Normal Distribution The Bell Curve Questions What are

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 0 0 Graph illustrates. 1915 z = 1. The shaded. 5 portion is about 1. 3413 16 percent of the 1. 96. 4750 area under the curve. Larger F(a). 5. 6915. 8413. 9750 Smaller. 5. 3085. 1587. 0250

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. 436857 1. 597028 - 1. 529100 - [1] • 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