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1 1 Slide Slides Prepared by JOHN S. LOUCKS St. Edwards University © 2002 South-Western /Thomson Learning

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2 2 Slide Chapter 3 Descriptive Statistics: Numerical Methods n Measures of Location n Measures of Variability n Measures of Relative Location and Detecting Outliers n Exploratory Data Analysis n Measures of Association Between Two Variables n The Weighted Mean and Working with Grouped Data Working with Grouped Data x x % %

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3 3 Slide Measures of Location n Mean n Median n Mode n Percentiles n Quartiles

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4 4 Slide Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.

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5 5 Slide Mean n The mean of a data set is the average of all the data values. n If the data are from a sample, the mean is denoted by. If the data are from a population, the mean is denoted by (mu). If the data are from a population, the mean is denoted by (mu).

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6 6 Slide Example: Apartment Rents n Mean

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7 7 Slide Median n The median is the measure of location most often reported for annual income and property value data. n A few extremely large incomes or property values can inflate the mean.

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8 8 Slide Median n The median of a data set is the value in the middle when the data items are arranged in ascending order. n For an odd number of observations, the median is the middle value. n For an even number of observations, the median is the average of the two middle values.

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9 9 Slide Example: Apartment Rents n Median Median = 50th percentile Median = 50th percentile i = ( p /100) n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

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10 Slide Mode n The mode of a data set is the value that occurs with greatest frequency. n The greatest frequency can occur at two or more different values. n If the data have exactly two modes, the data are bimodal. n If the data have more than two modes, the data are multimodal.

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11 Slide Example: Apartment Rents n Mode 450 occurred most frequently (7 times) 450 occurred most frequently (7 times) Mode = 450 Mode = 450

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12 Slide Percentiles n A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. n Admission test scores for colleges and universities are frequently reported in terms of percentiles.

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13 Slide 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. 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 i = ( p /100) n If i is not an integer, round up. The p th percentile is the value in the i th position. If i is not an integer, round up. The p th percentile is the value in the i th position. If i is an integer, the p th percentile is the average of the values in positions i and i +1. If i is an integer, the p th percentile is the average of the values in positions i and i +1. Percentiles

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14 Slide Example: Apartment Rents n 90th Percentile i = ( p /100) n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585 90th Percentile = (580 + 590)/2 = 585

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15 Slide Quartiles n Quartiles are specific percentiles n First Quartile = 25th Percentile n Second Quartile = 50th Percentile = Median n Third Quartile = 75th Percentile

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16 Slide Example: Apartment Rents n Third Quartile Third quartile = 75th percentile Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525 Third quartile = 525

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17 Slide Measures of Variability n It is often desirable to consider measures of variability (dispersion), as well as measures of location. n 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.

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18 Slide Measures of Variability n Range n Interquartile Range n Variance n Standard Deviation n Coefficient of Variation

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19 Slide Range n The range of a data set is the difference between the largest and smallest data values. n It is the simplest measure of variability. n It is very sensitive to the smallest and largest data values.

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20 Slide Example: Apartment Rents n Range Range = largest value - smallest value Range = largest value - smallest value Range = 615 - 425 = 190 Range = 615 - 425 = 190

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21 Slide Interquartile Range n The interquartile range of a data set is the difference between the third quartile and the first quartile. n It is the range for the middle 50% of the data. n It overcomes the sensitivity to extreme data values.

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22 Slide Example: Apartment Rents n Interquartile Range 3rd Quartile ( Q 3) = 525 3rd Quartile ( Q 3) = 525 1st Quartile ( Q 1) = 445 1st Quartile ( Q 1) = 445 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80 Interquartile Range = Q 3 - Q 1 = 525 - 445 = 80

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23 Slide Variance n The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample, for a population). It is based on the difference between the value of each observation ( x i ) and the mean ( x for a sample, for a population).

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24 Slide Variance n The variance is the average of the squared differences between each data value and the mean. n If the data set is a sample, the variance is denoted by s 2. If the data set is a population, the variance is denoted by 2. If the data set is a population, the variance is denoted by 2.

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25 Slide Standard Deviation n The standard deviation of a data set is the positive square root of the variance. n It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. n If the data set is a sample, the standard deviation is denoted s. If the data set is a population, the standard deviation is denoted (sigma). If the data set is a population, the standard deviation is denoted (sigma).

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26 Slide Coefficient of Variation n The coefficient of variation indicates how large the standard deviation is in relation to the mean. n If the data set is a sample, the coefficient of variation is computed as follows: n If the data set is a population, the coefficient of variation is computed as follows:

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27 Slide Example: Apartment Rents n Variance n Standard Deviation n Coefficient of Variation

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28 Slide Measures of Relative Location and Detecting Outliers n z-Scores n Chebyshevs Theorem n Empirical Rule n Detecting Outliers

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29 Slide z-Scores n The z-score is often called the standardized value. n It denotes the number of standard deviations a data value x i is from the mean. n A data value less than the sample mean will have a z- score less than zero. n A data value greater than the sample mean will have a z-score greater than zero. n A data value equal to the sample mean will have a z- score of zero.

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30 Slide n z-Score of Smallest Value (425) Standardized Values for Apartment Rents Example: Apartment Rents

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31 Slide Chebyshevs Theorem At least (1 - 1/ k 2 ) of the items in any data set will be At least (1 - 1/ k 2 ) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. At least 75% of the items must be within At least 75% of the items must be within k = 2 standard deviations of the mean. At least 89% of the items must be within At least 89% of the items must be within k = 3 standard deviations of the mean. At least 94% of the items must be within At least 94% of the items must be within k = 4 standard deviations of the mean. At least (1 - 1/ k 2 ) of the items in any data set will be At least (1 - 1/ k 2 ) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. At least 75% of the items must be within At least 75% of the items must be within k = 2 standard deviations of the mean. At least 89% of the items must be within At least 89% of the items must be within k = 3 standard deviations of the mean. At least 94% of the items must be within At least 94% of the items must be within k = 4 standard deviations of the mean.

