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Business Statistics: A Decision-Making Approach 6th Edition Chapter Describing Data Using Numerical Measures Business Statistics: A Decision-Making Approach, 6e © 2005 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- 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- 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 Variance Standard Deviation Coefficient of Variation Business Statistics: A Decision- Chap 3-4 Measures of Center and Location Overview Center and Location Mean Median Mode Weighted Mean n x= ∑x i=1 i XW n µ= i=1 i i i N ∑x wx ∑ = ∑w wx ∑ = ∑w i N Business Statistics: A Decision- µW i i i Chap 3-5 Mean (Arithmetic Average) The Mean is the arithmetic average of data values Sample mean n = Sample Size n x= Population mean ∑x n x1 + x + + x n = n N N = Population Size i=1 i ∑x x1 + x + + x N µ= = N N i=1 i Business Statistics: A Decision- 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- 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- 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- 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- 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- 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 = μ ⋅ 100% Business Statistics: A Decision- Sample s ⋅ 100% CV = x Chap 3-32 Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 s CVA = x $5 ⋅ 100% = ⋅ 100% = 10% $50 Stock B: Average price last year = $100 Standard deviation = $5 s CVB = x $5 ⋅ 100% = ⋅ 100% = 5% $100 Business Statistics: A Decision- 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- 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- 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- 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- 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- 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- 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- Chap 3-40 Using Excel Use menu choice: tools / data analysis / descriptive statistics Business Statistics: A Decision- Chap 3-41 Using Excel (continued ) Enter dialog box details Check box for summary statistics Click OK Business Statistics: A Decision- 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- 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- Chap 3-44 Chapter Summary (continued ) Illustrated distribution shapes Symmetric, skewed Discussed Tchebysheff’s Theorem Calculated standardized data values Business Statistics: A Decision- Chap 3-45 ... either vertical or horizontal format Business Statistics: A Decision- Chap 3-19 Distribution Shape and Box and Whisker Plot Left-Skewed Q1 Q2 Q3 Symmetric Q1 Q2 Q3 Business Statistics: A Decision-... quartile Business Statistics: A Decision- Chap 3-26 Interquartile Range Example: X minimum Q1 25% 12 Median (Q2) 25% 30 25% 45 X Q3 maximum 25% 57 70 Interquartile range = 57 – 30 = 27 Business. .. Interquartile Range Quartiles Weighted Mean Variance Standard Deviation Coefficient of Variation Business Statistics: A Decision- Chap 3-4 Measures of Center and Location Overview Center and Location
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