GOALS. Describing Data: Numerical Measures. Why a Numeric Approach? Concepts & Goals. Characteristics of the Mean. Graphic of the Arithmetic Mean

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1 GOALS Describig Data: umerical Measures Chapter 3 Dr. Richard Jerz Calculate the arithmetic mea, weighted mea, media, ad mode Explai the characteristics, uses, advatages, ad disadvatages of each measure of locatio. Compute ad iterpret the rage, mea deviatio, variace, ad stadard deviatio. Uderstad the characteristics, uses, advatages, ad disadvatages of each measure of dispersio. Why a umeric Approach? Cocepts & Goals Graphic approach Quick overview of data Geeral relatioships ad treds Easy for people to iterpret umerical approach Detailed aalysis of data Powerful relatioships ad treds More difficult for people to iterpret Helps i calculatig probabilities Cetral Tedecy (oe umber) Mea (Average) Media Mode Dispersio (spread) Rage Mea deviatio Variace Stadard deviatio Be able to do this i Excel! 3 4 Characteristics of the Mea The arithmetic mea is the most widely used measure of locatio. It requires the iterval or ratio scales. Its major characteristics are: All values are used. It is uique. The sum of the deviatios from the mea is 0 The mea is affected by uusually large or small data values. It is calculated by summig the values ad dividig by the umber of values. Graphic of the Arithmetic Mea 5 6

2 Populatio Mea Example: Populatio Mea For ugrouped data, the populatio mea, μ, is the sum of all the populatio values divided by the total umber of populatio values: å i= µ = X i 7 8 Sample Mea Weighted Mea For ugrouped data, the sample mea is the sum of all the sample values divided by the umber of sample values: X = å i= X i The weighted mea of a set of umbers X, X,..., X, with correspodig weights w, w,...,w, is computed from the followig formula: ote: This is grouped data. 9 0 Example: Weighted Mea The Carter Costructio Compay pays its hourly employees $6.50, $9.00, or $5.00 per hour. There are 6 hourly employees, 4 of which are paid at the $6.50 rate, 0 at the $9.00 rate, ad at the $5.00 rate. What is the mea hourly rate paid the 6 employees? The Arithmetic Mea of grouped Data Example I Chapter, a frequecy distributio for the vehicle sellig prices was costructed. The iformatio is repeated below. Determie the arithmetic mea vehicle sellig price.

3 The Arithmetic Mea of Grouped Data The Media The media is the midpoit of the values after they have bee ordered from the smallest to the largest. Properties of the media: There is a uique media for a data set. There are as may values above the media as below it i the data array. With a eve set of values, the media will be the arithmetic average of the two middle umbers. It is ot affected by extremely large or small values. It ca be computed for ratio-level, iterval-level, ad ordial-level data. 3 4 The Mode The mode is the value of the observatio that appears most frequetly. Mea, Media, & Mode Shape of Distributio 5 6 Dispersio Dispersios with Same Mea Why Study Dispersio? The mea or the media oly describes the ceter of the data, but it does ot tell us aythig about the spread of the data. A secod reaso for studyig the dispersio i a set of data is to compare the spread i two or more distributios

4 Measures of Dispersio Example: Rage Rage Mea Deviatio å X -X MD = å( X - µ ) Variace s = Stadard Deviatio s = å( X - µ ) The umber of cappuccios sold at the Starbucks locatio i the Orage Coutry Airport betwee 4 ad 7 p.m. for a sample of 5 days last year were 0, 40, 50, 60, ad 80. Determie the rage for the umber of cappuccios sold. Rage = Largest Smallest value = 80 0 = Example: Mea Deviatio The umber of cappuccios sold at the Starbucks locatio i the Orage Coutry Airport betwee 4 ad 7 p.m. for a sample of 5 days last year were 0, 40, 50, 60, ad 80. Determie the mea deviatio for the umber of cappuccios sold. Example: Variace ad Stadard Deviatio The umber of traffic citatios issued durig the last five moths i Beaufort Couty, South Carolia, is 38, 6, 3, 4, ad. What is the populatio variace? Example: Sample Variace The hourly wages for a sample of part-time employees at Home Depot are: $, $0, $6, $8, ad $9. What is the sample variace? s = å( X - X) - Why Use the Stadard Deviatio Ad Variace? Iterestig observatios: Chebyshev s Theorem Empirical Rule 3 4 4

5 Chebyshev s Theorem The Empirical Rule The arithmetic mea biweekly amout cotributed by the Dupree Pait employees to the compay s profit-sharig pla is $5.54, ad the stadard deviatio is $7.5. At least what percet of the cotributios lie withi plus 3.5 stadard deviatios ad mius 3.5 stadard deviatios of the mea? 5 6 How do we do these calculatios i Excel? 7 5