μ 1σ contains about 68% of the values in the population or the sample X 68% μ μ 1σ Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-34 The Empirical Rule μ 2σ contains about 95% of the values in the population or the sample μ 3σ contains about 99.7% of the values in the population or the sample 95% 99.7% μ 2σ μ 3σ Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-35 Tchebysheff’s Theorem Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean Examples: At least within (1 - 1/12) = 0% …… k=1 (μ ± 1σ) (1 - 1/22) = 75% … k=2 (μ ± 2σ) (1 - 1/32) = 89% ……… k=3 (μ ± 3σ) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-36 Standardized Data Values A standardized data value refers to the number of standard deviations a value is from the mean Standardized data values are sometimes referred to as z-scores Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-37 Standardized Population Values x μ z σ where: x = original data value μ = population mean σ = population standard deviation z = standard score (number of standard deviations x is from μ) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-38 Standardized Sample Values x x z s where: x = original data value x = sample mean s = sample standard deviation z = standard score (number of standard deviations x is from μ) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-39 Using Microsoft Excel Descriptive Statistics are easy to obtain from Microsoft Excel Use menu choice: tools / data analysis / descriptive statistics Enter details in dialog box Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-40 Using Excel Use menu choice: tools / data analysis / descriptive statistics Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-41 Using Excel (continued ) Enter dialog box details Check box for summary statistics Click OK Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-42 Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000,000 500,000 300,000 100,000 100,000 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-43 Chapter Summary Described measures of center and location Mean, median, mode, geometric mean, midrange Discussed percentiles and quartiles Described measure of variation Range, interquartile range, variance, standard deviation, coefficient of variation Created Box and Whisker Plots Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-44 Chapter Summary (continued ) Illustrated distribution shapes Symmetric, skewed Discussed Tchebysheff’s Theorem Calculated standardized data values Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-45 ... Approach, 6e © 2010 PrenticeHall, Inc Chap 3-21 Measures of Variation Variation Range Interquartile Range Variance Standard Deviation Population Variance Population Standard Deviation Sample Variance... Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 3-36 Standardized Data Values A standardized data value refers to the number of standard deviations a value is from the mean... original data value μ = population mean σ = population standard deviation z = standard score (number of standard deviations x is from μ) Business Statistics: A Decision- Making Approach,