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Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved.

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3-2 Numerical Descriptive Measures Chapter 3 Learning Objectives LO 3.1Calculate and interpret the mean, the median, and the mode. LO 3.2Calculate and interpret percentiles and a box plot. LO 3.3Calculate and interpret the range, the mean absolute deviation, the variance, the standard deviation, and the coefficient of variation. LO 3.4Explain mean-variance analysis and the Sharpe ratio. LO 3.5Apply Chebyshevs Theorem, the empirical rule, and z- scores. LO 3.6Calculate the mean and the variance for grouped data. LO 3.7Calculate and interpret the covariance and the correlation coefficient.

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3-3 Numerical Descriptive Measures The arithmetic mean is a primary measure of central location. LO 3.1 Calculate and interpret the arithmetic mean, the median, and the mode. 3.1 Measures of Central Location Sample Mean Population Mean

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3-4 Numerical Descriptive Measures The mean is sensitive to outliers. Consider the salaries of employees at Acetech. LO Measures of Central Location This mean does not reflect the typical salary!

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3-5 Numerical Descriptive Measures The median is another measure of central location that is not affected by outliers. When the data are arranged in ascending order, the median is the middle value if the number of observations is odd, or the average of the two middle values if the number of observations is even. LO Measures of Central Location

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3-6 Numerical Descriptive Measures The mode is another measure of central location. The most frequently occurring value in a data set Used to summarize qualitative data A data set can have no mode, one mode (unimodal), or many modes (multimodal). Consider the salary of employees at Acetech The mode is $40,000 since this value appears most often. LO Measures of Central Location

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3-7 Numerical Descriptive Measures Weighted Mean Let w 1, w 2,..., w n denote the weights of the sample observations x 1, x 2,..., x n such that w 1 + w w n = 1, then LO Measures of Central Location

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3-8 Numerical Descriptive Measures In general, the pth percentile divides a data set into two parts: Approximately p percent of the observations have values less than the pth percentile; Approximately (100 p ) percent of the observations have values greater than the pth percentile. LO 3.2 Calculate and interpret percentiles and a box plot. 3.2 Percentiles and Box Plots

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3-9 Numerical Descriptive Measures Calculating the pth percentile: First arrange the data in ascending order. Locate the position, L p, of the pth percentile by using the formula: We use this position to find the percentile as shown next. LO 3.2 Percentiles and Box Plots

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3-10 Numerical Descriptive Measures Calculating the pth percentile Once you find L p, observe whether or not it is an integer. If L p is an integer, then the L p th observation in the sorted data set is the pth percentile. If L p is not an integer, then interpolate between two corresponding observations to approximate the pth percentile. LO 3.2 Percentiles and Box Plots

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3-11 Numerical Descriptive Measures Both L 25 = 2.75 and L 75 = 8.25 are not integers, thus The 25th percentile is located 75% of the distance between the second and third observations, and it is The 75th percentile is located 25% of the distance between the eighth and ninth observations, and it is LO 3.2 Percentiles and Box Plots

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3-12 A box plot allows you to: Graphically display the distribution of a data set. Compare two or more distributions. Identify outliers in a data set. Numerical Descriptive Measures Box * WhiskersOutliers LO 3.2 Percentiles and Box Plots

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3-13 Numerical Descriptive Measures The box plot displays 5 summary values: Min = smallest value Max = largest value Q1 = first quartile = 25th percentile Q2 = median = second quartile = 50th percentile Q3 = third quartile = 75th percentile LO 3.2 Percentiles and Box Plots MinMax

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3-14 Detecting outliers Calculate IQR, then multiply by 1.5 There are outliers if Q1 – Smallest Value > 1.5*IQR Largest Value – Q3 > 1.5*IQR Numerical Descriptive Measures LO 3.2 Percentiles and Box Plots

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3-15 Numerical Descriptive Measures Measures of dispersion gauge the variability of a data set. Measures of dispersion include: Range Mean Absolute Deviation (MAD) Variance and Standard Deviation Coefficient of Variation (CV) LO 3.4 Calculate and interpret the range, the mean absolute deviation, the variance, the standard deviation, and the coefficient of variation. 3.3 Measures of Dispersion

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3-16 Numerical Descriptive Measures Range It is the simplest measure. It is focused on extreme values. LO Measures of Dispersion

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3-17 Numerical Descriptive Measures Mean Absolute Deviation (MAD) MAD is an average of the absolute difference of each observation from the mean. LO Measures of Dispersion

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3-18 Numerical Descriptive Measures Variance and standard deviation For a given sample, For a given population, LO Measures of Dispersion

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3-19 Numerical Descriptive Measures Coefficient of variation (CV) CV adjusts for differences in the magnitudes of the means. CV is unitless, allowing easy comparisons of mean- adjusted dispersion across different data sets. LO Measures of Dispersion

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3-20 Numerical Descriptive Measures Mean-variance analysis: The performance of an asset is measured by its rate of return. The rate of return may be evaluated in terms of its reward (mean) and risk (variance). Higher average returns are often associated with higher risk. The Sharpe ratio uses the mean and variance to evaluate risk. LO 3.5 Explain mean-variance analysis and the Sharpe Ratio. 3.4 Mean-Variance Analysis and the Sharpe ratio

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3-21 Numerical Descriptive Measures Sharpe Ratio Measures the extra reward per unit of risk. For an investment І, the Sharpe ratio is computed as : whereis the mean return for the investment is the mean return for a risk-free asset is the standard deviation for the investment LO Mean-Variance Analysis and the Sharpe Ratio

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3-22 Numerical Descriptive Measures Chebyshevs Theorem For any data set, the proportion of observations that lie within k standard deviations from the mean is at least 1 1/k 2, where k is any number greater than 1. Consider a large lecture class with 280 students. The mean score on an exam is 74 with a standard deviation of 8. At least how many students scored within 58 and 90? With k = 2, we have 1 (1/2) 2 = At least 75% of 280 or 210 students scored within 58 and 90. LO 3.5 Apply Chebyshevs Theorem and the empirical rule. 3.5 Analysis of Relative Location

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3-23 Numerical Descriptive Measures The Empirical Rule: Approximately 68% of all observations fall in the interval Approximately 95% of all observations fall in the interval Almost all observations fall in the interval LO 3.5 Analysis of Relative Location

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3-24 Numerical Descriptive Measures LO 3.5 Analysis of Relative Location z-Scores Often useful to use z-score to denote relative location A sample values z-score indicates that sample values distance from the mean z-score calculated as A z-score of 1.25 indicates that that sample value is 1.25 standard deviations above the mean. A z-score of –2.43 would indicate a sample value that is 2.43 standard deviations below the mean.

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3-25 Numerical Descriptive Measures When data are grouped or aggregated, we use these formulas: where m i and i are the midpoint and frequency of the ith class, respectively. LO 3.6 Calculate the mean and the variance for grouped data. 3.6 Summarizing Grouped Data

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3-26 Numerical Descriptive Measures The covariance (s xy or xy ) describes the direction of the linear relationship between two variables, x and y. The correlation coefficient (r xy or xy ) describes both the direction and strength of the relationship between x and y. LO 3.7 Calculate and interpret the covariance and the correlation coefficient. 3.7 Covariance and Correlation

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3-27 Numerical Descriptive Measures The sample covariance s xy is computed as The population covariance xy is computed as LO 3.7 Covariance and Correlation

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3-28 Numerical Descriptive Measures The sample correlation r xy is computed as The population correlation xy is computed as Note, 1 < r xy < +1 or 1 < xy < +1 LO 3.7 Covariance and Correlation

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