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1 1 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. John Loucks St. Edward’s University...................... SLIDES. BY

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2 2 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 3, Part A Descriptive Statistics: Numerical Measures n Measures of Location n Measures of Variability

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3 3 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Location If the measures are computed for data from a sample, for data from a sample, they are called sample statistics. If the measures are computed for data from a sample, for data from a sample, they are called sample statistics. If the measures are computed for data from a population, for data from a population, they are called population parameters. If the measures are computed for data from a population, for data from a population, they are called population parameters. A sample statistic is referred to as the point estimator of the corresponding population parameter. A sample statistic is referred to as the point estimator of the corresponding population parameter. n Mean n Median n Mode n Percentiles n Quartiles

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4 4 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mean n The mean of a data set is the average of all the data values. The sample mean is the point estimator of the population mean . The sample mean is the point estimator of the population mean . n Perhaps the most important measure of location is the mean. n The mean provides a measure of central location.

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5 5 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Mean Number of observations in the sample Number of observations in the sample Sum of the values of the n observations Sum of the values of the n observations

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6 6 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Population Mean Number of observations in the population Number of observations in the population Sum of the values of the N observations Sum of the values of the N observations

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7 7 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seventy efficiency apartments were randomly Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below. Sample Mean Example: Apartment Rents Example: Apartment Rents

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8 8 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Mean Example: Apartment Rents Example: Apartment Rents

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9 9 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Median Whenever a data set has extreme values, the median Whenever a data set has extreme values, the median is the preferred measure of central location. is the preferred measure of central location. A few extremely large incomes or property values A few extremely large incomes or property values can inflate the mean. can inflate the mean. The median is the measure of location most often The median is the measure of location most often reported for annual income and property value data. reported for annual income and property value data. The median of a data set is the value in the middle The median of a data set is the value in the middle when the data items are arranged in ascending order. when the data items are arranged in ascending order.

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10 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Median 121419262718 27 For an odd number of observations: For an odd number of observations: in ascending order 26182712142719 7 observations the median is the middle value. Median = 19

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11 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 121419262718 27 Median For an even number of observations: For an even number of observations: in ascending order 26182712142730 8 observations the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 19 30

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12 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Median Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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13 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mode The mode of a data set is the value that occurs with The mode of a data set is the value that occurs with greatest frequency. greatest frequency. The greatest frequency can occur at two or more The greatest frequency can occur at two or more different values. different values. If the data have exactly two modes, the data are If the data have exactly two modes, the data are bimodal. bimodal. If the data have more than two modes, the data are If the data have more than two modes, the data are multimodal. multimodal. Caution: If the data are bimodal or multimodal, Caution: If the data are bimodal or multimodal, Excel’s MODE function will incorrectly identify a Excel’s MODE function will incorrectly identify a single mode. single mode.

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14 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Mode 450 occurred most frequently (7 times) Mode = 450 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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15 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Percentiles A percentile provides information about how the A percentile provides information about how the data are spread over the interval from the smallest data are spread over the interval from the smallest value to the largest value. value to the largest value. Admission test scores for colleges and universities Admission test scores for colleges and universities are frequently reported in terms of percentiles. are frequently reported in terms of percentiles. n The p th percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p ) percent of the items take on this value or more.

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16 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Percentiles Arrange the data in ascending order. Arrange the data in ascending order. Compute index i, the position of the p th percentile. Compute index i, the position of the p th percentile. i = ( p /100) n If i is not an integer, round up. The p th percentile If i is not an integer, round up. The p th percentile is the value in the i th position. is the value in the i th position. If i is not an integer, round up. The p th percentile If i is not an integer, round up. The p th percentile is the value in the i th position. is the value in the i th position. If i is an integer, the p th percentile is the average If i is an integer, the p th percentile is the average of the values in positions i and i +1. of the values in positions i and i +1. If i is an integer, the p th percentile is the average If i is an integer, the p th percentile is the average of the values in positions i and i +1. of the values in positions i and i +1.

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17 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 80 th Percentile i = ( p /100) n = (80/100)70 = 56 Averaging the 56 th and 57 th data values: 80th Percentile = (535 + 549)/2 = 542 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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18 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 80 th Percentile “At least 80% of the items take on a items take on a value of 542 or less.” “At least 20% of the items take on a value of 542 or more.” 56/70 =.8 or 80%14/70 =.2 or 20% Example: Apartment Rents Example: Apartment Rents

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19 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Quartiles Quartiles are specific percentiles. Quartiles are specific percentiles. First Quartile = 25th Percentile First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile Third Quartile = 75th Percentile

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20 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Third Quartile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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21 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Variability It is often desirable to consider measures of variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we For example, in choosing supplier A or supplier B we might consider not only the average delivery time for might consider not only the average delivery time for each, but also the variability in delivery time for each. each, but also the variability in delivery time for each.

