Question

What percentage of cases are lower than those with each Z score: -1, -2, -1.75, 1, 2, 1.75

For a data set with Mean = 20, SD = 3 find the Z scores for each of the following raw scores:

23, 17, 15, 22, 30. Would you consider any of these cases an outlier? Explain your reasoning.

Answer 1

We will have to use standard normal distribution table for finding the probabilities for the given z. Percentage of cases lower than given Z is basically the probability of getting Z lower than the given value. We can find this directly in standard normal distribution table. If you don't know how to use the table then you can either comment or use the excel formula that I will write after each answer.

P ( Z < -1) = NORM.DIST(-1,0,1,TRUE) { in excel} = 0.15

P( Z < -2) = NORM.DIST(-2,0,1,TRUE) = 0.0275

P( Z < -1.75) = NORM.DIST(-1.75,0,1,TRUE) = 0.04

P( Z < 1 ) = NORM.DIST(1,0,1,TRUE) = 0.85

P( Z < 2 ) = NORM.DIST(2,0,1,TRUE) = 0.97725

P ( Z < 1.75 ) = 0.96

Z score is ( X - mean)/ SD

Z for 23 is (23-20)/3 = 1

Z for 17 is (17-20)/3 = -1

Z for 22 is (22-20) / 3 = 0.66

Z for 15 is (15-20)/3 = -1.66

Z for 30 is (30-20) /3 = 3.33

P( Z > 3.33 ) = 0.00135 which is less than 1 %, this can be thus treated as outlier. Probability of things happening equal or greater than 30 is very very less thus this is an outlier.