Probability Density Function in a Normal Distribution is given by :-
P(x) = ( 1 / σ*[2PI]1/2 )e- (x-m)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 m and σ
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
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(2*PI)1/2]
Thus Probability is inversely proportional to Standard Deviation .
9 . 68.27% of area lie between x = - σ to x = + σ
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*[2PI]1/2 ) e -z²/2
Properties of Standard Normal Distribution
1 . mean = 0 ; σ = 1