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22 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Variability Range Range Interquartile Range Interquartile Range Variance Variance Standard Deviation Standard Deviation Coefficient of Variation Coefficient of Variation

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23 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Range The range of a data set is the difference between the The range of a data set is the difference between the largest and smallest data values. largest and smallest data values. It is the simplest measure of variability. It is the simplest measure of variability. It is very sensitive to the smallest and largest data It is very sensitive to the smallest and largest data values. values.

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24 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Range Range = largest value - smallest value Range = 615 - 425 = 190 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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25 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interquartile Range The interquartile range of a data set is the difference The interquartile range of a data set is the difference between the third quartile and the first quartile. between the third quartile and the first quartile. It is the range for the middle 50% of the data. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. It overcomes the sensitivity to extreme data values.

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26 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interquartile Range 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80 Note: Data is in ascending order. Example: Apartment Rents Example: Apartment Rents

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27 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The variance is a measure of variability that utilizes The variance is a measure of variability that utilizes all the data. all the data. The variance is a measure of variability that utilizes The variance is a measure of variability that utilizes all the data. all the data. Variance It is based on the difference between the value of It is based on the difference between the value of each observation ( x i ) and the mean ( for a sample, each observation ( x i ) and the mean ( for a sample, for a population). for a population). It is based on the difference between the value of It is based on the difference between the value of each observation ( x i ) and the mean ( for a sample, each observation ( x i ) and the mean ( for a sample, for a population). for a population). The variance is useful in comparing the variability The variance is useful in comparing the variability of two or more variables. of two or more variables. The variance is useful in comparing the variability The variance is useful in comparing the variability of two or more variables. of two or more variables.

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28 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Variance The variance is computed as follows: The variance is computed as follows: The variance is the average of the squared The variance is the average of the squared differences between each data value and the mean. differences between each data value and the mean. The variance is the average of the squared The variance is the average of the squared differences between each data value and the mean. differences between each data value and the mean. for a sample population

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29 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Standard Deviation The standard deviation of a data set is the positive The standard deviation of a data set is the positive square root of the variance. square root of the variance. The standard deviation of a data set is the positive The standard deviation of a data set is the positive square root of the variance. square root of the variance. It is measured in the same units as the data, making It is measured in the same units as the data, making it more easily interpreted than the variance. it more easily interpreted than the variance. It is measured in the same units as the data, making It is measured in the same units as the data, making it more easily interpreted than the variance. it more easily interpreted than the variance.

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30 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The standard deviation is computed as follows: The standard deviation is computed as follows: for a sample population Standard Deviation

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31 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The coefficient of variation is computed as follows: The coefficient of variation is computed as follows: Coefficient of Variation The coefficient of variation indicates how large the The coefficient of variation indicates how large the standard deviation is in relation to the mean. standard deviation is in relation to the mean. The coefficient of variation indicates how large the The coefficient of variation indicates how large the standard deviation is in relation to the mean. standard deviation is in relation to the mean. for a sample population

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32 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. the standard deviation is about 11% of the mean the standard deviation is about 11% of the mean Variance Variance Standard Deviation Standard Deviation Coefficient of Variation Coefficient of Variation Sample Variance, Standard Deviation, And Coefficient of Variation Example: Apartment Rents Example: Apartment Rents

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33 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Five Number Summaries and Box Plots n Measures of Association Between Two Variables n Data Dashboards: Adding Numerical Measures to Improve Effectiveness

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34 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Symmetric (not skewed) Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness = 0 Skewness = 0 Skewness is zero. Skewness is zero. Mean and median are equal. Mean and median are equal.

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35 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relative Frequency.05.10.15.20.25.30.35 0 0 Distribution Shape: Skewness n Moderately Skewed Left Skewness = .31 Skewness = .31 Skewness is negative. Skewness is negative. Mean will usually be less than the median. Mean will usually be less than the median.

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36 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Moderately Skewed Right Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness =.31 Skewness =.31 Skewness is positive. Skewness is positive. Mean will usually be more than the median. Mean will usually be more than the median.

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37 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Distribution Shape: Skewness n Highly Skewed Right Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness = 1.25 Skewness = 1.25 Skewness is positive (often above 1.0). Skewness is positive (often above 1.0). Mean will usually be more than the median. Mean will usually be more than the median.

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38 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Seventy efficiency apartments were randomly Seventy efficiency apartments were randomly sampled in a college town. The monthly rent prices for the apartments are listed below in ascending order. Distribution Shape: Skewness n Example: Apartment Rents

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39 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relative Frequency.05.10.15.20.25.30.35 0 0 Skewness =.92 Skewness =.92 Distribution Shape: Skewness n Example: Apartment Rents

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40 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The z-score is often called the standardized value. The z-score is often called the standardized value. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. z-Scores Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score. Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score.

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41 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z-Scores A data value less than the sample mean will have a A data value less than the sample mean will have a z-score less than zero. z-score less than zero. A data value greater than the sample mean will have A data value greater than the sample mean will have a z-score greater than zero. a z-score greater than zero. A data value equal to the sample mean will have a A data value equal to the sample mean will have a z-score of zero. z-score of zero. An observation’s z-score is a measure of the relative An observation’s z-score is a measure of the relative location of the observation in a data set. location of the observation in a data set.

