Percentile – Calculating Percentile
This is widely used in calculating the positions of students participated in entrance exams, though the calculations vary slightly.
The pth percentile of a data set is a value such that at least p % of the items in
the dataset have a value equal to or less than the value of the data item. It also means that
(100 - p) % of the items have a value more than the value of the data item.
Calculating Percentile
Percentile can be calculated using the following approach:
Arrange the data in ascending order.
The position of the pth percentile item, i = (p/100) * n where n is the total numbers in the data sample.
If i is not an integer, round up. The pth percentile is the value in the ith position.
If i is an integer, the pth percentile is the average of the values in positions ‘i’ and ‘i +1’.
Calculating 90th Percentile
Consider the below sample of data.
5,6,2,3,4,4,4,5,8,9
Arrange the data in ascending order
2,3,4,4,4,5,5,6,8,9
i = (p/100) * n = (90/100) x 10 = 9
Averaging the 9th and 10th data value you get 90th percentile = (8 +9)/2 = 8.5
Median and Mode
Median: The median of a data set is the value in the middle when the data items are arranged in ascending or descending order. If there are an odd number of items, the median is the value of the middle item. If there is an even number of items, the median is the average of the values for the middle two items.
Consider the below sample of data.
5,6,2,3,4,4,4,5,8,9
Arrange the data in ascending order
2,3,4,4,4,5,5,6,8,9
Median = (4 + 5 )/2 = 4.5
Mode
Definition: The mode of a data set is the value that occurs with greatest frequency or is the value that is repeated most often in the data set.
In the above example 4 is repeated thrice. So 4 is the Mode of the sample data.
Arithmetic Mean
Mean: The mean of a data set is the average of all the data values. Mean is one of the most widely used measures of central tendency for various data distributions.
If the data are from a sample, the mean is denoted by
x = ∑xi
___
n
For example arithmetic mean of sample 2,3,4,4,4,4,5,6 is 32/8 = 4
If the data are from a population, the mean is denoted by Calculating mean
µ = ∑xi
____
N

