# PY 211 Lecture Notes - Lecture 8: Community Reinvestment Act, Normal Distribution, Standard Deviation

Chapter 6

Probability and Normal Distributions

What Does Standard Deviation Tell Us?

A. Empirical rule

a. For normal distributions with an mean and any variance:

i. Mean +/- one SD – includes at least 68% of all scores

ii. Mean +/- two SDs – at least 95% of all scores

iii. Mean +/- three SDs – at least 99.7% of all scores

From Frequency Distributions to Probability

Frequency polygon of a normal distribution (proportion)

o Describes what it is

o Can use to estimate probable scores

How does the standard deviation affect the confidence in your estimate?

o Smaller SDs

o Larger SDs

The Beauty of the Normal Distribution

No matter what the mean is

No matter what the SD is

Proportions remain the same (68%, 95%, 99.7%)

Normal Distribution in Behavioral Science

I. Measurements of behavior, attitudes, thoughts

a. Form normal distributions (approximately)

b. Mean = median = mode

II. Use properties of normal distribution to estimate likelihood (probability)

a. Given a score of x, how similar is it to other scores in distribution?

b. Is this a likely score, given a particular distribution?

Characteristics of Normal Distribution

I. Theory

a. Mathematically defined

b. Theoretical

c. Tails are asymptotic (tails never reach 0 frequency)

d. Area under curve is 1.0

II. Properties

a. Mean, median, mode are all at 50% percentile

b. Symmetrical

c. Mean and standard deviation can have any values (positive values only)

Clicker: relative frequency is on the y-axis

Standard Normal Distribution

I. Transform all x scores to z scores

a. Z = x-M / SD

b. Z-score transformation

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## Document Summary

Frequency polygon of a normal distribution (proportion: describes what it is, can use to estimate probable scores. How does the standard deviation affect the confidence in your estimate: smaller sds, larger sds. Proportions remain the same (68%, 95%, 99. 7%) Normal distribution in behavioral science: measurements of behavior, attitudes, thoughts, form normal distributions (approximately, mean = median = mode. Characteristics of normal distribution: theory, mathematically defined, theoretical, tails are asymptotic (tails never reach 0 frequency, area under curve is 1. 0. Properties: mean, median, mode are all at 50% percentile, symmetrical, mean and standard deviation can have any values (positive values only) Standard normal distribution: transform all x scores to z scores, z = x-m / sd, z-score transformation, z-score distribution (standard normal distribution, mean always translate to a z score of 0. Transformed mean and standard deviation: new mean: m = 0, new sd = 1.