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42 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z-Score of Smallest Value (425) z-Score of Smallest Value (425) z-Scores Example: Apartment Rents Example: Apartment Rents Standardized Values for Apartment Rents

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43 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule When the data are believed to approximate a When the data are believed to approximate a bell-shaped distribution … bell-shaped distribution … The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean.

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44 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 68.26%68.26% +/- 1 standard deviation of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 95.44%95.44% +/- 2 standard deviations of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. 99.72%99.72% +/- 3 standard deviations

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45 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rule x – 3 – 1 – 2 + 1 + 2 + 3 68.26% 95.44% 99.72%

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46 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Detecting Outliers An outlier is an unusually small or unusually large An outlier is an unusually small or unusually large value in a data set. value in a data set. A data value with a z-score less than -3 or greater A data value with a z-score less than -3 or greater than +3 might be considered an outlier. than +3 might be considered an outlier. It might be: It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the a data value that was incorrectly included in the data set data set a correctly recorded data value that belongs in a correctly recorded data value that belongs in the data set the data set

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47 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Detecting Outliers The most extreme z-scores are -1.20 and 2.27 The most extreme z-scores are -1.20 and 2.27 Using | z | > 3 as the criterion for an outlier, there Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. are no outliers in this data set. Standardized Values for Apartment Rents Example: Apartment Rents Example: Apartment Rents

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48 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summaries and Box Plots Summary statistics and easy-to-draw graphs can be Summary statistics and easy-to-draw graphs can be used to quickly summarize large quantities of data. used to quickly summarize large quantities of data. Summary statistics and easy-to-draw graphs can be Summary statistics and easy-to-draw graphs can be used to quickly summarize large quantities of data. used to quickly summarize large quantities of data. Two tools that accomplish this are five-number Two tools that accomplish this are five-number summaries and box plots. summaries and box plots. Two tools that accomplish this are five-number Two tools that accomplish this are five-number summaries and box plots. summaries and box plots.

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49 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summary 11 Smallest Value Smallest Value First Quartile First Quartile Median Median Third Quartile Third Quartile Largest Value Largest Value 22 33 44 55

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50 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Five-Number Summary Lowest Value = 425 First Quartile = 445 Median = 475 Third Quartile = 525 Largest Value = 615 Example: Apartment Rents Example: Apartment Rents

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51 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. Box plots provide another way to identify outliers. Box plots provide another way to identify outliers.

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52 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 400 425 450 475 500 525 550 575 600 625 A box is drawn with its ends located at the first and A box is drawn with its ends located at the first and third quartiles. third quartiles. Box Plot A vertical line is drawn in the box at the location of A vertical line is drawn in the box at the location of the median (second quartile). the median (second quartile). Q1 = 445 Q3 = 525 Q2 = 475 Example: Apartment Rents Example: Apartment Rents

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53 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot Limits are located (not drawn) using the interquartile range (IQR). Limits are located (not drawn) using the interquartile range (IQR). Data outside these limits are considered outliers. Data outside these limits are considered outliers. The locations of each outlier is shown with the symbol *. The locations of each outlier is shown with the symbol *.continued

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54 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Box Plot Whiskers (dashed lines) are drawn from the ends Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values of the box to the smallest and largest data values inside the limits. inside the limits. 400 425 450 475 500 525 550 575 600 625 Smallest value inside limits = 425 Largest value inside limits = 615 Example: Apartment Rents Example: Apartment Rents

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55 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Association Between Two Variables Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient.

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56 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance Positive values indicate a positive relationship. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Negative values indicate a negative relationship. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables.

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57 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance The covariance is computed as follows: The covariance is computed as follows: forsamples forpopulations

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58 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Correlation Coefficient Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation.

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59 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The correlation coefficient is computed as follows: The correlation coefficient is computed as follows: forsamplesforpopulations Correlation Coefficient

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60 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Correlation Coefficient Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship.

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61 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A golfer is interested in investigating the A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score. 277.6 259.5 269.1 267.0 255.6 272.9 69 71 70 70 71 69 Average Driving Distance (yds.) Average 18-Hole Score Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

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62 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Covariance and Correlation Coefficient 277.6259.5269.1267.0255.6272.9 697170707169 xy 10.65 10.65 -7.45 -7.45 2.15 2.15 0.05 0.05-11.35 5.95 5.95 1.0 1.0 0 0 -10.65 -10.65 -7.45 -7.45 0 0-11.35 -5.95 -5.95 Average Std. Dev. 267.070.0-35.40 8.2192.8944 Total Example: Golfing Study Example: Golfing Study

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63 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sample Covariance Sample Covariance Sample Correlation Coefficient Sample Correlation Coefficient Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

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64 Slide © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter

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