The Normal Distribution

1. The Normal Distribution

2. The Normal Distribution • Distribution – any collection of scores, from either a sample or population • Can be displayed in any form, but is usually represented as a histogram • Normal Distribution – specific type of distribution that assumes a characteristic bell shape and is perfectly symmetrical

3. The Normal Distribution • Can provide us with information on likelihood of obtaining a given score • 60 people scored a 6 – 60/350 = .17 = 17% • 9 people scored a 1 – 3%

4. The Normal Distribution • Why is the Normal Distribution so important? • Almost all statistical tests that we will be covering throughout the course assume that the population distribution, that our sample is drawn from (but for the variable we are looking at), is normally distributed • Many variables that psychologists and health professionals look at are normally distributed

5. The Normal Distribution • Ordinate • Density – what is measured on the ordinate • Abscissa

6. The Normal Distribution • Mathematically defined as: • Since  and e are constants, we only have to determine μ (the population mean) and σ (the population standard deviation) to graph the mathematical function of any variable we are interested in • No, this will not be on the test

7. The Normal Distribution • Using this formula, mathematicians have determined the probabilities of obtaining every score on a “standard normal distribution” • To determine these probabilities for the variable you’re interested in we must plug in your variable to the formula • Note: This assumes that your variable fits a normal distribution, if not, your results will be inaccurate

8. The Normal Distribution • However, your data probably doesn’t exactly fit a standard normal distribution • μ = 0; σ = 1 • How do you get your variable to fit?

9. The Normal Distribution • Z-Scores • Range from +∞ to -∞ • Distribution: μ = 0; σ = 1 • Represent the number of standard deviations your score is from the mean • i.e. z = +1 is a score that is 1 standard deviation above the mean and z = -3 is a score 3 standard deviations below the mean

10. Reminder: Z-Scores represent # of standard deviations from the mean • For this distribution, if μ = 50 and σ = 10, what score does z = -3 represent? z = +2.5?

11. The Normal Distribution • What are the scores that lie in the middle 50% of a distribution of scores with μ = 50 and σ = 10? • z = ± .67 = .2500 • Solve for X using z-score formula • Scores = 56.7 and 43.3

12. The Normal Distribution

13. The Normal Distribution • We can do the same thing for the middle 95% to identify the scores in the extreme 2.5% of the distribution • z = 1.96 corresponds to p ≈ .025 • Keep this in mind, we’ll come back to it later

14. The Normal Distribution • Other uses for z-scores: • Converting variables to a standard metric • You took two exams, you got an 80 in Statistics and a 50 in Biology – you cannot say which one you did better in without knowing about the variability in scores in each • If the class average in Stats was a 90 and the s = 15, what would we conclude about your score now? How is it different than just using the score itself? • If the mean in Bio was a 30 and the s = 5, you did 4 s’ above the mean (a z-score of +4) or much better than everyone else

15. The Normal Distribution • Other uses for z-scores: • Converting variables to a standard metric • This also allows us to compare two scores on different metrics • i.e. two tests scored out of 100 = same metric one test out of 50 vs. one out of 100 = two different metrics • Is 20/50 better than 40/100? Is it better when compared to the class average? • Allows for quick comparisons between a score and the rest of the distribution it is a part of

16. The Normal Distribution • Standard Scores – scores with a predetermined mean and standard deviation, i.e. a z-score • Why convert to standard scores? • You can compare performance on two different tests with two different metrics • You can easily compute Percentile ranks • butthey are population-relative!