How can you use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages?

We use the mean and standard deviation to standardising a value, in other words, to find the z-score. In a normally distributed data, the value of mean, μ, represent 50% of the population. 

A z-score will allow us to 'read' the probability on a z-table

For example, we have a normally distributed data with mean 12 and standard deviation 2.1. We can evaluate the probability of any given value, X, by finding the z-score then reading the score on the z-table. Say that we want to find the probability when X < 15.

The standardising formula is (X-μ) / σ
Where μ is the mean and σ is the standard deviation

Using μ and σ from our example, we have
z-score = (15-12) / 2.1 = 1.43 (rounded to two decimal places)

p-value is P(z<1.43) = 0.9236

The value 0.9326 is a probability value, and as percentage, it's 93.26