## Z-Scores and Percentiles from the Normal Model

When working with the

The first way to use normalcdf is to plug in the left z-score and right z-score and the calculator will give you the percentile between those z-scores. Frequently we look at percentiles above or below given values, so we usually use 9E99 for infinity.

The second way to use normalcdf is by using the values instead of the z-scores. The calculator will give you the percentile between those two values. Again, when you want to know percentiles above or below given values, we usually use 9E99 for infinity.

Now to go from percentiles to z-scores we use the invNorm function. The inv comes from the “inverse” of the normalcdf function. If you then want the value, you will need to plug the z-score into the formula.

The second way to invNorm is by using the mean and standard deviation and what you are given is the boundary point.

Make sure that the distribution you are working with is normal or nearly normal (symmetric and unimodal). You need to check this (ex: histogram, normal probability plot) before you can use it.

**Overview: ** When working with the

**Normal Model**we need to go from z-scores to percentiles and from percentiles to z-scores. The easiest way to do this is using the normalcdf and invNorm functions in the DISTR menu on your calculator.**normalcdf(left z-score, right z-score) = percentile**The first way to use normalcdf is to plug in the left z-score and right z-score and the calculator will give you the percentile between those z-scores. Frequently we look at percentiles above or below given values, so we usually use 9E99 for infinity.

**normalcdf(left value, right value, mean, standard deviation) = percentile**The second way to use normalcdf is by using the values instead of the z-scores. The calculator will give you the percentile between those two values. Again, when you want to know percentiles above or below given values, we usually use 9E99 for infinity.

*Note: If using this on an FRQ, you should state the parameters you are using separately. For example “mean = 5, and standard deviation = 1.7”)***invNorm(percentile) = z-score**Now to go from percentiles to z-scores we use the invNorm function. The inv comes from the “inverse” of the normalcdf function. If you then want the value, you will need to plug the z-score into the formula.

**invNorm(percentile, mean, standard deviation) = boundary value**The second way to invNorm is by using the mean and standard deviation and what you are given is the boundary point.

*Note: If using this on an FRQ, you should state the parameters you are using separately. For example “mean = 5, and standard deviation = 1.7”)***Final Reminders:**Make sure that the distribution you are working with is normal or nearly normal (symmetric and unimodal). You need to check this (ex: histogram, normal probability plot) before you can use it.

__Example 1:__Suppose that the distribution of the weighs of a bag of plain M&Ms is 7 oz and the standard deviation is 0.25 oz. If your bag weighs of 7.05 oz. What percent of bags weigh less than your bag?*Solution: The distribution is normal, so I will use the “normcdf” function on my calculator. The mean is 7 oz. and the standard deviation is 0.25 oz. normalcdf(-9E99, 7.05, 7, 0.25) = .5793. 57.93% of bags weigh less than my bag.*__Example 2:__90% of the bags of M&Ms have at most what weight?*Solution: The distribution is normal, so I will use the “invNorm” function on my calculator. The mean is 7 oz and the standard deviation is .25 oz. invNorm(.9,7,.25)= 7.32. So 90% of bags have at most 7.32 oz.*