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Normal Distribution

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?

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Normal Distribution

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  1. Normal Distribution The Bell Curve

  2. 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?

  3. 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.

  4. 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.)

  5. Areas and Probabilities • Cumulative probability:

  6. Areas and Probabilities (2) • Probability of an Interval

  7. Areas and Probabilities (3) • Howell Table 3.1 shows a table with cumulative and split proportions Graph illustrates z = 1. The shaded portion is about 16 percent of the area under the curve.

  8. 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.

  9. 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?

  10. 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

  11. Computer Exercise • Get data from class (e.g., height in inches) • Compute mean, SD, StErr of Mean in Excel • Compute same in SAS PROC UNIVARIATE • Show plots (stem-leaf & Boxplot) • Show test of normality

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