Percentiles in statistics are used to divide a data set into 100 equal parts, or percentiles. The nth percentile is the value in a data set that is greater than or equal to n% of the data points and less than or equal to (100 – n)% of the data points.

For example, if the 75th percentile of a data set is 10, this means that 75% of the data points in the set are less than or equal to 10 and 25% of the data points are greater than 10.

Percentiles are useful for understanding the distribution of data within a set and for comparing the relative standing of different values within the data. For example, if a student scored in the 95th percentile on a standardized test, this means that they scored higher than or equal to 95% of the other students who took the test.

Percentiles can be calculated using a formula that involves ordering the data from lowest to highest, finding the position of the percentile in the data set, and determining the value at that position. The formula for calculating percentiles is:

**nth percentile = (n / 100) * (number of data points)**

Where n is the percentile you want to find and the number of data points is the size of the data set. For example, to find the 75th percentile of a data set with 100 data points, the formula would be:

75th percentile = (75 / 100) * 100 = 75

So the 75th percentile would be the 75th data point in the set when it is ordered from lowest to highest.

Here is an example of how to calculate the 75th percentile of a data set:

- Order the data from lowest to highest. For example, let’s say we have the following data set: 4, 8, 9, 10, 12, 15, 18, 20, 22, 24, 25
- Find the position of the 75th percentile in the data set. The formula for this is:

nth percentile = (n / 100) * (number of data points)

So for the 75th percentile, the formula is:

75th percentile = (75 / 100) * 11 = 8.25

Since 8.25 is not a whole number, we need to round it up to the nearest whole number, which is 9.

- Determine the value at the 9th position in the data set. Since the data set is ordered from lowest to highest, the 9th position is the value of 20.

So the 75th percentile of this data set is 20. This means that 75% of the data points in the set are less than or equal to 20 and 25% of the data points are greater than 20.

**There are several properties of percentiles in statistics:**

- Percentiles are used to divide a data set into 100 equal parts.
- The nth percentile is the value in a data set that is greater than or equal to n% of the data points and less than or equal to (100 – n)% of the data points.
- Percentiles are useful for understanding the distribution of data within a set and for comparing the relative standing of different values within the data.
- Percentiles can be calculated using a formula that involves ordering the data from lowest to highest, finding the position of the percentile in the data set, and determining the value at that position.
- If the position of the percentile is not a whole number, it must be rounded up to the nearest whole number.
- Percentiles are often used in conjunction with other statistical measures, such as mean, median, and mode, to get a more complete understanding of the data.