Question

Why is L75 different than the Third Quartile? I can easily find 3rd Quartile using excel. I'm not sure how to find L75 though? Thank you!!  

The average number of minutes Americans commute to work is 27.7 minutes (Sterling's Best Places, April 13, 2012). The average commute time in minutes for 48 cities are as follows: Click on the datafile logo to reference the data. DATA file Albuquerque 23.3 Jacksonville 26.2 Phoenix 28.3 Atlanta 28.3 Kansas City 23.4 Pittsburgh 25.0 Austin 24.6 Las Vegas 28.4 Portland 26.4 Baltimore 32.1 Little Rock 20.1 Providence 23.6 Boston 31.7 Los Angeles 32.2 Richmond 23.4 Charlotte 25.8 Louisville 21.4 Sacramento 25.8 Chicago 38.1 Memphis 23.8 Salt Lake City 20.2 Cincinnati 24.9 Miami 30.7 San Antonio Cleveland 26.8 Milwaukee 24.8 San Diego 24.8 Columbus 23.4 Minneapolis 23.6 San Francisco Dallas 28.5 Nashville 25.3 San Jose 28.5 Denver 28.1 New Orleans 31.7 Seattle 27.3 Detroit 29.3 New York 43.8 St. Louis 26.8 El Paso 24.4 Oklahoma City 22.0 Tucson 24.0 Fresno 23.0 Orlando 27.1 Tulsa 20.1 Indianapolis 24.8 Philadelphia 34.2 Washington, D.C. 32.8 26.1 32.6 a. What is the mean commute time for these 48 cities (to 2 decimals)? 26.91 minutes b. Compute the median commute time (to 2 decimals). 25.95 minutes c. Compute the mode (to 1 decimal). Enter the values in ascending order. The data are bimodal 7 . The modes are 23.4 and 24.8 . d. Compute the third quartile using the location of the 75th percentile formula shown on page 118 of text textbook (to 2 decimals). L75 is 28.5 , so the third quartile is 28.5 .

Answer 1

L75 is the estimated position of 3rd quartile or 75th percentile, but not the value itself. If there are n number of data points, then L75 = i(n+1) Here n = 48, then L75 = i(48 + 1) = 36.75 We illustrate this with following more simple examples - Example 1 : Let the given data set is : 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5 Here number of data points, n = 7 Then L75 = 3(7 + 1) = 6 :. 3rd quartile, Q3 = Value of the 6th position in the sorted data set = 6.5 Example 2 : Let the given data set is: 2, 4, 4, 1, 6, 8, 3, 6, 5, 10, 9, 4, 1 Then n = 13 L75 = {(13+1) = 10.5 We sort the data in ascending order : 1, 1, 2, 3, 4, 4, 4, 5, 6, 6, 8, 9, 10 Since we don't exactly have a 10.5 th position, we can only estimate the value using different methods.

I am using simple linear interpolation method, please follow the formula your textbook or instructor has provided. Using linear interpolation method we get f(L75]) - f(LL75]) 3rd quartile, Q3 = f(LL75]) + (L75 - [L75]) x (L75] - [L75] Where [2] denote smallest integer greater than or equal to z and largest integer less than or equal to z and fi) denotes value at ith position Then here Q3 = f(10) + (10.5 – 10) x (11)-f(10) = 6 +0.5 x 8,6 = 7 11-10