Percentile Calculator

A percentile rank is the position of a value within a data set. The kth percentile is the value for which k percent of observations are less than or equal to that value. So the 90th percentile means that 90 percent of the data have values less than or equal to this value.

This calculator provides two functions: it returns the value for a given percentile and calculates the percentile rank for a given value. It also displays quartiles, median, inter-quartile range, and a complete five-number summary.

What is a percentile?

A percentile divides sorted data into hundredths. If a test score is in the 80th percentile, it means that the score is better than 80 percent of the participants.

The most common percentiles have special names. The 50th percentile is the median, the 25th percentile is the first quartile and the 75th percentile is the third quartile. Quartiles divide data into four equal parts.

Percentiles do not indicate distance but position. They show the rank of a value relative to other values, which is why they are often used in situations where rank is more important than actual difference between numeric values, such as test scores, salaries, reaction times or growth charts.

Calculating percentiles

There is not one universally accepted method for calculating percentiles and slight differences in results can occur. This calculator uses linear interpolation by default, which corresponds to the PERCENTILE.INC function in Excel but also offers an option of a rank-based method.

Linear interpolation

The data is sorted in ascending order and the position of the rank that corresponds to the desired percentile is determined.

where p is the percentile and n is the number of values. If r is a whole number, then the percentile is the value at that position. Otherwise, interpolation is used to find the percentile between the two neighboring values.

Example

These seven values are considered: 42, 54, 65, 47, 59, 40, 53. This is the data set that was loaded into this calculator. After sorting, the values become 40, 42, 47, 53, 54, 59, 65.

For the 90th percentile, if n is equal to 7, then rank r = (90 / 100) * (7 - 1) + 1 = 6.4. This position falls between the value of 59 in 6th place and the value of 65 in 7th place.

So the 90th percentile is equal to 61.4. The median, or 50th percentile, is exactly the value 53 in the 4th position, while the first quartile and third quartile are respectively the values 44.5 and 56.5.

Method of next ranking

The next rank method does not interpolate and always returns an actual data point.

For the same data set, the 90th percentile is the value at position (0.9 * 7), which rounds up to 7, so the result would be 65 rather than 61.4. The next-rank method is appropriate when an actual data value must be selected and there can be no intermediate values.

Percentile rank (inverse calculation)

Sometimes you know a value and want to know what percentile it is in. This is called percentile rank and is the reverse of the problem we've been working on.

To determine the rank, count the number of values that are less than the given value, add half the number of values that are equal to the given value, divide by the total number and multiply by 100. In the loaded data there are three values below a score of 53 and one additional value that is equal, so the rank is (3 + 0.5) / 7 x 100 or 50 percent.

As both directions use a standard formula, they are not completely inverse procedures but closely related combinations. Therefore there may be slight changes in the result when a value is converted to rank and then back to a value. This is normal and works the same way with all statistical software packages.

Quartiles, IQRs and outliers

The quartiles divide the data into 25th, 50th and 75th percentiles. The difference between the first and third quartile is called interquartile range (IQR). It measures the spread of the middle half of the data.

The interquartile range (IQR) is often used to detect outliers. According to the Tukey rule, values that are farther than a multiple of the IQR from the first or third quartiles are marked as outliers.

In the loaded data, the IQR is 56.5 and 44.5, so it's 12. So the lower limit is 26.5 and upper limit is 74.5. All values are within these limits, so there are no outliers in this data according to that rule. You can check the relevant limits for your values by opening the "Outlier Check".

Common mistakes.

I always forget to sort data.All the different methods of calculating percentiles assume that the values are arranged in ascending order. This calculator will automatically sort the values for you, but if you want to calculate it manually then you'll need to sort the values yourself first.

The confusion of percentiles and percents.A position at the 90th percentile does not mean that you got 90 percent of your answers correct; it only means that your performance was better than 90 percent of those in this group.

You expect a response that is applicable in all cases.Even with the same small data sets, different methods and software can give slightly different percentiles, so it is helpful to understand what method was used.

Use of percentiles:

In education, percentiles are used to report standardized test results so that a student can be compared to all other students who took the same test.

In healthcare, percentiles are used to assess a child's growth in terms of their height and weight on a growth chart, tracking development against a reference group. In this medical context, published reference data is relied upon rather than small samples, but the concept of percentiles remains the same.

In the fields of business and engineering percentiles are used to summarize response times or service levels. For example, a 95th percentile latency indicates that 95% of requests can be completed within this time. This is much more meaningful than an average value.

Used symbols:

Every symbol that appears in the formulas above is defined here.

This calculator is a general tool for education and analysis purposes. For growth curves and clinical percentiles please use the published reference charts that are appropriate for your target population.