POLI 205

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12/14/2018

Normal Distributions

• Normal distribution – Theoretical distribution based on an infinite number of scores used to estimate a population

– Generated from mathematical formulas, not from collected data

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Characteristics of Normal Distributions

• Left and right tails continue to infinity without touching the X‐axis

• Shape: unimodal, symmetric, “bell‐ shaped”

• Mean: population mean () • Standard deviation: population standard deviation ()

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

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Importance of Normal Distributions

• Researchers believe many variables are normally distributed in the population

• Because of this, many inferential statistics (used to test hypotheses about populations) are based on normal distributions

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Normal Distributions Still Vary

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Distributions with same standard deviation but different means

Distributions with same mean but different standard deviations

The Standard Normal Distribution

• Has properties of other normal distributions (unimodal, symmetric, “bell‐shaped)

• In addition: – Mean = 0 – Standard deviation = 1 – Scores (z‐scores) measured in standard deviation units

• # standard deviations above or below the mean Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

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The Standard Normal Distribution

• Value of z‐score indicates distance from the mean in standard deviation units

– Positive z‐scores: above the mean

– Negative z‐scores: below the mean

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

The Standard Normal Distribution

• z‐scores also indicate position relative to the entire distribution

– Can determine percentage of distribution associated with z‐score

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Using the Normal Curve Table

• % of distribution between the mean and a z‐score

• % of distribution less or greater than a z‐score

• % of distribution between two z‐scores

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

12/14/2018

Applying the Standard Normal Distribution to

Normal Distributions

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

• Transforming scores in normal distributions to z‐scores

Howard T. Tokunaga, Fundamental Statistics for the Social and Behavioral Sciences © SAGE Publications, 2016

Example: SAT Math ( = 500,  = 100)

Using the standard normal distribution table, What percentage of scores is less than 630? What percentage of scores is between 450 and 630?

Applying z‐scores to Normal Distributions

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Standardizing Frequency Distributions

• When population parameters are unknown, scores from a sample can be transformed into standardized scores

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Standardizing Frequency Distributions

• Standardized scores help compare and combine scores from different distributions

• Imagine you want to compare your score of 80 on a statistics quiz to your roommate’s score of 75 on a political science quiz

Standardizing Frequency Distributions

Statistics quiz Political science quiz

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Standardizing Frequency Distributions

• Standardizing a frequency distribution does not change the shape of the distribution (it’s a linear transformation)

– e.g., transforming a temperature in Fahrenheit to Celsius

Looking Ahead

• How do normal distributions relate to probability?

• How is the role of probability in testing research hypotheses?