- MathematicsStatistics & Probability: Normal Distribution
**Normal Distribution****Normal Distribution**(Gaussian distribution) is a continuous probability distribution for a real-valued random variable defined by the probability density function

(*f*) = 1σ*x*√2π*e*^{ - 12 ( x - μσ )2}

and the**Cumulative Distribution Function**:

(*F*) =*x*∫x-∞*f*(*t*)t*d**Please note:*- By specifying one input value and pressing "Calculate" button, all other values will be calculated.
- If, as an input, you specify the value of the density function
(*f*) ,then all the values will be calculated based on the corresponding*x*

value ofthat is smaller than the mean*x***μ**value. - If, as an input, you specify the value of the probability of the random variable
being above (below) the mean μ :*X*(μ <*P*<*X*) , or*x*

the value of the probability that thescore will be above (below) the 0 :*Z*(0 <*P*<*X*) ,*Z*

then all the values will be calculated based on the corresponding value ofthat is greater than the mean*x***μ**value. - If, as an input, you specify the value of the probability that the random variable
will be closer to the 0 than the*X*score:*Z*(-*P*<*Z*<*X*) , or*Z*

the value of the probability that the random variablewill be further from the 0 than the*X*score:*Z*(*P*< -*X*∨*Z*<*Z*)*X*

then all the values will be calculated based on the corresponding value ofthat is greater than the mean*x***μ**value. - Even though we calculate all of the numbers with the precission of 15 significant digits, for clarity, we are displaying the results rounded to 12 decimal places.

Please select if the Normal Distribution is the**Standard Normal Distribution**with the mean value**μ**= 0 and the standard deviation**σ**= 1,

or the**Generic Normal Distribution**with the mean value**μ**and the standard deviation**σ**that you will provide:

DistributionMean **μ**=Standard Deviation **σ**=*x*= (Sample, Data Point, Raw Data, Raw Score) Density (*f*)*x*= (Probability Density, Probability Distribution) Probability (*F*) =*x*(*P*<*X*)*x*= (Cumulative Distribution Function, CDF) (*P*>*X*)*x*= (*P*<*μ*<*X*)*x*= Z-Score *Z*= (Standard Score, Number of Standard Deviations, Pull) Probability (*P*<*X*)*Z*= (*P*>*X*)*Z*= (*P***-**<*Z*<*X*)*Z*= (*P*<*X***-**∨*Z*<*Z*)*X*= (0 <*P*<*X*)*Z*= Calculate probability of random variablebeing between two values*X*and*a*:*b**a*= *b*= (*P*<*a*<*X*) =*b*