Education 793 Class Notes

Normal Distribution24 September 2003

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Today’s agenda

• Class and lab announcements

• What questions do you have?

• The normal distribution– Its properties– Identifying area under the curve

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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

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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  

Standard score = Raw score - mean scoreStandard deviation

z = X - Xs

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Standard score properties

1. Shape of distribution unchanged

2. Mean of z-score distribution equals

zero

3. Variance of z-score distribution equals one

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Calculating Standard Scores

Raw SAT score

Deviation score

Std. Deviation Z

340 111.8450 111.8510 111.8550 111.8580 111.8600 111.8620 111.8660 111.8670 111.8710 111.8

Sum 5,690 0 0

z = X - Xs

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Graphing scores

SAT Math scores

l l l l l l l l l l

200 300 400 500 600 700 800

l l l l l l l l l l

-3 -2 -1 0 1 2

Standardized SAT Math scores

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Family Traditions

1. Unimodal, symmetrical, and bell-shaped2. Continuous3. Asymptotic

Standard Normal Distribution

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Area under the standard normal

Defined by mathematical equation, that indicates:50% of the area falls below the mean34% falls between the mean and one standard deviation above16% falls beyond one standard deviation above the mean

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Moving beyond eyeballing

• Direct calculation / calculators• Table look-ups

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Navigating a Standard Normal Probability Table

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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?

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Group exercise

• See handout

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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.

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Next week

• Chapter 6 p. 145-179, Correlation

• Chapter 7 p. 181-204 Linear Regression