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Business Statistics: A Decision-Making Approach 7th Edition Chapter Describing Data Using Numerical Measures Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, 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 whisker graph 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 Weighted mean, percentiles, quartiles Measures of Variation Mean, median, mode Range, interquartile range, variance and standard deviation, coefficient of variation Using the mean and standard deviation together Coefficient of variation, z-scores 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 Population mean N = Population Size N Sample mean ∑x x1 + x + + x N µ= = N N i=1 i n = Sample Size n x= ∑x i=1 n i Business Statistics: A Decision- x1 + x + + x n = n 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 In an ordered array, the median is the “middle” number, i.e., the number that splits the distribution in half The median is not affected by extreme values 10 10 Median = Median = Business Statistics: A Decision- Chap 3-8 Median (continued) To find the median, sort the n data values from low to high (sorted data is called a data array) Find the value in the i = (1/2)n position The ith position is called the Median Index Point If i is not an integer, round up to next highest integer Business Statistics: A Decision- Chap 3-9 Median Example (continued) Data array: 4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24 Note that n = 13 Find the i = (1/2)n position: i = (1/2)(13) = 6.5 Since 6.5 is not an integer, round up to The median is the value in the 7th position: Md = 12 Business Statistics: A Decision- Chap 3-10 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 3-36 Comparing Coefficients 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 3-37 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 68% μ μ ± 1σ Business Statistics: A Decision- Chap 3-38 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 3-39 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 3-40 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 3-41 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 3-42 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 3-43 Standardized Value Example IQ scores in a large population have a bell-shaped distribution with mean μ = 100 and standard deviation σ = 15 Find the standardized score (z-score) for a person with an IQ of 121 Answer: x − μ 121 − 100 z= = = 1.4 σ 15 Someone with an IQ of 121 is 1.4 standard deviations above the mean Business Statistics: A Decision- Chap 3-44 Using Microsoft Excel Descriptive Statistics are easy to obtain from Microsoft Excel Use menu choice: Data / data analysis / descriptive statistics Enter details in dialog box Business Statistics: A Decision- Chap 3-45 Using Excel Select: Data / data analysis / descriptive statistics Business Statistics: A Decision- Chap 3-46 Using Excel (continued) Enter dialog box details Check box for summary statistics Click OK Business Statistics: A Decision- Chap 3-47 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 3-48 Chapter Summary Described measures of center and location Mean, median, mode, weighted mean Discussed percentiles and quartiles Created Box and Whisker Plots Illustrated distribution shapes Symmetric, skewed Business Statistics: A Decision- Chap 3-49 Chapter Summary (continued) Described measure of variation Range, interquartile range, variance, standard deviation, coefficient of variation Discussed Tchebysheff’s Theorem Calculated standardized data values Business Statistics: A Decision- Chap 3-50 ... i=1 n i Business Statistics: A Decision- x1 + x + + x n = n Chap 3-6 Mean (Arithmetic Average) (continued) The most common measure of central tendency Mean = sum of values divided by the... 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... the number that splits the distribution in half The median is not affected by extreme values 10 10 Median = Median = Business Statistics: A Decision- Chap 3-8 Median (continued) To find the
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