MGMT305-Lec1

MGMT305-Lec1 - Chapter 3 Descriptive Statistics: Numerical...

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Unformatted text preview: Chapter 3 Descriptive Statistics: Numerical Methods s s s s Population Vs. Sample Measures of Location Measures of Variability Exploratory Data Analysis µ σ% x Slide 1 Population vs. Sample s s Population: The set of all elements of interest in a particular study. Sample: A subset of the population. Example: Purdue Students We might be interested in finding out the relationship between SAT scores, high school GPAs and current GPAs of all Purdue students (population or sample?), but given the difficulty of dealing with approximate 36,000 bits and pieces of info. We can choose 200 Purdue students randomly and compare their info (population or sample?). s Slide 2 Measures of Location s s s s s Mean Median Mode Percentiles Quartiles Slide 3 Example: Apartment Rents Given below is a sample of monthly rent values ($) for one­bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order. 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 Slide 4 Mean If the data are from a sample, the mean is denoted by . x ∑ xi x= s n s If the data are from a population, the mean is denoted by µ (mu). ∑ xi µ= N Slide 5 Example: Apartment Rents s Mean ∑ xi 34 , 356 x= = = 490.80 n 70 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 Slide 6 Median s s s s s The median 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. 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. Slide 7 Example: Apartment Rents s Median 425 440 450 465 480 510 575 Median = 50th percentile i = (p/100)n = (50/100)70 = 35 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 430 430 435 435 435 435 435 440 440 440 445 445 445 445 445 450 450 450 450 450 460 460 460 465 470 470 472 475 475 475 480 480 485 490 490 490 500 500 500 500 515 525 525 525 535 549 550 570 575 580 590 600 600 600 600 615 440 450 465 480 510 570 615 Slide 8 Mode s s s s 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. Slide 9 Example: Apartment Rents Mode 450 occurred most frequently (7 times) Mode = 450 s 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 Slide 10 Percentiles s 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 next integer greater than i is the position of the p th percentile. If i is an integer, the p th percentile is the average of the values in positions i and i +1. Slide 11 Example: Apartment Rents s 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 480 510 575 470 485 515 575 470 490 525 580 472 490 525 590 475 490 525 600 475 500 535 600 475 500 549 600 480 500 550 600 480 500 570 615 480 510 570 615 Slide 12 Quartiles s s s s Quartiles are specific percentiles First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile Slide 13 Example: Apartment Rents s Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 Slide 14 Measures of Variability s s s s s Range Interquartile Range Variance Standard Deviation Coefficient of Variation Slide 15 Example: Apartment Rents s Range Range = largest value ­ smallest value Range = 615 ­ 425 = 190 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 Slide 16 Interquartile Range s s s 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 as for the range . Slide 17 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 s 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 Slide 18 Variance s s 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 denoted by s2. ∑ ( xi − x ) n −1 2 2 2 ∑ xi − n ( x ) = n −1 s= s 2 If the data set is a population, the variance is denoted by σ 2. ∑ ( xi − µ ) σ2 = N 2 Slide 19 Standard Deviation s s s The standard deviation 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 deviation is denoted s. If the data set is a population, the standard deviation is denoted σ (sigma). s s = s2 σ = σ2 Slide 20 Notation Mean Variance Standard Deviation Population µ Sample s2 s σ σ 2 Slide 21 Why Mean, Variance, and SD? s It is often desirable to consider measures of location, as well as measures of variability (dispersion). 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. Which supplier would you choose? Supplier A Mean of Delivery Times SD of Delivery Times 8 3.5 Supplier B 10 3 Slide 22 s s Coefficient of Variation s s 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 as follows: s (100) x s If the data set is a population, the coefficient of variation is computed as follows: σ (100) µ Slide 23 Example: Apartment Rents s Variance s s 2 ∑ = ( xi − x ) 2 n −1 = 2 , 996.16 Standard Deviation s = s2 = 2996. 47 = 54. 74 s Coefficient of Variation s 54. 74 × 100 = × 100 = 11.15 x 490.80 Slide 24 Example s Example: Police records show the following numbers of daily crime reports for a sample of days during the summer months. Summer 28 18 24 32 18 (a) Compute the sample mean. Slide 25 Example (continued) (b) Compute the sample variance. (use the first formula) (c) Compute the sample coefficient of variation. Slide 26 Exploratory Data Analysis s s Five­Number Summary Box Plot Slide 27 Five­Number Summary s s s s s Smallest Value First Quartile Median Third Quartile Largest Value Slide 28 Example: Apartment Rents s Five­Number Summary Lowest Value = 425 First Quartile = 450 Median = 475 Third Quartile = 525 Largest Value = 615 425 430 430 435 435 435 435 435 440 440 440 440 440 445 445 445 445 445 450 450 450 450 450 450 450 460 460 460 465 465 465 470 470 472 475 475 475 480 480 480 480 485 490 490 490 500 500 500 500 510 510 515 525 525 525 535 549 550 570 570 575 575 580 590 600 600 600 600 615 615 Slide 29 Box Plot s s s 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 Slide 30 Box Plot (Continued) s 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 * . s Slide 31 Example: Apartment Rents s Box Plot Lower Limit: Q1 ­ 1.5(IQR) = Upper Limit: Q3 + 1.5(IQR) = There are no outliers. 37 5 40 0 42 5 45 0 47 5 50 0 52 5 550 575 600 625 Slide 32 ...
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