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A percentile measures the relative standing of a value within a dataset. It indicates the proportion of data points below a certain value. For instance, the 38th percentile means that 38% of the data falls below that value. Percentiles are very useful in understanding distributions in various fields like education, healthcare, and finance.
In our example, we find the value corresponding to the 38th percentile in a normally distributed dataset with a mean (\(\mu\)) of 50 and a standard deviation (\(\sigma\)) of 7. To do this, we convert the 38th percentile to a Z-score using the standard normal distribution.

A Z-score, also known as a standard score, measures how many standard deviations an element is from the mean. The Z-score is a way to compare individual data points relative to the overall distribution. It is calculated using the formula:

Z = (X - \(\mu\))/\(\sigma\)

Here, we already have a pre-calculated Z-score for the 38th percentile, which is approximately -0.31. This negative value indicates that the 38th percentile is below the mean of the data. For our purposes, we need to convert this Z-score back into the original units of our distribution, which is done using the formula:

X = \(\mu\) + Z\(\sigma\)

A standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. These values are represented by a Z-score. This distribution is useful because it allows for standardizing different datasets, making it easier to perform comparisons and calculations.
In our example, we use the standard normal distribution to find the Z-score corresponding to a given percentile. Once the Z-score is determined, it can be converted using the original distribution's mean and standard deviation to find the specific value within that dataset.

We saw that the Z-score for the 38th percentile is -0.31. Using the given mean (\(\mu = 50\)) and standard deviation (\(\sigma = 7\)), we find the exact value in our original normal distribution:

X = 50 + (-0.31)(7) = 50 - 2.17 = 47.83. So, the 38th percentile corresponds to approximately 47.83 in our dataset.