Normal Distribution The Bell Curve

Questions What are the parameters that drive the normal distribution? What does each control? Draw a picture to illustrate. Identify proportions of the normal, e.g., what percent falls above the mean? Between 1 and 2 SDs above the mean? What is the 95 percent confidence interval for the mean? How can the confidence interval be computed?

Function The Normal is a theoretical distribution specified by its two parameters. It is unimodal and symmetrical. The mode, median and mean are all just in the middle.

Function (2) There are only 2 variables that determine the curve, the mean and the variance. The rest are constants. 2 is 2. Pi is about 3.14, and e is the natural exponent (a number between 2 and 3). In z scores (M=0, SD=1), the equation becomes: (Negative exponent means that big |z| values give small function values in the tails.)

Areas and Probabilities Cumulative probability:

Areas and Probabilities (2) Probability of an Interval

Areas and Probabilities (3) Howell Table 3.1 shows a table with cumulative and split proportions z Mean to z Larger F(a) Smaller .5 .1915 .6915 .3085 1 .3413 .8413 .1587 1.96 .4750 .9750 .0250 Graph illustrates z = 1. The shaded portion is about 16 percent of the area under the curve.

R functions for the Normal ‘p’ returns cumulative density (probability) ‘q’ returns quantile or inverse of density ‘r’ returns random numbers ‘d’ returns density or height at the point pnorm(0) = .5; pnorm(1) =.8413 qnorm(.5)=0; qnorm(.13)= -1.13 > rnorm(3) [1] -1.529100 -1.597028 1.436857 dnorm(0) = .3989 (not used much)

Areas and Probabilities (3) Using the unit normal (z), we can find areas and probabilities for any normal distribution. Suppose X=120, M=100, SD=10. Then z=(120-100)/10 = 2. About 98 % of cases fall below a score of 120 if the distribution is normal. In the normal, most (95%) are within 2 SD of the mean. Nearly everybody (99%) is within 3 SD of the mean.

Review What are the parameters that drive the normal distribution? What does each control? Draw a picture to illustrate. Identify proportions of the normal, e.g., what percent falls below a z of .4? What part falls below a z of –1?

Importance of the Normal Errors of measures, perceptions, predictions (residuals, etc.) X = T+e (true score theory) Distributions of real scores (e.g., height); if normal, can figure much Math implications (e.g., inferences re variance) Will have big role in statistics, described after the sampling distribution is introduced

Computer Exercise Open Davis Exercise Follow the instructions it it – Using R to describe distributions