A percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found.
Contents
- Applications
- The normal distribution and percentiles
- Definitions
- The Nearest Rank method
- Worked examples of the Nearest Rank method
- The Linear Interpolation Between Closest Ranks method
- Commonalities between the Variants of this Method
- First Variant C 1 2 displaystyle C12
- Worked Example of the First Variant
- Second Variant C 1 displaystyle C1
- Worked Examples of the Second Variant
- Third Variant C 0 displaystyle C0
- Worked Example of the Third Variant
- Definition of the Weighted Percentile method
- References
The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. For example, if a score is at the 86th percentile, where 86 is the percentile rank, it is equal to the value below which 86% of the observations may be found (carefully contrast with in the 86th percentile, which means the score is at or below the value of which 86% of the observations may be found - every score is in the 100th percentile). The 25th percentile is also known as the first quartile (Q1), the 50th percentile as the median or second quartile (Q2), and the 75th percentile as the third quartile (Q3). In general, percentiles and quartiles are specific types of quantiles.
Applications
When ISPs bill "burstable" internet bandwidth, the 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then bills at the nearest rate. In this way infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time, the usage is below this amount. Just the same, the remaining 5% of the time, the usage is above that amount.
Physicians will often use infant and children's weight and height to assess their growth in comparison to national averages and percentiles which are found in growth charts.
The 85th percentile speed of traffic on a road is often used as a guideline in setting speed limits and assessing whether such a limit is too high or low.
The normal distribution and percentiles
The methods given in the Definitions section are approximations for use in small-sample statistics. In general terms, for very large populations following a normal distribution, percentiles may often be represented by reference to a normal curve plot. The normal distribution is plotted along an axis scaled to standard deviations, or sigma units. Mathematically, the normal distribution extends to negative infinity on the left and positive infinity on the right. Note, however, that only a very small proportion of individuals in a population will fall outside the −3 to +3 range. For example, with human heights very few people are above the +3 sigma height level.
Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3
Definitions
There is no standard definition of percentile, however all definitions yield similar results when the number of observations is very large. In the limit, as the sample size approaches infinity and the data points become so densely spaced they appear continuous, the 100pth percentile (0<p<1) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p, as p approximates the CDF. Some methods for calculating the percentiles are given below.
The Nearest Rank method
One definition of percentile, often given in texts, is that the P-th percentile
Note the following:
Worked examples of the Nearest Rank method
Example 1:
Consider the ordered list {15, 20, 35, 40, 50}, which contains five data values. What are the 5th, 30th, 40th, 50th and 100th percentiles of this list using the Nearest Rank method?
So the 30th, 40th, 50th and 100th percentiles of the ordered list {15, 20, 35, 40, 50} using the Nearest Rank method are {20, 20, 35, 50}
Example 2:
Consider an ordered population of 10 data values {3, 6, 7, 8, 8, 10, 13, 15, 16, 20}. What are the 25th, 50th, 75th and 100th percentiles of this list using the Nearest Rank method?
So the 25th, 50th, 75th and 100th percentiles of the ordered list {3, 6, 7, 8, 8, 10, 13, 15, 16, 20} using the Nearest Rank method are {7, 8, 15, 20}
Example 3:
Consider an ordered population of 11 data values {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20}. What are the 25th, 50th, 75th and 100th percentiles of this list using the Nearest Rank method?
So the 25th, 50th, 75th and 100th percentiles of the ordered list {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20} using the Nearest Rank method are {7, 9, 15, 20}
The Linear Interpolation Between Closest Ranks method
An alternative to rounding used in many applications is to use linear interpolation between adjacent ranks.
Commonalities between the Variants of this Method
All of the following variants have the following in common. Given the order statistics
we seek a linear interpolation function that passes through the points
where
There are two ways in which the variant approaches differ. The first is in the linear relationship between the rank
There is the additional requirement that the midpoint of the range
and our revised function now has just one degree of freedom, looking like this:
The second way in which the variants differ is in the definition of the function near the margins of the
First Variant, C = 1 / 2 {displaystyle C=1/2}
(Sources: Matlab "prctile" function,)
where
Furthermore, let
The inverse relationship is restricted to a narrower region:
Worked Example of the First Variant
Consider the ordered list {15, 20, 35, 40, 50}, which contains five data values. What are the 5th, 30th, 40th and 95th percentiles of this list using the Linear Interpolation Between Closest Ranks method? First, we calculate the percent rank for each list value.
Then we take those percent ranks and calculate the percentile values as follows:
So the 5th, 30th, 40th and 95th percentiles of the ordered list {15, 20, 35, 40, 50} using the Linear Interpolation Between Closest Ranks method are {15, 20, 27.5, 50}
Second Variant, C = 1 {displaystyle C=1}
(Source: Some software packages, including NumPy and Microsoft Excel (up to and including version 2013 by means of the PERCENTILE.INC function). Noted as an alternative by NIST)
Note that the
Worked Examples of the Second Variant
Example 1:
Consider the ordered list {15, 20, 35, 40, 50}, which contains five data values. What is the 40th percentile of this list using this variant method?
First we calculate the rank of the 40th percentile:
So, x=2.6, which gives us
Example 2:
Consider the ordered list {1,2,3,4} which CONTAINS four data values. What is the 75th percentile of this list using the Microsoft Excel method?
First we calculate the rank of the 75th percentile as follows:
So, x=3.25, which gives us an integral part of 3 and a fractional part of 0.25. So, the value of the 75th percentile is
Third Variant, C = 0 {displaystyle C=0}
(The primary variant recommended by NIST. Adopted by Microsoft Excel since 2010 by means of PERCENTIL.EXC function. However, as the "EXC" suffix indicates, the Excel version excludes both endpoints of the range of p, i.e.,
The inverse is restricted to a narrower region:
Worked Example of the Third Variant
Consider the ordered list {15, 20, 35, 40, 50}, which contains five data values. What is the 40th percentile of this list using the NIST method?
First we calculate the rank of the 40th percentile as follows:
So x=2.4, which gives us
So the value of the 40th percentile of the ordered list {15, 20, 35, 40, 50} using this variant method is 26.
Definition of the Weighted Percentile method
In addition to the percentile function, there is also a weighted percentile, where the percentage in the total weight is counted instead of the total number. There is no standard function for a weighted percentile. One method extends the above approach in a natural way.
Suppose we have positive weights
the
and
The 50% weighted percentile is known as the weighted median.