Lesson 3 3 Measures of Central Tendency and

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 - x (xi – x)²fi 0 – 1. 99 1 2 2 -5. 1 52. 02 2 – 3. 99 3 5 15 -3. 1 48. 05 4 – 5. 99 5 6 30 -1. 1 7. 26 6 – 7. 99 7 8 56 0. 9 6. 48 8 – 9. 99 9 9 81 2. 9 75. 69 Σfi = 30 Σxifi = 184 Total Mean Σxifi = 184 x ≈ -------- = 6. 1 Σfi = 30 Σ(xi – x)²fi = 189. 5 Variance Σ(xi – x)² fi = 189. 5 s² ≈ ------------- = 6. 53 Σfi – 1 = 29

Descriptive Stats from Summary Stats Class Midpt xi Freq fi xi - x (xi – x)²fi 0 – 1. 99 1 2 2 -5. 1 52. 02 2 – 3. 99 3 5 15 -3. 1 48. 05 4 – 5. 99 5 6 30 -1. 1 7. 26 6 – 7. 99 7 8 56 0. 9 6. 48 8 – 9. 99 9 9 81 2. 9 75. 69 Σfi = 30 Σxifi = 184 Total 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 – 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