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Statistics 800: Quantitative Business Analysis for Decision Making Measures of Locations and Variability

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403.42 Lecture Outline n n A. Discussion of Problems From Chapter 3 n n B. Population and Sample n n C. Averages (Mean, Median, Mode) n n D. Box Plot ; Cumulative Distribution Curves n n E. Variance and Standard Deviation ; Coefficient of Variation

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403.43 Normal Bell Curve

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403.44 Bell Curves Cont’d

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403.45 Portion of the population is called sample Sample aspects are called statistics Population aspects are called parameters Population mean is denoted by µ Population standard deviation by Population and Sample

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403.46 Notations

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7 Notations n n Capital letters (X, Y, Z, etc.) denote variables n n Lower case letters (x 1, x 2, x 3, etc.) denote observations for X. n n n = random sample size

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403.48 Averages (Mean, Median, Mode) n n Mean of x 1, x 2, ….x 300 for sample size n = 300. n n T = total, n n Mean =

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403.49 Mean (con’t) n n Formula for computing weighted mean:

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403.410 Median & Mode n n Median – –has property that 50% of the numbers in the data set are less, and 50% are greater in magnitude – –The median need not be a number in the data set n n Mode – –most frequently occurring number in the data set

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403.411 Ranks and Percentiles n n Ranks expressed as percentages are called Percentiles of a data set are. 0 th percentile 100 th percentile 50th Percentile 25th percentile (lower quartile) 75th Percentile (upper quartile)

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403.412 Box Plot n n Five number summary: – –smallest value (0 th percentile) – –lower quartile (25 th percentile) – –median (50 th percentile) – –upper quartile (75 th percentile) – –largest value (100 th percentile) n n Box Plot is a graphical representation of the five number summary

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Two Box Plots

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403.414 Cumulative Distribution Curves n n Plots data against their percentages n n Easy to estimate any percentile value

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403.415 One More Cumulative Distribution Curve

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403.416 Standard Deviation and Coefficient of Variation n n Sampling Variation or Sampling Error: Extent to which repeated samples may differ from each other. n n Reliability of an estimate is measured by its sampling error n n Standard deviation of a sample quantifies the extent to which its values vary from their mean

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403.417 Notation and Formula for Standard Deviation n n s = sample standard deviation n n Formula:

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403.418 An Empirical Rule n n Nearly 67% of the population is inside the one S.D. interval around the mean: n n Nearly 95% of the population is inside the two S.D. interval around the mean: n n Nearly 99.7% of the population is inside the three S.D. interval around the mean:

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403.419 Coefficient of Variation n n CV is a relative measure of variability; it is the standard deviation divided by the mean n n Useful when the variation is better understood as a percentage

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403.420 Effects of Adding to or Re- Scaling Data

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