Normal+Distribution

//**Normal Distribution**// (also called Gaussian distribution) is the most common probability distribution.

The graph of a normal distribution is a //**bell curve**//.




 * Some properties of the bell curve:**
 * It is perfectly symmetrical
 * It is unimodal (has a single mode)
 * Its domain is all real numbers
 * Area under the curve = 1

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 * The Empirical Rule (68-95-99 Rule)**
 * Two essential things to understand are standard deviation and mean.
 * One can use the Empirical Rule to estimate where certain data lies based on the known values of standard deviation and mean.
 * The rule states that certain percentages of the data fall within 1, 2, or 3 standard deviations of the mean:
 * 68% → 1 deviation
 * 95% → 2 deviations
 * 99.7% → 3 deviations
 * This can be seen in the following bell curve graph:
 * In the set of data represented by this graph, the mean is 490 and the standard deviation is 100.
 * The blue section represents 68% of the data (1 deviation).
 * The blue and pink sections combined are 95% (2 deviations).
 * The blue, pink, and red sections altogether are about 99.7% (3 deviations).


 * Normal Density**

> >> >>
 * Different values of the mean and standard deviation also determine another factor: **density**. Mean specifically determines the height of a bell curve and standard deviation relates to the wideness or spread of the graph.
 * The height of the graph at any //x// value can be found through this equation:
 * To picture this visually, the narrowness or wideness of the bell curve depends upon the value of the standard deviation. The larger the standard deviation, the wider the graph. The smaller it is, the narrower the graph. Out of these two graphs, **GRAPH A** and **GRAPH B**, which one represents a set of data with a larger standard deviation?
 * **GRAPH A**
 * **GRAPH B**
 * The correct answer is **GRAPH A**, due to its shape.


 * Here are some examples of how to solve normal distribution problems:**

1. A set of data representing PSAT scores has a mean of 130 and a standard deviation of 15. The majority of these scores (68%) fall within one standard deviation of that mean. Between which two scores will the majority of the students' scores fall?


 * Following the Empirical Rule, one standard deviation of the mean would be 15 less than 130 or 15 more than 130.
 * Most of the PSAT scores in this set of data would fall somewhere between 115 and 145.

2. An annual community fund raiser concert usually collects an average of $4,500 with a standard deviation of $500. If it is a bad year for the event, and the outcome is $3,000 at its worst, how much money would the concert collect on its best year?


 * 3,000 is 3 standard deviations less than 4,500, so 3 standard deviations more than that would be 6,000.
 * On its best year, the fund raiser would bring in about $6,000.

3. A physics class takes their midterm. The class mean is a 71, and the standard deviation is 8. What percentile would a student who earned an 88 and a student who earned a 65 fall under?


 * 68% of the class scores would fall between 63 and 79.
 * 95% of the class would fall between 55 and 87.
 * 99.7% of the class would fall between 47 and 95.

Therefore, an 88 would be in the top 2.65% of the class, relatively a very good score. A student who earned a 65 would fall within 1 standard deviation of the mean, and would be an average score.

Sources: [|NKU] [|NETMBA] [|Wikipedia] [|StatTrek] [|Stanford]