The Normal Distribution NORMAL CURVES Opening Question Are

The Normal Distribution

NORMAL CURVES Opening Question: • Are NORMAL CURVES really NORMAL ?

Most data is NOT normal!

WHAT ARE NORMAL CURVES? • Normal curves are: – SYMMETRIC – SINGLE-PEAKED – BELL-SHAPED – LACK OUTLIERS – HAVE IDENTICAL MEANS AND MEDIANS

If they’re NOT NORMAL, why are they IMPORTANT? • 1. ) They are good descriptions for some real data – SAT and IQ tests – Measurements (diameter of tennis ball) – Characteristics of biological populations (like yields of corn or animal pregnancies) • 2. ) They are good descriptions of chance outcomes such as tossing coins. • 3. ) Many statistical inferences procedures are based on Normal Curves

ESTIMATING STANDARD DEVIATION for a normal curve • Imagine you are skiing down a mountain that has the shape of a normal curve. • At first, you descend at an even-steeper angle as you go out from the peak. • Then the slope begins to grow flatter rather than steeper as you go down. • The points at which the curve changes from steep to flat are one standard deviation from the mean.

NORMAL CURVES—Standard Deviation

Estimating Standard Deviation

YOU TRY! Estimate the standard deviation of the Normal curve below.

You Try: Estimating Standard Deviation

Changing mean & standard deviation Does changing the mean affect the shape of the curve? Does changing the standard deviation affect the mean of the curve?

BASIC FACTS For normal density curves • 1. ) We can describe a Normal curve with mean and standard deviation • 2. ) The mean determines the center of the curve • 3. ) The standard deviation determines the shape of the curve.

THE 68 -95 -99. 7 RULE In any normal distribution: • 68% of the observations fall within one standard deviation of the mean. • 95% of the observations fall within two standard deviations of the mean. • 99. 7% of observations fall within three standard deviations of the mean.

NORMAL CURVE!

LET’S TRY! • Jennie scored 600 on the Reading section of the SAT. The test had a mean of 500 and a standard deviation of 100. How well did she do? That depends on where a score of 600 lies in the distribution of all scores. Think about percentiles and standard deviation.

Example: • A truck is loaded with cartons of eggs that weigh an average of 2 pounds each with a standard deviation of 0. 1 pound. A histogram of these weights looks very much like a Normal distribution. Use the 68 -95 -99. 7 rule to answer the following question: What percent of the cartons weigh less than 2. 1 pounds?

On Your Whiteboards • A truck is loaded with cartons of eggs that weigh an average of 2 pounds each with a standard deviation of 0. 1 pound. A histogram of these weights looks very much like a Normal distribution. Use the 68 -95 -99. 7 rule to answer the following: a. What percent of the cartons weigh less than 1. 8 pounds? b. What percent of the cartons weigh more than 1. 9 pounds?