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Normal Distribution: Probability Density Function
  Enter the Mean Value( m)
Enter Standard Deviation(sd)
Enter Variable x( - infinity to + infinity)
Find Probability Density Function P(x)
Find Mean Deviation(md)
Find Maximum Value of P(x)
Find Value of z
Find area from z= 0 to z=z
 
Probability Density Function in a Normal Distribution is given by :-
P(x) = ( 1 / σ*√[2π])e- (x-μ)² /2σ²
where σ is the standard deviation , m is mean value
Properties of Normal Distribution
1 . It is a continuous probability distribution having parameters μ and σ .  μ is called the location parameter and σ  the scaling parameter. The change in value  of   μ shifts the curve to the right or left of x axis. μ=0 puts the maximum value of the curve at x=0; the change in value of σ to greater than 1 makes it flatter where the peak is lower and the spread is more about the mean such that the total area remains the same. σ less than 1 makes the curve with maxima higher and squeezed towards the mean. At σ=0, it becomes a spike at the mean.
2 . The Normal curve is perfectly symmetrical about the mean(m) and is bell shaped . The two tails of the curve on either side of mean extend to infinity .
3 . Mean = Mode = Median=m=μ  and σ is the standard deviation . σ2= variance
4 . Skewness = 0
5 . It has only 1 mode .
6 . Quartile Deviation = 0.67 * standard deviation
7 . Mean deviation = 4 / 5 *standard deviation
8 . Maximum value of Ordinate or Probability = 1 / (σ√[2π])
Thus Probability is inversely proportional to Standard Deviation .
9 . 50% of the area lie in between   x = -0.6745 σ to x = +0.6745 σ

    68.27% of area lie between x = - 1σ to x = +1σ
    95.45% of area lie between x = - 2σ to x = +2σ
    99.73% of area lie between x = - 3σ to x = +3σ
10 . x can lie anywhere between - infinity to + infinity .
11 . if we take( x - m ) / σ = z , then
P(x) = ( 1 / σ*√[2π] ) e -z*z;/2
Properties of Standard Normal Distribution
1 . mean =μ= 0 ; σ = 1
2.P(x) = ( 1/√[2π])   e- z*z/2
3 .∫ P(x) dx = 1 where limit is from -infinity to +infinity. The integral does not have a closed formula  but has to be numerically computed  The largest value of the function is inversely proportional to standard deviation  σ
4. The Normal/Gaussian distribution is the limit of Binomial distribution  when sample size n approaches infinity. Otherwise, when value of p and q in the binomial are close to 0.5, it can be approximated to a gaussian distribution with finite small sample size.