ELEMENTARY STATISTICS, BLUMAN Mean and Standard Deviation for a Probability Distribution © 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom.  No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

Objectives for this PowerPoint Calculate the mean for a discrete probability distribution Calculate the standard deviation for a discrete probability distribution

Example - Average Let x = number of girls in a family with 3 children 1 2 3 P(x) 1/8 3/8 What is the average number of girls in a family with 3 children? It seems like this average would be 1.5, but it is not possible through experimentation to determine this average.

Formula Such a determination would require infinitely many trials of the experiment. Therefore it is necessary that this average would be determined theoretically. The formula for calculating the theoretical mean for a discrete probability distribution shows that we must sum the products of the values of the random variable and the respective probabilities. μ= x∙P(x)

Calculation of Mean We will multiply each value of the random variable by its associated probability and add them all together. μ= x∙P x =0∙ 1 8 +1∙ 3 8 +2∙ 3 8 +3∙ 1 8 =1.5

Expected Value The mean of a probability distribution is also the expected value of the discrete random variable. The expected value is expressed as E(x). We can also interpret this result by saying that if infinitely many families with three children were observed, we would expect an average of 1.5 girls per family.

Variance Let x = number of girls in a family with 3 children x 1 2 3 1 2 3 P(x) 1/8 3/8 What is the variance for the number of girls in a family with 3 children? The formula for the variance is similar to the formula for the mean, except that the values of the random variable are squared and then the square of the mean is subtracted from the entire sum.

Calculation of Variance σ 2 = x 2 ∙P x − μ 2 = 0 2 ∙ 1 8 + 1 2 ∙ 3 8 + 2 2 ∙ 3 8 + 3 2 ∙ 1 8 − 1.5 2 =0.75

Standard Deviation The standard deviation of a probability distribution is found by taking the square root of the variance. 𝜎= 𝜎 2 = 0.75 =0.9 The rounding rule for the mean and a standard deviation for a discrete probability distribution is to round to one more decimal place than is found in the values of the random variable. Our random variable values are whole numbers. We should round our mean and standard deviation to the tenths place.

Summary In this PowerPoint we learned how to calculate the mean for a discrete probability distribution How to calculate the standard deviation for a discrete probability distribution