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Normal distribution
 

Normal distribution

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Normal Disribution

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    Normal distribution Normal distribution Document Transcript

    • Normal Distribution Submitted to submitted by Mohd. Atif Harish vasdev Assistant professor Cuhp13MBA29
    • 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
    • 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
    • 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.
    • 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_.
    • 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
    • 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":
    • 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
    • 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.