Determine percentiles from normal distributions

Percentiles in a normal model are cumulative areas: the $p$ th percentile is the value below which $p\%$ of observations are expected to fall, with the median at the mean and symmetry relating corresponding lower and upper percentiles. Given a mean and standard deviation, the learner finds the associated $z$ -score using a standard-normal table or inverse-normal technology and converts it with $x=\mu+z\sigma$ , distinguishing percentile rank from a percentage of the measured value; multivariate and non-normal methods are outside this scope.

Detailed Explanation: Determine percentiles from normal distributions

A percentile tells you the value below which a certain percentage of observations fall. For example, the 90th percentile is the value that is greater than about $(90\%)$ of the observations.

For a normal distribution, use these steps:

Convert the percentile to a cumulative area.
Find the corresponding $z$ -score using a standard-normal table or inverse-normal technology.
Convert the $z$ -score to the original measurement using

x=\mu+z\sigma,

where $\mu$ is the mean and $\sigma$ is the standard deviation.

Example

Test scores are approximately normally distributed with mean $(70)$ and standard deviation $8$ . Find the score at the 90th percentile.

Step 1: Identify the cumulative area

The 90th percentile means that $(90\%)$ of scores are below the desired score:

P(X\leq x)=0.90.

Step 2: Find the $z$ -score

Use a standard-normal table to find the $z$ -score with cumulative area $(0.9000)$ . This gives approximately

z=1.28.

This means the 90th percentile is $(1.28)$ standard deviations above the mean.

Step 3: Convert to the original score

Substitute $\mu=70$ , $(z=1.28)$ , and $\sigma=8$ into the conversion formula:

x=\mu+z\sigma

x=70+(1.28)(8)

x=80.24.

Therefore, the score at the 90th percentile is approximately

\boxed{80.2}.

This means about $(90\%)$ of the scores are below $(80.2)$ , not that $(80.2\%)$ of the score is being measured. The 50th percentile is the median, which equals the mean in a normal distribution.

Learn by doing: Determine percentiles from normal distributions

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Statistics - Standard Deviation - Mean, SD and Z-Table Value to Top/Bottom X Percent Score

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