Slides Prepared by JOHN S. LOUCKS St. Edwards University 2002 South-Western /Thomson Learning 1 Chapter 3 Descriptive Statistics: Numerical Methods Measures of Location Measures of Variability Measures of Relative Location and Detecting Outliers Exploratory Data Analysis Measures of Association Between Two Variables The Weighted Mean and Working with Grouped Data

x % 2 Measures of Location Mean Median Mode Percentiles Quartiles 3 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 425

430 430 435 435 435 435 435 440 440 presented 440 440 445 445 445 445 445 450 450 in440 ascending order. 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 4 Mean The mean of a data set is the average of all the data values. xIf the data are from a sample, the mean is denoted by xi . x n

If the data are from a population, the mean is denoted by (mu). xi N 5 Example: Apartment Rents Mean xi 34 , 356 x 490.80 n 70 425 440 450 465 480 510 575 430 440 450 470

485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460

475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450

465 480 510 570 615 6 Median The median is the measure of location most often reported for annual income and property value data. A few extremely large incomes or property values can inflate the mean. 7 Median The median of a data set is the value in the middle when the data items are arranged in ascending order. For an odd number of observations, the

median is the middle value. For an even number of observations, the median is the average of the two middle values. 8 Example: Apartment Rents Median values: 425 440 450 465 480 510 575 430 440 450 470 485 515 575 Median = 50th percentile i = (p/100)n = (50/100)70 = 35.5

Averaging the 35th and 36th data Median (475 435 + 475)/2 430 435 = 435 435 = 435475440 440 445 445 445 445 445 450 450 450 450 460 460 460 465 470 472 475 475 475 480 480 490 490 490 500 500 500 500 525 525 525 535 549 550 570 580 590 600 600 600 600 615 440 450 465 480 510 570 615 9 Mode

The mode of a data set is the value that occurs with greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data are bimodal. If the data have more than two modes, the data are multimodal. 10 Example: Apartment Rents Mode 450 occurred most frequently (7 times) Mode = 450 425 440 450 465 480 510 575 430 440 450 470

485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460

475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450

465 480 510 570 615 11 Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universities are frequently reported in terms of percentiles. 12 Percentiles The pth 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. Compute index i, the position of the pth percentile.

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 an integer, the p th percentile is the average of the values in positions i and i +1. 13 Example: Apartment Rents 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585 425 440 450 465 480 510 575 430 440 450 470 485 515 575

430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535

600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510

570 615 14 Quartiles Quartiles are specific percentiles First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile 15 Example: Apartment Rents Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 425 440 450 465 480 510 575

430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525

600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500

570 615 440 450 465 480 510 570 615 16 Measures of Variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. 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. 17 Measures of Variability

Range Interquartile Range Variance Standard Deviation Coefficient of Variation 18 Range The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values. 19 Example: Apartment Rents Range Range = largest value - smallest value Range = 615 - 425 = 190

425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525

590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500

550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 20 Interquartile Range The interquartile range of a data set is the difference between the third quartile and the first quartile. It is the range for the middle 50% of the data.

It overcomes the sensitivity to extreme data values. 21 Example: Apartment Rents Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 22 Variance The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation (xi) and the mean (x for a sample, for a population).

23 Variance The variance is the average of the squared differences between each data value and the mean. If the data set is a sample, the variance is 2 denoted by s2. ( x x ) s2 i n 1 If the data set is a population, the variance is denoted by 2. 2 ( x

) i 2 N 24 Standard Deviation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. If the data set is a sample, the standard 2 s s deviation is denoted s. If the data set is a population, the standard deviation is denoted (sigma).

2 25 Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. If the data set is a sample, the coefficient of variation is computed s as follows: x (100) If the data set is a population, the coefficient of variation is computed as follows: (100)

26 Example: Apartment Rents Variance 2 s ( xi x ) 2 n 1 2 ,996.16 Standard Deviation s s2 2996. 47 54. 74 Coefficient of Variation s 54. 74 100 100 11.15 x

490.80 27 Measures of Relative Location and Detecting Outliers z-Scores Chebyshevs Theorem Empirical Rule Detecting Outliers 28 z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the xi mean. x zi

s A data value less than the sample mean will have a z-score less than zero. A data value greater than the sample mean will have a z-score greater than zero. A data value equal to the sample mean will have a z-score of zero. 29 Example: Apartment Rents z-Score of Smallest Value (425) xi x 425 490.80 z 1. 20 s 54. 74 Standardized Values for Apartment Rents -1.20 -0.93

-0.75 -0.47 -0.20 0.35 1.54 -1.11 -0.93 -0.75 -0.38 -0.11 0.44 1.54 -1.11 -0.93 -0.75 -0.38 -0.01 0.62 1.63 -1.02 -0.84 -0.75 -0.34 -0.01 0.62 1.81 -1.02

-0.84 -0.75 -0.29 -0.01 0.62 1.99 -1.02 -0.84 -0.56 -0.29 0.17 0.81 1.99 -1.02 -0.84 -0.56 -0.29 0.17 1.06 1.99 -1.02 -0.84 -0.56 -0.20 0.17 1.08 1.99

-0.93 -0.75 -0.47 -0.20 0.17 1.45 2.27 -0.93 -0.75 -0.47 -0.20 0.35 1.45 2.27 30 Chebyshevs Theorem At least (1 - 1/k22) 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 k = 2 standard deviations of the mean. 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 k = 4 standard deviations of the mean. 31

