Measures of Central Tendency and Dispersion from Grouped Data Lesson 3 - 3 Measures of Central Tendency and Dispersion from Grouped Data

Objectives Approximate the mean of a variable from grouped data Compute the weighted mean Approximate the variance and standard deviation of a variable from grouped data

Vocabulary Weighted Mean – mean of a variable value times its weighted value

Calculating Stats from Summary Data Each class is assumed to have all of its data at the class midpoint Each classes observations serve as the weighted value GPA example: ∑ class hours * class grade = GPA This is a weighted average!

Descriptive Stats from Summary Stats Class Midpt xi Freq fi xi fi xi - x (xi – x)²fi 0 – 1.99 1 2 -5.1 52.02 2 – 3.99 3 5 15 -3.1 48.05 4 – 5.99 6 30 -1.1 7.26 6 – 7.99 7 8 56 0.9 6.48 8 – 9.99 9 81 2.9 75.69 Total Σfi = 30 Σxifi = 184 Σ(xi – x)²fi = 189.5 Mean Variance Σxifi = 184 x ≈ --------------- = 6.1 Σfi = 30 Σ(xi – x)² fi = 189.5 s² ≈ -------------------------- = 6.53 Σfi – 1 = 29

Descriptive Stats from Summary Stats Class Midpt xi Freq fi xi fi xi - x (xi – x)²fi 0 – 1.99 1 2 -5.1 52.02 2 – 3.99 3 5 15 -3.1 48.05 4 – 5.99 6 30 -1.1 7.26 6 – 7.99 7 8 56 0.9 6.48 8 – 9.99 9 81 2.9 75.69 Total Σfi = 30 Σxifi = 184 Σ(xi – x)²fi = 189.5 n/2 – CF 30/2 - 13 Median = M ≈ L + ------------ (ci) = 6 + ------------- (2) = 6 + (2/8) 2 = 6.25 f 8 L is the lower class limit of the class containing the median n is the total number of data values in the frequency distribution CF is the cum freq of the class preceding the median class f is the frequency of the median class ci is the class width of the median class

Summary & Homework Summary Homework: pg 161 - 165: 3, 4, 5, 21, 25 Use raw data whenever possible If grouped (summarized data) is the only data available, estimates for mean and standard deviation can still be obtained using class midpoints and numbers in each class Homework: pg 161 - 165: 3, 4, 5, 21, 25