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z Scores and Normal Distributions PSY 360 Introduction to Statistics for the Behavioral Sciences z Scores The aspect of the data we want to describe/measure is relative position z scores tell us how many standard deviations above or below the mean a given score falls z scores are statistics that describe the relative position of something in its distribution Verbal formula: z is something minus its mean divided by its standard deviation Formulas: For X in sample, z = (X-X) s For X in population, z = (X-µ) σ z Scores Characteristics: The mean of a distribution of z scores is zero The variance of a distribution of z scores is one The shape of a distribution of z scores is reflective, the shape is the same as the shape of the distribution of the Xs Example: Compute z for a sample when X=34, X=40, and s²=9 z= (X-X) = (34-40) = -6 = -2 s 3 3 Compute z for a population when X=10, µ=8, and σ²=16 z = (X-µ)/σ = (10-8)/4 = 2/4 = .5 1 Normal Distributions Family of theoretical distributions There are many different normal distributions Characteristics: Symmetric, continuous, unimodal Bell-shaped Scores range from -∞ to +∞ Mean, median, and mode are all the same value Each distribution has two parameters, µ and σ² Normal Distributions Examples: IQ is normally distributed with µ=100 and σ²=225 Height of American males is normally distributed with µ=69 and σ²=9 The standard normal (or unit normal) distribution has µ=0 and σ²=1 We can transform any normal distribution to the standard normal distribution by computing z scores: the resulting distribution of z scores will have a shape that is normal. WHY? Standard Normal Distribution We use this distribution to get probabilities associated with a z score (probability, proportion, and area under the curve are synonymous) Example: If Joe is 73 inches tall, what is the probability that any randomly selected man is his height or taller? For height, µ=69 and σ²=9, so z =(X-µ)/σ=(73-69)/3=4/3=1.33 From Table B.1 (pp. 687-690), -3 p(z>1.33)=.0918 -2 -1 0 1 2 3 2 Standard Normal Distribution Using Table B.1, there are two key facts: Total area equals one Symmetry Steps Example: p(z>2) Tail Small p(z>2)=.0228 (from Table B.1) 1 Draw a picture with zero and z Locate desired area: is it small or large? Is area in tail or middle? Use symmetry and/or total area=1 -3 -2 -1 0 1 2 3 Picture -3 -2 -1 0 1 2 3 Standard Normal Distribution More examples: if z is normal and p(z>1.645)=.05 Compute p(z<1.645) Want all the area to the left of 1.645, a large area. The area is in the middle and left tail. Use total area=1 to get p(z<1.645) = 1-.05 = .95 0 1.645 Standard Normal Distribution More examples: if z is normal and p(z>1.645)=.05 Compute p(z<-1.645) Want all the area to the left of -1.645, a small area. The area is in the left tail. Use symmetry to get p(z<-1.645) = .05 -1.645 0 3 Standard Normal Distribution More examples: if z is normal and p(z>1.645)=.05 Compute p(z>-1.645) Want all the area to the right of -1.645, a large area The area is in the middle and right tail Use symmetry and total area=1 to get p(z>-1.645) = .95 -1.645 0 Standard Normal Distribution More examples: if z is normal and p(z>1.645)=.05 Compute p(-1.645<z<1.645) Want all the area between -1.645 and 1.645, a large area The area is in the middle Use symmetry and total area=1 to get p(-1.645<z<1.645) = 1-2(.05) = 1-.10 = .90 -1.645 0 1.645 Standard Normal Distribution More examples: if z is normal and p(z>1.1)=.1357 Compute p(z<1.1) Want all the area to the left of 1.1, a large area The area is in the middle and left tail Use total area=1 to get p(z<1.1) = 1-.1357 = .8643 0 1.1 4 Standard Normal Distribution More examples: if z is normal and p(z>1.1)=.1357 Compute p(z<-1.1) Want all the area to the left of -1.1, a small area The area is in the left tail Use symmetry to get p(z<-1.1) = .1357 -1.1 0 Standard Normal Distribution More examples: if z is normal and p(z>1.1)=.1357 Compute p(z>-1.1) Want all the area to the right of -1.1, a large area The area is in the middle and right tail Use symmetry and total area=1 to get p(z>-1.1) = .8643 -1.1 0 Standard Normal Distribution More examples: if z is normal and p(z>1.1)=.1357 Compute p(-1.1<z<1.1) Want all the area between –1.1 and 1.1, a large area The area is in the middle Use symmetry and total area=1 to get p(-1.1<z<1.1) = -1.1 0 1.1 1-2(.1357) = .7286 5 Standard Normal Distribution Problem: Joe took a standardized test for which the norms are µ = 24 and σ2 = 4. His score was X = 22. When he computed his z score, he got z = (22–24)/4. Knowing that he is not doing as well in statistics as you are, Joe asked you to check his work. What did Joe get for his z score? Was he correct? Why? Solution: He should have divided by σ rather than σ2. His z score was really -1, not -.5 Standard Normal Distribution Data: 4, 3, 6, 5, 8, 3, 2, 4, 7, 8 Compute X Compute s2 Compute the z score for the person who has X=6 If you computed all ten z scores, what would the mean be for these ten z scores? If you computed all ten z scores, what would the variance be for these ten z scores? 6