# Normal random variables

Description of activity and Excel instructions: Normal random variables

Dataset: none

## Calculate a normal probability from a given z-score

• Open Calc with a new worksheet.
• Create a table with the given values:
• Make the row and column labels: pick a cell and type "Statistic" and one cell to the right, type "Value".
• In each successive cell underneath Statistic, type "Population Mean", "Population Standard Deviation", "X", "z-score", "P(X=700)".
• Enter the population mean, 507, in the Value column.
• Enter the population standard deviation, 111, and the X value, 700.

To calculate the z-score,

• Enter the equation z = (X - mean)/stdev. (In this case, we get z = (700-507)/111, or 1.738739.)

To calculate P(X=700), we'll use the Calc function called NORMSDIST (note the spelling--be careful, there is also a function called "NORMDIST"), that takes a z-score and uses it to calculate a probability. This step takes the place of having to look up the probability in a normal table.

• In the cell next to P(X=700), type "=NORMSDIST(1.738739)" (or use the cell reference instead of typing the z-score).

Calc tells us that the probability is 0.95896.

Return to the OLI's Normal random variables page to complete the interpretive exercises on this part of the "Learn by doing" exercise.

## Calculate a z-score from a given probability

We continue with the table created above, but this time we're looking for: z-score and X. Basically, we're going to do the inverse of the procedure we described above. Instead of being given an X and asked to find the probability, we're given a probability and asked to find the corresponding X value.

To calculate the z-score given a probability, we'll use the CALC function called NORMSINV, that takes a probability and uses it to calculate a z-score. This function is the inverse of the NORMSDIST function that we used previously, and takes the place of us having to look up the z-score in a normal table.

• In the column labeled Statistic', type "P(Z=z)".
• Under Value, type ".98".
• In the next row, under Statistic, type "z-score".
• Under Value, type "=NORMSINV(0.98)". (Calc tells us that the z-score is 2.053748)

To calculate the X value that corresponds to that z-score, solve for X in the equation z = (X - mean)/stdev.

• In the next row, under Statistic, type "X".
• Under Value, enter the formula for X. (Solving for X, we get X = 2.053748 * 111 + 507, or 734.966.)

Return to the OLI's Normal random variables page to complete the interpretive exercises.