Percentiles

Percentiles are values in a given set of observations that divide the data into 100 equal parts. These values can be denoted by P₁, P₂,....., P₉₉, where

1 % of the data falls below (is less than or equal to) P₁
2 % of the data falls below P₂

:
:

99 % of the data falls below P₉₉

Percentiles can be calculated using a sorted list of observations or the cumulative frequency distribution table corresponding to the observations. In the latter method, it is assumed that the values in a class interval are uniformly distributed within it; extrapolation is then used to calculate the percentiles. As this assumption is often untrue, percentile values can differ depending on whether raw data or frequency distributions were used in the computation. Therefore, percentiles are often treated as estimates for the value below which certain percentages of the observations fall.

EX. Given the following sorted list of observations:

P₇₅ = 6.1, since 40 x 75 % = 30 and 6.1 is the 30th ranked value.
P₄₅ = 4.0, since 40 x 45 % = 18 and 4.0 is the 18th ranked value.
P₆₂ = 5.2, since 40 x 62 % = 24.8 and 5.2 is the 25th ranked value.

This set of observations has the following cumulative frequency distribution:

The percentiles can also be calculated from the cumulative frequency distribution table, using extrapolation to arrive at estimates:

P₇₅ = 6.0 + 1.0 * ((0.750 - 0.725) / (0.850 - 0.725)) = 6.0 + 0.025 / 0.125 = 6.2

where 6.0 is the upper class limit of interval 5.0 - 6.0 with cumulative frequency 0.725, and 0.850 is the cumulative frequency of the next interval, 6.0 - 7.0, with class width 1.0.

P₄₅ = 4.0, since the interval 3.0 - 4.0 has a cumulative frequency of 0.45

P₆₂ = 5.0 + 1.0 × ((0.620 - 0.600) / (0.725 - 0.600)) = 5.0 + 0.020 / 0.125 = 5.16

where 5.0 is the upper class limit of interval 4.0 - 5.0 with cumulative frequency 0.600, and 0.725 is the cumulative frequency of the next interval, 5.0 - 6.0, with class width 1.0.

The values of P₇₅ and P₆₂ differ between the two methods of calculation, while the values of P₄₅ for both methods are the same.

Deciles are values in a given set of observations that divide the data into 10 equal parts. These values can be denoted by D₁, D₂,....., D₉ , where

10 % of the data falls below D₁
20 % of the data falls below D₂

:
:

90 % of the data falls below D₉

It is easy to see that

Quartiles are values in a given set of observations that divide the data in 4 equal parts. These values can be denoted by Q₁, Q₂ and Q₃, where

25 % of the data falls below Q₁
50 % of the data falls below Q₂
75 % of the data falls below Q₃

Again, it is obvious that Q₁ = P₂₅, Q₂ = P₅₀, and Q₃ = P₇₅ .

Deciles and quartiles are calculated in the same manner as percentiles.