Business and Finance Colleges Principles of Statistics Eng. Heba Hamad week

Introduction to Statistics The Weighted Mean and Working with Grouped Data Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data

Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. Introduction to Statistics

Weighted Mean where: x i = value of observation i x i = value of observation i w i = weight for observation i w i = weight for observation i

Introduction to Statistics GPA Example Grade#Credits (Weight)Product A4416 B339 B326 C (sum of above) A = 4, B = 3, C = 2 GPA = 33/10 = 3.3

Grouped Data The weighted mean computation can be used to The weighted mean computation can be used to obtain approximations of the mean, variance, and obtain approximations of the mean, variance, and standard deviation for the grouped data. standard deviation for the grouped data. To compute the weighted mean, we treat the To compute the weighted mean, we treat the midpoint of each class as though it were the mean midpoint of each class as though it were the mean of all items in the class. of all items in the class. We compute a weighted mean of the class midpoints We compute a weighted mean of the class midpoints using the class frequencies as weights. using the class frequencies as weights. Similarly, in computing the variance and standard Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. deviation, the class frequencies are used as weights. Introduction to Statistics

Mean for Grouped Data where: f i = frequency of class i f i = frequency of class i M i = midpoint of class i M i = midpoint of class i Sample Data Population Data Introduction to Statistics

Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the form of a frequency distribution. Sample Mean for Grouped Data Sample Mean for Grouped Data

This approximation differs by $2.41 from the actual sample mean of $

Variance for Grouped Data For sample data For population data Introduction to Statistics

Sample Variance for Grouped Data Sample Variance for Grouped Datacontinued Introduction to Statistics

s 2 = 208,234.29/(70 – 1) = 3, This approximation differs by only $.20 from the actual standard deviation of $ Sample Variance for Grouped Data Sample Variance for Grouped Data Sample Variance Sample Standard Deviation Introduction to Statistics

General Examples Introduction to Statistics

Example 1 Fine mean, median, mode. Introduction to Statistics

Solution Introduction to Statistics

Example 2 Fine Standard deviation, variance for each of the two sample Introduction to Statistics

Fine Standard deviation, variance for each of the two sample Introduction to Statistics

Example 3 Introduction to Statistics

Example 4 Fine the indicated quartile or percentile a) Q1, b) Q3, c) P80, d) P33 Introduction to Statistics

Exercise Fine mean, median, mode, midrange, range, standard deviation, variance, P30 Age of US President Introduction to Statistics