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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. Edwards 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 n Weighted Mean n Geometric Mean

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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 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. Weighted Mean n The weights might be the number of credit hours earned for each grade, as in GPA. n In other weighted mean computations, quantities such as pounds, dollars, or volume are frequently used. n In some instances the mean is computed by giving each observation a weight that reflects its relative importance. n The choice of weights depends on the application.

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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. Weighted Mean Denominator: sum of the weightsDenominator: weights Numerator: sum of the weighted data values Numerator: sum of the weighted data values If data is from a population, r eplaces x. r eplaces x. If data is from a population, r eplaces x. r eplaces x. where: x i = value of observation i x i = value of observation i w i = weight for observation i w i = weight for observation i

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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. Weighted Mean Example: Construction Wages Example: Construction Wages Ron Butler, a custom home builder, is looking over the Expenses he incurred for a house he just completed constructing. For the purpose of pricing future projects, he would like to know the average wage ($/hour) he paid the workers he employed. (The cost of materials is estimated in advance by the architect.) Listed below are the categories of worker he employed, along with their respective wage and total hours worked. Ron Butler, a home builder, is looking over the expenses he incurred for a house he just built. For the purpose of pricing future projects, he would like to know the average wage ($/hour) he paid the workers he employed. Listed below are the categories of worker he employed, along with their respective wage and total hours worked. Ron Butler, a home builder, is looking over the expenses he incurred for a house he just built. For the purpose of pricing future projects, he would like to know the average wage ($/hour) he paid the workers he employed. Listed below are the categories of worker he employed, along with their respective wage and total hours worked.

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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. Weighted Mean Example: Construction Wages Example: Construction Wages FYI, equally-weighted (simple) mean = $21.21 FYI, equally-weighted (simple) mean = $21.21

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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. 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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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. 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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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. 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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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. 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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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. Trimmed Mean It is obtained by deleting a percentage of the It is obtained by deleting a percentage of the smallest and largest values from a data set and then smallest and largest values from a data set and then computing the mean of the remaining values. computing the mean of the remaining values. For example, the 5% trimmed mean is obtained by For example, the 5% trimmed mean is obtained by removing the smallest 5% and the largest 5% of the removing the smallest 5% and the largest 5% of the data values and then computing the mean of the data values and then computing the mean of the remaining values. remaining values. Another measure, sometimes used when extreme Another measure, sometimes used when extreme values are present, is the trimmed mean. values are present, is the trimmed mean.

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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. Geometric Mean It is often used in analyzing growth rates in It is often used in analyzing growth rates in financial data (where using the arithmetic mean financial data (where using the arithmetic mean will provide misleading results). will provide misleading results). It should be applied anytime you want to determine It should be applied anytime you want to determine the mean rate of change over several successive the mean rate of change over several successive periods (be it years, quarters, weeks,...). periods (be it years, quarters, weeks,...). The geometric mean is calculated by finding the n th The geometric mean is calculated by finding the n th root of the product of n values. root of the product of n values. Other common applications include: changes in Other common applications include: changes in populations of species, crop yields, pollution levels, populations of species, crop yields, pollution levels, and birth and death rates. and birth and death rates.

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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. Geometric Mean

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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. Geometric Mean Example: Rate of Return Example: Rate of Return Average growth rate per period is (.97752 - 1) (100) = -2.248% Average growth rate per period is (.97752 - 1) (100) = -2.248%

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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. 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, Excels MODE function will incorrectly identify a Excels MODE function will incorrectly identify a single mode. single mode.

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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. 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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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. The standard deviation is computed as follows: The standard deviation is computed as follows: for a sample population Standard Deviation

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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. 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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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 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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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. End of Chapter 3, Part A

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Measures of Variability

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