Help Building Useful Normal Distribution Calculator

I want to develop my own Normal Distribution calculator, to provide to a developer to perform a task within an app. I want a user with little knowledge of statistics to be able to perform, and understand Normal Distribution calculations. Here is what I want it to accomplish . . .

Given the mean and the std deviation (which will be in the app's data table), I want the user to be able to enter a value, call it X. I want the calculator to calculate the cumulative probability of a value being < X.

In my case, this is for overhead reach statistic. The mean is 83.5", and the std dev is 3.3". The user of the app should be able to enter a dimension, X, and find out what percentage of people have a reach < X. For example, if the user entered 78" (the value of X), I want the output to tell me what percentage of people have a reach < 78".

I plug in the mean & std dev and X = 78 into the attached Normal Equation, and I get 0.03014 (which does not agree with the value from using value from Std Normal table, roughly 0.0475)

Could someone help me? Did I do calculation wrong? Am I missing something?

Would really appreciate any help. I am new to this site.


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The formula attached is for the density function - not the cumulative density function. There isn't a nice closed form for the CDF (which is what you want) but there are good approximations. What language are you using? There are libraries in almost every language that do this for you.
Not sure what language. It will be an app for Apple (I assume iOS). I'm not a programmer though. I am building out the design of the app, and was trying my best to give them exact instructions - to define exactly what calculation needed to be performed.

So you are telling me that it shouldn't be that big of a deal to get this? I would appreciate any specifics I could tell the developer - website to search for code, etc.

Thank you so much!

Now it makes sense why we always use the Normal Distribution table w/ Z statistics (no easy way to calculate directly). I have had several statistics course, Six Sigma training, etc. But was quite puzzled by this. Obviously I'm no wiz in statistics. I know enough to be dangerous :)