So far, we have talked about Z score and finding it for a normal distribution. Suppose, you are watching oil prices and you found out that it follows a normal distribution as shown.

Suppose, you need to find out the probability that prices will be below $40 per barrel. As taught previously, you will simply find the Z score. Here, mean is $30 as shown. Let's assume that standard deviation is $5. Hence, Z score is (40-30)/5 = 2.

From Z-table, we get following information -

Hence, the probability that prices will be below $40 is 0.9772.

Now, suppose I ask you - for what value will the probability that oil prices are equal to or less than that value will be 68% or 0.68? Let's discuss what it means.

We have drawn a blue line to indicate a value. Area towards left of it is 0.68. Thus, for this value, probability that oil prices are equal or less than is 0.68.

So, how do we find this value?

First, we go to the z-table.

We look up a number in the table closest to 0.68. We see that for the row 0.4 and column 0.07, we get a suitable value. Hence, z-score is 0.47.

Now, we need to find a oil price for which z-score is 0.47.

We can do so by multiplying z-score by standard deviation and then adding mean to it.

Hence, the value is 0.47 * 5 + 30 = 32.35

Thus, our desired value is $32.35.