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# Normal distribution

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Normal Disribution

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### Normal distribution

1. 1. Normal Distribution Submitted to submitted by Mohd. Atif Harish vasdev Assistant professor Cuhp13MBA29
2. 2. A function that tells the probability of a number in some context falling between any two real numbers is called Normal Distribution. The term bell curve is used to describe the mathematical concept called normal distribution , and it also sometimes referred to as Gaussian distribution. The important things to note about a normal distribution is the curve is concentrated in the center and decreases on either side. This is significant in that the data has less of a tendency to produce unusually extreme values, called outliers, as compared to other distributions. A bell curve graph depends on two factors, the mean and the standard deviation. The mean identifies the position of the center and the
3. 3. standard deviation determines the the height and width of the bell. Many things closely follow a Normal Distribution:      Heights of people Size of things produced by machines Errors in measurements Blood pressure Marks on a test We say the data is "normally distributed". The Normal Distribution  Mean=median=mode Symmetry about the center 50% of values less than the mean and 50% greater than the mean
4. 4. Features:  Normal distributions are symmetric around their mean.  The mean, median, and mode of a normal distribution are equal.  The area under the normal curve is equal to 1.0.  Normal distributions are denser in the center and less dense in the tails.  Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ).  68% of the area of a normal distribution is within one standard deviation of the mean.  Approximately 95% of the area of a normal distribution is within two standard deviations of the mean. This means that a variable whose values can be described as normally distributed should have the following Characteristics: 1) If graphed in a frequency polygon, the polygon will be essentially bell shaped and symmetrical. 2) When computed, the mean, median, and mode will be similar.
5. 5. 3) Most values will fall between 1 and 1 standard deviations from the mean; a few values may fall below or above three standard deviations from the mean. Properties of the Normal Probability Curve:  The highest point occurs at x=μ.  It is symmetric about the mean, μ. One half of the curve is a mirror image of the  other half, i.e., the area under the curve to the right of μ is equal to the area under  the curve to the left of μ equals ½.  It has inflection points at μ-_ and μ+_.  The curve is asymptotic to the horizontal axis at the extremes.  The total area under the curve equals one.  Empirical Rule: o Approximately 68% of the area under the curve is between μ-_ and μ+_. o Approximately 95% of the area under the curve is between μ-2_ and μ+2_. o Approximately 99.7% of the area under the curve is between μ-3_ and μ+3_.
6. 6. A normal curve has two characteristics: mean (μ) and standard deviation (_). Example 1:—normal curves for two populations with different means: Population #1 Population #2 μ1 = 50 μ2 = 70 _=4_=4 Draw the normal curves for both populations. Summary: The two curves are exactly the same, except one curve is to the right of the other curve. Example 2:—normal curves for two populations with different standard deviations. Population #1 Population #2
7. 7. μ1 = 50 μ2 = 50 _1 = 4 _2 = 7 Draw the normal curves for both populations. Summary: Increasing the standard deviation causes the curve for Population #2 to become flatter and more spread out. Comparing the two normal curves:  For Population #1, there is more area under the curve within a given distance of the mean;  For Population #2, there is more area under the curve away from the mean. Standard Scores: The number of standard deviations from the mean is also called the "Standard Score", "sigma" or "z-score". Convert a value to a Standard Score ("z-score"):   first subtract the mean, then divide by the Standard Deviation And doing that is called "Standardizing":
8. 8. Convert the values to z-scores ("standard scores"): Formula for z-score that we have been using: o o o o z is the "z-score" (Standard Score) x is the value to be standardized μ is the mean σ is the standard deviation
9. 9. References:o www.mathsisfun.com/data/standard-normaldistribution.html o http://www3.nd.edu/~rwilliam/stats1/x21.pdf o http://en.wikipedia.org/wiki/Normal_distribution o Ken black 5th Edition o http://www.mathsrevision.net/advanced-levelmaths-revision/statistics/normal-distribution.