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.The answer is 2.376365899787781…Here’s the Probability Mass Function for a normal distribution. You can find it online or in any good stats textbook:If you integrate the PDF from x=3 to x=5 it will give the area under the normal curve between x=3 and x=5. If you then subtract that value from 1/2, it will give the area for x<=3, which your problem says is 0.2 :So the equation for your problem isDoing the integration (using a free CAS, like WxMaxima) yields:… where erf() is the Gauss error function.With some simple algebra, you can solve for sigma:… and then evaluate to get a decimal answer:2.376365899787781

The Z distribution is a Gaussian distribution with mean [math]\mu=0[/math] and standard deviation [math]\sigma=1[/math]. This is known as the Standard Normal Distribution.See this answer for more detail.The Probability Density Function of the Standard Normal Distribution (Z distribution) is given by:… where the domain of x is [math]-\infty[/math] < x < [math]\infty[/math].If you want the probability between two Z values, say Z1=0 and Z2=1, you find the area under the curve pdf(x) between x=Z1 and x=Z2 by integrating pdf(x) from x=Z1 to x=Z2:and you get:Notice the “erf”. That’s the error function:Most Computer Algebra Systems (CAS), such as WxMaxima, have built-in capability to evaluate erf.So setting Z1=0 and Z2=1, we get:which evaluates to:which looks correct because that’s what we would expect for one standard deviation from the mean..Another way to do the same example is to use the Cumulative Distribution Function (CDF) for the Standard Normal Distribution (Z distribution).The CDF gives the area (probability) under the PDF from [math]-\infty[/math] to x:So for the above example, the answer would beOne final word:The quickest way to get a very accurate answer if you have a range Z1..Z2 that you want the probability for is to just type it into WolframAlpha.Let’s say Z1=0 and Z2=1. Just type this:integrate(1/sqrt(2*pi)*e^(-x^2/2),x,0,1)into WolframAlpha and you’ll get the answer:

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