Education 793 Class Notes Normal Distribution 24 September 2003

2 Today’s agenda Class and lab announcements What questions do you have? The normal distribution –Its properties –Identifying area under the curve

3 Properties of the normal distribution Symmetrical, with one mode, (mean, median and mode are all equal) – the classic bell- shaped curve Really a family of distributions with similar shape, but varying in terms of two parameters mean and standard deviations Of most use to us is the standard normal distribution, with a mean of zero and a standard deviation of 1

4 Standard scores Transformation of raw scores to a standard scale that reflects the position of each score relative to the distribution of all scores being considered

5 Standard score properties 1. Shape of distribution unchanged 2. Mean of z-score distribution equals zero 3. Variance of z-score distribution equals one

6 Calculating Standard Scores

7 Graphing scores SAT Math scores Standardized SAT Math scores

8 Family Traditions 1.Unimodal, symmetrical, and bell-shaped 2. Continuous 3.Asymptotic Standard Normal Distribution

9 Area under the standard normal Defined by mathematical equation, that indicates: 50% of the area falls below the mean 34% falls between the mean and one standard deviation above 16% falls beyond one standard deviation above the mean

10 Moving beyond eyeballing Direct calculation / calculators Table look-ups

11 Navigating a Standard Normal Probability Table

12 Probability questions about the normal distribution What percent of a standard normal distribution falls between one and two standard deviations below the mean? What percent falls above three standard deviations above the mean? If there were 100,000 people in a sample, how many would be expected to fall more than three standard deviations above the mean on any normally distributed characteristic? What percent of the normal distribution falls below a point.675 standard deviations above the mean? What percent of the normal distribution falls above a point 1.96 standard deviations above the mean?

13 Group exercise See handout

14 Some final points about the normal distribution Standard scores can be calculated for any distribution of numerical scores. In short, if we can calculate meaningful values for mean and standard deviation we can calculate standard scores. Standard scores, regardless of other factors (such as shape, skewness, and kurtosis), reflect the position of each score relative to the distribution of all scores being considered. We cannot, however, make precise statements about percentages associated with certain regions of a given distribution unless it represents a standard normal curve.

15 Next week Chapter 6 p , Correlation Chapter 7 p Linear Regression