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- 1. Slides Prepared by JOHN S. LOUCKS St. Edward’s University
- 2. Chapter 3 Descriptive Statistics: Numerical Methods Part A <ul><li>Measures of Location </li></ul><ul><li>Measures of Variability </li></ul>x %
- 3. Measures of Location <ul><li>Mean </li></ul><ul><li>Median </li></ul><ul><li>Mode </li></ul><ul><li>Percentiles </li></ul><ul><li>Quartiles </li></ul>
- 4. Example: Apartment Rents <ul><li>Given below is a sample of monthly rent values ($) </li></ul><ul><li>for one-bedroom apartments. The data is a sample of 70 </li></ul><ul><li>apartments in a particular city. The data are presented </li></ul><ul><li>in ascending order. </li></ul>
- 5. Mean <ul><li>The mean of a data set is the average of all the data values. </li></ul><ul><li>If the data are from a sample, the mean is denoted by </li></ul><ul><li>. </li></ul><ul><li>If the data are from a population, the mean is denoted by (mu). </li></ul>
- 6. Example: Apartment Rents <ul><li>Mean </li></ul>
- 7. Median <ul><li>The median is the measure of location most often reported for annual income and property value data. </li></ul><ul><li>A few extremely large incomes or property values can inflate the mean. </li></ul>
- 8. Median <ul><li>The median of a data set is the value in the middle when the data items are arranged in ascending order. </li></ul><ul><li>For an odd number of observations, the median is the middle value. </li></ul><ul><li>For an even number of observations, the median is the average of the two middle values. </li></ul>
- 9. Example: Apartment Rents <ul><li>Median </li></ul><ul><li> Median = 50th percentile </li></ul><ul><li>i = ( p /100) n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: </li></ul><ul><li>Median = (475 + 475)/2 = 475 </li></ul>
- 10. Mode <ul><li>The mode of a data set is the value that occurs with greatest frequency. </li></ul><ul><li>The greatest frequency can occur at two or more different values. </li></ul><ul><li>If the data have exactly two modes, the data are bimodal . </li></ul><ul><li>If the data have more than two modes, the data are multimodal . </li></ul>
- 11. Example: Apartment Rents <ul><li>Mode </li></ul><ul><li> 450 occurred most frequently (7 times) </li></ul><ul><li> Mode = 450 </li></ul>
- 12. Percentiles <ul><li>A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. </li></ul><ul><li>Admission test scores for colleges and universities are frequently reported in terms of percentiles. </li></ul>
- 13. <ul><li>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. </li></ul><ul><ul><li>Arrange the data in ascending order. </li></ul></ul><ul><ul><li>Compute index i , the position of the p th percentile. </li></ul></ul><ul><ul><li> </li></ul></ul><ul><ul><li> i = ( p /100) n </li></ul></ul><ul><ul><li>If i is not an integer, round up. The p th percentile is the value in the i th position. </li></ul></ul><ul><ul><li>If i is an integer, the p th percentile is the average of the values in positions i and i +1. </li></ul></ul>Percentiles
- 14. Example: Apartment Rents <ul><li>90th Percentile </li></ul><ul><li>i = ( p /100) n = (90/100)70 = 63 </li></ul><ul><li>Averaging the 63rd and 64th data values: </li></ul><ul><ul><li> 90th Percentile = (580 + 590)/2 = 585 </li></ul></ul>
- 15. Quartiles <ul><li>Quartiles are specific percentiles </li></ul><ul><li>First Quartile = 25th Percentile </li></ul><ul><li>Second Quartile = 50th Percentile = Median </li></ul><ul><li>Third Quartile = 75th Percentile </li></ul>
- 16. Example: Apartment Rents <ul><li>Third Quartile </li></ul><ul><li> Third quartile = 75th percentile </li></ul><ul><li> i = ( p /100) n = (75/100)70 = 52.5 = 53 </li></ul><ul><ul><li> Third quartile = 525 </li></ul></ul>
- 17. Measures of Variability <ul><li>It is often desirable to consider measures of variability (dispersion), as well as measures of location. </li></ul><ul><li>For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each. </li></ul>
- 18. Measures of Variability <ul><li>Range </li></ul><ul><li>Interquartile Range </li></ul><ul><li>Variance </li></ul><ul><li>Standard Deviation </li></ul><ul><li>Coefficient of Variation </li></ul>
- 19. Range <ul><li>The range of a data set is the difference between the largest and smallest data values. </li></ul><ul><li>It is the simplest measure of variability. </li></ul><ul><li>It is very sensitive to the smallest and largest data values. </li></ul>
- 20. Example: Apartment Rents <ul><li>Range </li></ul><ul><li> Range = largest value - smallest value </li></ul><ul><li> Range = 615 - 425 = 190 </li></ul>
- 21. Interquartile Range <ul><li>The interquartile range of a data set is the difference between the third quartile and the first quartile. </li></ul><ul><li>It is the range for the middle 50% of the data. </li></ul><ul><li>It overcomes the sensitivity to extreme data values. </li></ul>
- 22. Example: Apartment Rents <ul><li>Interquartile Range </li></ul><ul><li> 3rd Quartile ( Q 3) = 525 </li></ul><ul><li> 1st Quartile ( Q 1) = 445 </li></ul><ul><li> Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80 </li></ul>
- 23. Variance <ul><li>The variance is a measure of variability that utilizes all the data. </li></ul><ul><li>It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample, for a population). </li></ul>
- 24. Variance <ul><li>The variance is the average of the squared differences between each data value and the mean. </li></ul><ul><li>If the data set is a sample, the variance is denoted by s 2 . </li></ul><ul><li>If the data set is a population, the variance is denoted by 2 . </li></ul>
- 25. Standard Deviation <ul><li>The standard deviation of a data set is the positive square root of the variance. </li></ul><ul><li>It is measured in the same units as the data , making it more easily comparable, than the variance, to the mean. </li></ul><ul><li>If the data set is a sample, the standard deviation is denoted s . </li></ul><ul><li>If the data set is a population, the standard deviation is denoted (sigma). </li></ul>
- 26. Coefficient of Variation <ul><li>The coefficient of variation indicates how large the standard deviation is in relation to the mean. </li></ul><ul><li>If the data set is a sample, the coefficient of variation is computed as follows: </li></ul><ul><li>If the data set is a population, the coefficient of variation is computed as follows: </li></ul>
- 27. Example: Apartment Rents <ul><li>Variance </li></ul><ul><li>Standard Deviation </li></ul><ul><li>Coefficient of Variation </li></ul>
- 28. End of Chapter 3, Part A

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