Example: Apartment Rents Chebyshevs Theorem x Let k = 1.5 with 54.74 = 490.80 and s = At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56% x of the rent values must be between x - k(s) = 490.80 - 1.5(54.74) = 409 and + k(s) = 490.80 + 1.5(54.74) = 573 32 Example: Apartment Rents Chebyshevs Theorem (continued) Actually, 86% of the rent values are between 409 and 573. 425 440

450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435

445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600

440 450 465 480 500 570 615 440 450 465 480 510 570 615 33 Empirical Rule For data having a bell-shaped distribution: Approximately 68% of the data values will be within one standard deviation of the mean. 34 Empirical Rule For data having a bell-shaped distribution: Approximately 95% of the data values will

be within two standard deviations of the mean. 35 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. 36 Example: Apartment Rents Empirical Rule Interval % in Interval Within +/- 1s 436.06 to 545.54 Within +/- 2s 381.32 to 600.28 Within +/- 3s 326.58 to 655.02 425 430 430 435 435 435 435 100% 440 450

465 480 510 575 440 450 470 485 515 575 440 450 470 490 525 580 445 450 472 490 525 590 445 450 475 490 525

600 445 460 475 500 535 600 445 460 475 500 549 600 48/70 = 69% 68/70 = 97% 70/70 = 435 445 460 480 500 550 600 440 450 465 480

500 570 615 440 450 465 480 510 570 615 37 Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be an incorrectly recorded data value. It might be a data value that was incorrectly included in the data set.

It might be a correctly recorded data value that belongs in the data set ! 38 Example: Apartment Rents 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. Values -1.20 Standardized -1.11 -1.11 -1.02 -1.02 for -1.02Apartment -1.02 -1.02 Rents -0.93 -0.93 -0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75 -0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47 -0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20 -0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.35 0.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.45 1.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27 39 Exploratory Data Analysis

Five-Number Summary Box Plot 40 Five-Number Summary Smallest Value First Quartile Median Third Quartile Largest Value 41 Example: Apartment Rents Five-Number Summary Lowest Value = 425 First Quartile = 450 Median = 475 Third Quartile = 525 Largest Value = 425 430 430 435 435 435 435 435 440 440

615 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 42 Box Plot A box is drawn with its ends located at the first and third quartiles. A vertical line is drawn in the box at the location of the median. Limits are located (not drawn) using the interquartile range (IQR). The lower limit is located 1.5(IQR) below Q1. The upper limit is located 1.5(IQR) above Q3. Data outside these limits are considered outliers. continued

43 Box Plot (Continued) Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. The locations of each outlier is shown with the symbol * . 44 Example: Apartment Rents Box Plot Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 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 52 550 575 600 625 5 45 Measures of Association Between Two Variables Covariance Correlation Coefficient 46 Covariance

The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. 47 Covariance If the data sets are samples, the covariance is denoted by sxy. ( xi x )( yi y ) sxy n 1 If the data sets are populations, the covariance xy is denoted by . xy ( xi x )( yi y )

N 48 Correlation Coefficient The coefficient can take on values between -1 and +1. Values near -1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. If the data sets are samples, the coefficient is rxy. rxy sxy sx s y xy the coefficient is If the data sets are populations, xy x y

. xy 49 The Weighted Mean and Working with Grouped Data Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data 50 Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.

In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. 51 Weighted Mean x = wi xi wi where: xi = value of observation i wi = weight for observation i 52 Grouped Data The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data.

To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. 53 Mean for Grouped Data Sample Data fM x f i i i Population Data fM

i i N where: fi = frequency of class i Mi = midpoint of class i 54 Example: Apartment Rents Given below is the previous sample of monthly rents for one-bedroom apartments presented here as grouped Rent ($) Frequency data in the form of a frequency 420-439 8 distribution. 440-459 460-479 480-499 500-519 520-539 540-559 560-579 580-599 600-619

17 12 8 7 4 2 4 2 6 55 Example: Apartment Rents Mean for Grouped Data Rent ff ii M ff iiM Rent ($) ($) Mii Mii 420-439 88 429.5 3436.0 420-439 429.5 3436.0

440-459 17 449.5 7641.5 440-459 17 449.5 7641.5 460-479 12 469.5 5634.0 460-479 12 469.5 5634.0 480-499 88 489.5 3916.0 480-499 489.5 3916.0 approximation 500-519 77 509.5 3566.5 500-519 509.5 3566.5

differs by44$2.41 from 2118.0 520-539 529.5 520-539 529.5 2118.0 540-559 22 549.5 1099.0 540-559 549.5 1099.0 560-579 44 569.5 2278.0 560-579 569.5 2278.0 580-599 589.5 1179.0 580-599 589.5 1179.0 $490.80.22 600-619 66 609.5

3657.0 600-619 609.5 3657.0 Total 70 34525.0 Total 70 34525.0 34, 525 x 493. 21 70 This the actual sample mean of 56 Variance for Grouped Data Sample Data 2 f ( M

x ) i s2 i n 1 Population Data 2 f ( M ) i 2 i N 57 Example: Apartment Rents Variance for Grouped Data s 2 3, 017.89

Standard Deviation for Grouped Data s 3, 017.89 54. 94 This approximation differs by only $.20 from the actual standard deviation of $54.74. 58 End of Chapter 3 59