Normal Distribution

The normal distribution is a function that represents the distribution of many random variables as a symmetrical bell-shaped graph.

The probability function for the normal distribution will be

Standard Normal Distribution

The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1.

The probability function for the standard normal distribution will be

68-95-99.7 Rule

In normal distribution, 68%, 95%, and 99.7% of the values lie within one, two and three standard deviations of the mean.

Example

What is the probability for a random variable x to have a value greater than 147 in a normal distribution whose μ = 100 and σ = 16?

	NORMAL DISTRIBUTION


Your Input:

  Random variable x = 147
  Significance level =
  Mean = 100
  Standard deviation = 16


Probability Density Function (PDF)

      x     f(x)
 ----------------+-------------------+
   36.00  0.0000 |
   42.40  0.0000 |
   48.80  0.0001 |
   55.20  0.0005 |
   61.60  0.0014 |*
   68.00  0.0034 |***
   74.40  0.0069 |******
   80.80  0.0121 |**********
   87.20  0.0181 |***************
   93.60  0.0230 |******************
  100.00  0.0249 |********************
  106.40  0.0230 |******************
  112.80  0.0181 |***************
  119.20  0.0121 |**********
  125.60  0.0069 |******
  132.00  0.0034 |***
  138.40  0.0014 |*
  144.80  0.0005 |
  151.20  0.0001 |
  157.60  0.0000 |
  164.00  0.0000 |
 ----------------+-------------------+


Cumulative Distribution Function (CDF)

       x    F(x)
 ----------------+-------------------+
   36.00  0.0000 |
   42.40  0.0002 |
   48.80  0.0007 |
   55.20  0.0026 |
   61.60  0.0082 |
   68.00  0.0228 |
   74.40  0.0548 |*
   80.80  0.1151 |**
   87.20  0.2119 |****
   93.60  0.3446 |*******
  100.00  0.5000 |**********
  106.40  0.6554 |*************
  112.80  0.7881 |****************
  119.20  0.8849 |******************
  125.60  0.9452 |*******************
  132.00  0.9772 |********************
  138.40  0.9918 |********************
  144.80  0.9974 |********************
  151.20  0.9993 |********************
  157.60  0.9998 |********************
  164.00  1.0000 |********************
 ----------------+-------------------+


             ___
          ---   ---
        --         --
      --           | --
___--              |   --___
--------------+----+---------
                   x

  x = 147

  P(X < 147) = 0.998346
  P(X > 147) = 0.001654 (1.65E-3)


Properties of the PDF

  Mean = μ = 100

  Median = μ = 100

  Mode = μ = 100

  Variance = σ² = 256

  Standard deviation = σ = 16

  Skewness = 0

  Ex. kurtosis = 0