Ibrahim Altubasi, PT, PhD The University of Jordan Normal Distribution Ibrahim Altubasi, PT, PhD The University of Jordan

Normal Distribution The Normal Distribution In simpler terms, the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther from the middle in either direction. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (relative frequency). The equation is (π and e are mathematical constants.)

Normal Distribution A mathematical idealization of a particular type of symmetric distribution A mathematical curve that provides a good model of relative frequency distributions found in research

Normal Distribution Properties of Normal Distribution: Unimodal Symmetric about its mean The mean, mode, and median of the distribution are all equal Asymptotic: The curve never touches the X axis

Normal Distribution The Standard normal distribution The normal distribution following a z-score transformation.

Normal Distribution The population mean in a test is 200 and the SD is 50. Let’s assume that the population distribution of the test scores is normal. Then we can say that 68.26% of the population will have z-scores between -1 and +1 (corresponding to 150 and 250). 95.44% of the population will have z-scores between -2 and +2 (corresponding to 100 and 300).

Normal Distribution The population mean in a test is 200 and the SD is 50. Let’s assume that the population distribution of the test scores is normal. If we randomly select a student from this population, the probability that the student’s score falls within 150 to 250 is 68.26%. The probability that the student’s score is more than 150 will be 84.13%. The probability that the student’s score is greater than 300 is only 2.28%. [The probability that one gets a score 300 or higher is so small that we may conclude that this person is not selected from the population with mean 200 and SD 50]

Normal Distribution Adult heights form a normal distribution with a mean of 68 inches and a SD of 6 inches. If we randomly select one from this population, the probability that his height will be 5 feet 2 inches (62 inches) or shorter is …. How about the probability that his height will be 6 feet 5 inches (77 inches) or higher? How about 70 inches or higher?

Normal Distribution Standard Normal Table B -- proportion in the body of the normal distribution up to the z-score value C -- proportion in the tail of the distribution beyond the z-score D -- proportion between the mean and the z-score value

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 1. Finding proportions/probabilities for specific z-score values What proportion of the normal distribution corresponding to z-score values greater than z=1.00? For a normal distribution, what is the probability of selecting a z-score less than z=1.50? How about the probability of selecting a z-score less than z= -.50?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 2. Finding a z-score corresponding to specific proportions For a normal distribution, what z-score separates the top 10% from the remainder of the distribution? For a normal distribution, what z-scores form the boundaries that separate the middle 60% of the distribution from the rest of the score?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 3. Finding percentile rank corresponding to a score from a normal distribution A distribution of scores is normal with μ=100 and σ=10. Sam had a score 92. What is the percentile rank for Sam’s score in the distribution?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 4. Finding percentiles from a normal distribution A distribution of scores is normal with μ=60 and σ=5. Based on the school report, Peter’s score is in the 34th percentile in the distribution. What is Peter’s actual score?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 5. Finding probabilities or proportions for scores from a normal distribution Adult heights form a normal distribution with a mean of 68 inches and a SD of 6 inches. If we randomly select one from this population, what is the proportion that his height will be 6 feet 5 inches (77 inches) or higher? How about 70 inches or higher? How about 65 inches or higher? z = (77-μ)/σ = (77-68)/6=1.50, p(z≥1.50) = ? z = (70-μ)/σ = (70-68)/6=0.33, p(z ≥ 0.33) = ? z = (65-μ)/σ = (65-68)/6=-0.5, p(z ≥ -0.5) = ?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 6. Finding probabilities or proportions between two scores The average speed of cars in a highway is μ=58 miles per hour with σ =10 and the distribution of the car speeds is normal. Given this information, if we randomly select a car driven in the highway, what is the probability that this car is traveling between 55 and 65 miles per hour? That is, p(55≤X ≤ 65)=? z = (55-μ)/σ = -.30, z = (65-μ)/σ= .70, p(-.30 ≤ z ≤.70) = ?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 6. Finding probabilities or proportions between two scores The average speed of cars in a highway is μ=58 miles per hour with σ =10 and the distribution of the car speeds is normal. Given this information, what is the proportion of the cars traveling between 70 and 80 miles per hour? That is, p(70 ≤ X ≤ 80)=?

Normal Distribution Standard Normal Table The use of the Standard Normal Table: 7. Finding raw scores corresponding to specific probabilities or proportions The scores in SAT is normally distributed with μ=500 and σ=100. Jane will take SAT and she wants her score to be in the top 15% of the SAT distribution. To achieve this goal, what is the minimum score Jane has to get? Top 15%  p=.15  z=? = (X-μ)/σ  X=zσ + μ