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32 Slide Example: Apartment Rents n Chebyshevs Theorem Let k = 1.5 with = 490.80 and s = 54.74 Let k = 1.5 with = 490.80 and s = 54.74 At least (1 - 1/(1.5) 2 ) = 1 - 0.44 = 0.56 or 56% of the rent values must be between of the rent values must be between - k ( s ) = 490.80 - 1.5(54.74) = 409 - k ( s ) = 490.80 - 1.5(54.74) = 409 and and + k ( s ) = 490.80 + 1.5(54.74) = 573 + k ( s ) = 490.80 + 1.5(54.74) = 573

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33 Slide n Chebyshevs Theorem (continued) Actually, 86% of the rent values Actually, 86% of the rent values are between 409 and 573. are between 409 and 573. Example: Apartment Rents

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34 Slide Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: Approximately 68% of the data values will be within one standard deviation of the mean. Approximately 68% of the data values will be within one standard deviation of the mean.

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35 Slide Empirical Rule For data having a bell-shaped distribution: Approximately 95% of the data values will be within two standard deviations of the mean. Approximately 95% of the data values will be within two standard deviations of the mean.

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36 Slide Empirical Rule For data having a bell-shaped distribution: Almost all (99.7%) of the items will be within three standard deviations of the mean. Almost all (99.7%) of the items will be within three standard deviations of the mean.

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37 Slide Example: Apartment Rents n Empirical Rule Interval % in Interval Interval % in Interval Within +/- 1 s 436.06 to 545.5448/70 = 69% Within +/- 2 s 381.32 to 600.2868/70 = 97% Within +/- 3 s 326.58 to 655.0270/70 = 100%

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38 Slide Detecting Outliers n An outlier is an unusually small or unusually large value in a data set. n A data value with a z-score less than -3 or greater than +3 might be considered an outlier. n It might be an incorrectly recorded data value. n It might be a data value that was incorrectly included in the data set. n It might be a correctly recorded data value that belongs in the data set !

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39 Slide Example: Apartment Rents n Detecting Outliers The most extreme z-scores are -1.20 and 2.27. Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

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40 Slide Exploratory Data Analysis n Five-Number Summary n Box Plot

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41 Slide Five-Number Summary n Smallest Value n First Quartile n Median n Third Quartile n Largest Value

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42 Slide Example: Apartment Rents n Five-Number Summary Lowest Value = 425 First Quartile = 450 Median = 475 Median = 475 Third Quartile = 525 Largest Value = 615

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43 Slide Box Plot n A box is drawn with its ends located at the first and third quartiles. n A vertical line is drawn in the box at the location of the median. n Limits are located (not drawn) using the interquartile range (IQR). The lower limit is located 1.5(IQR) below Q 1. The lower limit is located 1.5(IQR) below Q 1. The upper limit is located 1.5(IQR) above Q 3. The upper limit is located 1.5(IQR) above Q 3. Data outside these limits are considered outliers. Data outside these limits are considered outliers. … continued

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44 Slide Box Plot (Continued) n Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. n The locations of each outlier is shown with the symbol *.

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45 Slide Example: Apartment Rents n Box Plot Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 There are no outliers. 37 5 40 0 42 5 45 0 47 5 50 0 52 5 550 575 600 625

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46 Slide Measures of Association Between Two Variables n Covariance n Correlation Coefficient

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47 Slide Covariance n The covariance is a measure of the linear association between two variables. n Positive values indicate a positive relationship. n Negative values indicate a negative relationship.

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48 Slide n If the data sets are samples, the covariance is denoted by s xy. n If the data sets are populations, the covariance is denoted by. Covariance

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49 Slide Correlation Coefficient n The coefficient can take on values between -1 and +1. n Values near -1 indicate a strong negative linear relationship. n Values near +1 indicate a strong positive linear relationship. n If the data sets are samples, the coefficient is r xy. n If the data sets are populations, the coefficient is.

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50 Slide The Weighted Mean and Working with Grouped Data n Weighted Mean n Mean for Grouped Data n Variance for Grouped Data n Standard Deviation for Grouped Data

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51 Slide Weighted Mean n When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. n In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. n When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

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52 Slide Weighted Mean x = w i x i x = w i x i w i w iwhere: 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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53 Slide Grouped Data n The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. n To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. n We compute a weighted mean of the class midpoints using the class frequencies as weights. n Similarly, in computing the variance and standard deviation, the class frequencies are used as weights.

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54 Slide n Sample Data n Population Data where: f i = frequency of class i f i = frequency of class i M i = midpoint of class i M i = midpoint of class i Mean for Grouped Data

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55 Slide Example: Apartment Rents Given below is the previous sample of monthly rents for one-bedroom apartments presented here as grouped data in the form of a frequency distribution.

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56 Slide Example: Apartment Rents n Mean for Grouped Data This approximation differs by $2.41 from This approximation differs by $2.41 from the actual sample mean of $490.80. the actual sample mean of $490.80.

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57 Slide Variance for Grouped Data n Sample Data n Population Data

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58 Slide Example: Apartment Rents n Variance for Grouped Data n Standard Deviation for Grouped Data This approximation differs by only $.20 from the actual standard deviation of $54.74.

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59 Slide End of Chapter 3

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