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Normal Distribution
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2. Normal Distribution
The normal distribution is commonly encountered in practice, and is used throughout statistics.
The bean machine, a device invented by Francis Galton, can be called the first generator of normal random variables.
This machine consists of a vertical board with interleaved rows of pins.
Small balls are dropped from the top and then bounce randomly left or right as they hit the pins.
The balls are collected into bins at the bottom and settle down into a pattern resembling the bell-shaped curve.

4. All normal distributions look like a symmetric, bell-shaped curve.
The graph of the normal distribution depends on two factors - the mean and the standard deviation.

5. The Mean determines:
The mean of the distribution determines the location of the center of the graph.
when the standard deviation is small, the curve is tall and narrow.

6. The SD determines:
The standard deviation determines the height and width of the graph.
When the standard deviation is large, the curve is short and wide.

7. Rules:
About 68% of the area under the curve falls within 1 standard deviation of the mean.
About 95% of the area under the curve falls within 2 standard deviations of the mean.
About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

9. Example:
Suppose scores on an IQ test are normally distributed. The test has a mean of 100 and a standard deviation of 10.
We can be about 68% certain that a person’s score lies in the range of …
We can be about 95% certain that a person’s score lies in the range of …
We can be 99.7% certain that a person’s score lies in the range of …

10. Test Your Understanding of This Lesson
Molly earned a score of 95 on a test. The mean test score was 85 with a standard deviation of 10. What proportion (%) of students had a higher score than Molly? (Assume that test scores are normally distributed.)
Answer: about 16% (13,6 + 2, )