The Normal Distribution(Gaussian Distribution)

Honors Analysis

Learning Target: I can analyze data using the normal distribution.

Carl Friedrich Gauss (1777-1855)

German mathematicianInfluenced statistics, algebra, number theory, geometry, physics.Child prodigy!Constructed heptadecagonTriangular numbersProved Fundamental Theorem of AlgebraInfluenced development of statistics, including Normal Distribution(Gaussian Distribution)

Imagine you took a test in two different classes.

In the first class, you made a 93%. The class mean was a 96%, and the standard deviation was 3%.

In the second class, you made a 78%. The class mean was a 74%, and the standard deviation was 2%.

Which test performance was better?

Normal Distribution(Gaussian Distribution)

68-95-99.7 Rule

(Approximately) 68% within 1 std dev. of mean

95% within 2 std. deviations of mean

99.7% fall within 3 standard deviations of mean

Labeling a Simple Normal Curve

Calculate the mean (central value on curve)

Each region increases or decreases by one standard deviation from the mean

Ex: Test score mean: 74% Std. dev: 2%

So what happens if you want to calculate a percentage for a value that ISN’T on your normal curve?

Ex: PSAT math test with mean of 48 and a std. deviation of 3. What percent of scores are below 50?

Standard Normal Distribution

Normal distribution with a mean of 0 and a standard deviation of 1.

Total area under curve = 1Area to left of a given value on the curve

gives the percentile rank – percent of scores LOWER than a given score.

Z-Scores

You can convert values to standard normal distribution form by calculating a z-score:

Z-Score percentages can be looked up in a table or on a calculator.

Example

Solution

Example Part II

Types of Bias

Unrepresentative SampleUndercoverage (Convenience sample,

voluntary sample)Non-response BiasVoluntary response Bias

Measurement ErrorResponse Bias (Leading questions, social

desirability)