6. Percentiles and Quartiles

# 6.1 Discrete Data Percentiles and Quartiles

Before we get into how to calculate percentile in a data set, note that we can see percentiles directly on a cumulative frequency plot, see Figure 6.2.

*Computing percentile positions of discrete data. *Let be the *ordered* position of a data set of data points, then we define the percentile position of to be

(6.1)

This formula has the property that and . It is what we will use as a percentile formula but it is not the only one. Look at Figure 6.1. The way the histogram there is shaded the formula would be which would have the property that and . There are other, not necessarily wrong, ways to define the percentile position of discrete data but we will use Equation 6.1.

If you want to find the position, , of the data point corresponding to a given percentile then compute

(6.2)

Equation (6.2) is derived by solving Equation (6.1) for . Note that Equation (6.2) gives the *position* of the data point , not its value. To clarify that, let’s look at an example.

**Example 6.1** : Consider the dataset given below. Data would originally be given as the numbers in the first line. So the first step in answering any question about percentiles is to order the data, the same as what you need to to to determine the median of a dataset. Once the data are ordered, then you may assign a position number to each data point as shown in the third line.

original data | 18 | 15 | 12 | 6 | 8 | 2 | 3 | 5 | 20 | 10 |

ordered data | 2 | 3 | 5 | 6 | 8 | 10 | 12 | 15 | 18 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

Q : What is the percentile rank of ?

A : so percentile.

Q : What is the value corresponding to the percentile, ?

A :

The closest is 3 and . We can write .

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* Decile *:

The decile of data value in the *ordered* position is defined as

We will not make much use of decile except to see that quartile is defined in the same way.

* Quartile *:

The quartile of data value in the ordered position 1.

(6.3)

*Notation *: (This notation also applies to and .) We write :

&=& quartile

& = & quartile

& = & quartile

& = & quartile

& = & quartile

Quartiles are useful because we do not have to compute percentile first and then divide by 25 as given by Equation (6.3). Instead, we can use the following handy tricks after ordering our data:

**Example 6.2** : Example with an even number of data points. With the data *in order*, first find the median, then the medians of the two halves of the dataset :

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**Example 6.3** : Example with an even number of data points. With the data *in order*, first find the median, then the medians of the two halves of the dataset :

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