# Figuring Out Percentiles for a Normal Distribution

You might need to references the Z-table to solve the following questions.

## Sample questions

For the following questions, consider that in a population of adults ages 18 to 65, BMI (body mass index) is normally distributed with a mean of 27 and a standard deviation of 5.

1. What is the BMI score for which half of the population has a lower value?

In this case, using intuition is very helpful. If you have a normal distribution for the population, then half of the values lie below the mean (because it’s symmetrical and the total percentage is 100%). Here, the mean is 27, so 50%, or half, of the population of adults has a BMI lower than 27.

2. What BMI marks the bottom 25% of the distribution for this population?

You want to find the value of X (BMI) where 25% of the population lies below it. In other words, you want to find the 25th percentile of X. First, you need to find the 25th percentile for Z (using the Z-table) and then change the z-value to an x-value by using the z-formula:

To find the 25th percentile for Z (or the cutoff point where 25% of the population lies below it), look at the Z-table and find the probability that’s closest to 0.25.

The probabilities for the Z-table are the values inside the table. The numbers on the outsides that tell which row/column you’re in are actual z-values, not probabilities.

Searching the Z-table, you see that the closest probability to 0.25 is 0.2514.

Next, find what z-score this probability corresponds to. After you’ve located 0.2514 inside the table, find its corresponding row (–0.6) and column (0.07). Put these numbers together and you get the z-score of –0.67. This is the 25th percentile for Z. In other words, 25% of the z-values lie below –0.67.

To find the corresponding BMI that marks the 25th percentile, use the z-formula and solve for x. You know that

So 25% of the population has a BMI lower than 23.65.

3. What BMI marks the bottom 5% of the distribution for this population?

You want to find the value of X (BMI) where 5% of the population lies below it. In other words, you want to find the 5th percentile of X. First, you need to find the 5th percentile for Z (using the Z-table) and then change the z-value to an x-value by using the z-formula:

To find the 5th percentile for Z (or the cutoff point where 5% of the population lies below it), look at the Z-table and find the probability that’s closest to 0.05.

You see that the closest probability to 0.05 is either 0.0495 or 0.0505 (use 0.0505 in this case).

Next, find what z-score this probability corresponds to. After you’ve located 0.0505 inside the table, find its corresponding row (–1.6) and column (0.04). Put these numbers together and you get the z-score of –1.64. This is the 5th percentile for Z. In other words, 5% of the z-values lie below –1.64.

To find the corresponding BMI that marks the 5th percentile, use the z-formula and solve for x.

So 5% of the population has a BMI lower than 18.80.

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