Percentile

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. It represents the percentages of data values that are less than or equal to the p^th percentile. For example, 15% of data values are less than or equal to the 15th percentile.

• Low percentiles always correspond to lower data values.
• High percentiles always correspond to higher data values.

Percentiles divide ordered data into hundredths. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.

The following formula is used to find k^th percentile

k = the k^th percentile. It may or may not be part of the data.
i = the index (ranking or position of a data value)
n = the total number of data points or observations

If i is an integer, then the k^th percentile is the data value in the i^th position in the ordered data set. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set.

A percentile may or may not correspond to a value judgment about whether it is "good" or "bad." The interpretation of whether a certain percentile is "good" or "bad" depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered "good;" in other contexts, a high percentile might be considered "good." In many situations, there is no value judgment that applies.

Understanding how to interpret percentiles properly is important not only when describing data but also when calculating probabilities in later chapters of this text.

This text is adapted from Openstax, Introductory Statistics, Section 2.2  Measures of the Location of the Data