# Quartiles and Percentiles

## Definition

###### Definition

A **percentile** of a population or a sample of a population is a data point that represents a certain percentage of that data.

The **lower quartile** splits the data so that one quarter is below it and three quarters above. The **upper quartile** has three quarters below and one quarter above.

To find the $P$th percentile, calculate $\frac{P}{100} \times (n+1)$. If this is an integer?, the $P$th percentile is the value of that data point. If this is not an integer?, find the mean of the two data points either side.

To find a quartile, use $\frac{n+1}{4}$ for the lower quartile and $\frac{3(n+1)}{4}$ for the upper quartile.

When the quartile or percentile is not at a data point, a more accurate method is to use interpolation? between the data points at either side. When calculating by hand, the simpler method is used as the inaccuracy is not sufficient to justify the extra complication of the interpolation method.

## Grouped Data

When data is grouped, estimates for the percentiles and quartiles can be found using interpolation?. In this estimation, the data points in each group are considered to be equally spaced out in that group.

When dealing with grouped data, we use $n$ in place of $n+1$. This is because if we placed $12$ data points equally spaced in the interval from $0$ to $12$, they would actually be placed at $0.5$, $1.5$, $2.5$, …, $11.5$. The lower quartile would be half way between $2.5$ and $3.5$, which is at $3 = \frac{12}{4}$.

## Spreadsheet

In a spreadsheet, the formula to calculate a percentile or quartile of a range is:

```
=quartile(<range>,1)
=percentile(<range>,0.75)
```

The lower quartile is `1`

and upper quartile is `3`

. Note that the percentile is specified by giving the corresponding decimal.