NationMaster - Encyclopedia: Percentile

A percentile is the value of a variable below which a certain percent of observations fall. So the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in descriptive statistics as well as in the reporting of scores from norm-referenced tests. The percent sign. ... The percentile rank of a score is the percentage of scores in its frequency distribution which are lower. ... Descriptive statistics are used to describe the basic features of the data in a study. ...

The 25th percentile is also known as the first quartile; the 50th percentile as the median. In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. ... In probability theory and statistics, a median is a type of average that is described as the number dividing the higher half of a sample, a population, or a probability distribution, from the lower half. ...

1 Definition
2 Alternative methods
3 Relation between percentile, decile and quartile
4 Examples
5 See also
6 References
7 External links

Definition

There is no standard definition of percentile ^[1] ^[2] , however all definitions yield similar results when the number of observations is large. One definition, usually given in unsophisticated texts, is that the $p$ -th percentile of $N$ ordered values is obtained by first calculating the rank $n = frac{N}{100},p+frac{1}{2}$ , rounding to the nearest integer, and taking the value that corresponds to that rank.

An alternative method, used in many applications, is to use linear interpolation between the two nearest ranks instead of rounding. Specifically, if we have $N$ values $v 1$ , $v 2$ , $v 3$ ,..., $v N$ , ranked from least to greatest, define the percentile corresponding to the $n$ -th value as $p_n=frac{100}{N}(n-frac{1}{2}).$ In this way, for example, if $N = 5$ the percentile corresponding to the third value is $p_3=frac{100}{5}(3-frac{1}{2})=50.$ Suppose we now want to calculate the value $v$ corresponding to a percentile $p$ . If $p < p 1$ or $p > p N$ , we take $v = v 1$ or $v = v N$ respectively. Otherwise, we find an integer $k$ such that $p_kle p le p_{k+1}$ , and take $v=v_k+frac{N}{100}(p-p_k)(v_{k+1}-v_k).$ ^[3] When $p = 50$ , the formula gives the median. When $N$ is even and $p = 25$ , the formula gives the median of the first $frac{N}{2}$ values. Linear interpolation is a process employed in mathematics, and numerous applications including computer graphics. ...

Linked with the percentile function, there is also a weighted percentile, where the percentage in the total weight is counted instead of the total number. In most spreadsheet applications there is no standard function for a weighted percentile. One method for weighted percentile extends the method described above. Suppose we have positive weights $w 1$ , $w 2$ , $w 3$ ,..., $w N$ , associated respectively with our $N$ sample values. Let $S_n=sum_{k=1}^{n}w_k$ be the $n$ -th partial sum of these weights. Then the formulas above are generalized by taking $p_n=frac{100}{S_N}(S_n-frac{w_n}{2})$ and $v=v_k+frac{p-p_k}{p_{k+1}-p_k}(v_{k+1}-v_k).$

Alternative methods

Many software packages, such as Excel, use the following method to estimate the value, $v p$ , of the $p t h$ percentile of an ascending ordered dataset containing $N$ elements with values $v 1, v 2,..., v N$ ;

$n = frac{p}{100},({N}-1)+1$

$n$ is then split into its integer component, $k$ and decimal component, $d$ , such that $n = k + d$
If $k = 0$ , then the value for that percentile, $v p$ , is the first member of the ordered dataset, $v 1$ .
If $k = N$ , then the value for that percentile, $v p$ , is the $N t h$ member of the ordered dataset $v N$ .
Else $(0 < k < N)$ then $v p = v k + d (v k + 1 - v k).$
An alternative method, is as above, with $n$ calculated as $n = frac{p}{100},({N}+1)$

Relation between percentile, decile and quartile

P₂₅ = Q₁
P₅₀ = D₅ = Q₂ = median value
P₇₅ = Q₃
P₁₀₀ = D₁₀ = Q₄
P₁₀ = D₁
P₂₀ = D₂
P₃₀ = D₃
P₄₀ = D₄
P₆₀ = D₆
P₇₀ = D₇
P₈₀ = D₈
P₉₀ = D₉

Note: One quartile is equivalent to 25 percentile while 1 decile is equal to 10 percentile.

Examples

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. â€œISPâ€ redirects here. ... Burstable billing is a method of measuring your bandwidth based on peak utilization. ...

Physicians will often use infant and children's weight and height percentile as a gauge of relative health. Weight and height percentiles are determined by growth charts and body mass index charts to compare a childs measurements with those of other children in the same age group. ...

References

http://www.itl.nist.gov/div898/handbook/prc/section2/prc252.htm

^ Lane, David. Percentiles. Retrieved on 2007-09-15.
^ Pottel, Hans. Statistical flaws in Excel. Retrieved on 2006-03-22.
^ Matlab Statistics Toolbox - Percentiles. Retrieved on 2006-09-15.

Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 258th day of the year (259th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 81st day of the year (82nd in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 258th day of the year (259th in leap years) in the Gregorian calendar. ...

External links

Free Online Software (Calculator) computes Percentiles for any dataset according to 8 different percentile definitions.
Introductory tutorial on percentiles and other quantiles - aimed at the non-mathematically minded

Category: Summary statistics

Results from FactBites:

Percentile Rank (92 words)
	A percentile rank is typically defined as the proportion of scores in a distribution that a specific score is greater than or equal to.
	alternatively, percentile rank is sometimes defined simply as the proportion of a distribution that a score is greater than.
	Follow this link for a more complex definition of percentile.

More results at FactBites »

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