Percentile

[per-sen-tahyl, -til]

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 25th percentile is also known as the first quartile(Q₁); the 50th percentile as the median or second quartile(Q₂); the 75th percentile as the third quartile (Q₃).

Definition

There is no standard definition of percentile , 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 $p or p>p_N, we take v=v_1 or v=v_N respectively. Otherwise, we find an integer k such that p_kle p le p_{k+1}, and take v=v_k+frac{N}{100}(p-p_k)(v_{k+1}-v_k). When p=50, the formula gives the median. When N is even and p=25, the formula gives the median of the first frac{N}{2} values.$

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 formulae 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 Microsoft Excel, use the following method to estimate the value, $v_p$ , of the $p^{th}$ 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 $N^{th}$ member of the ordered dataset, $v_N$ .
Otherwise, $1 < k < N$ and $v_p=v_k+d(v_{k+1}-v_k).$
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. 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, your usage is below this amount. Just the same, the remaining 5% of the time, your usage is above that amount.

Physicians will often use infant and children's weight and height percentile as a gauge of relative health.

The normal curve and percentiles

Percentiles are often represented graphically, using a "normal curve". A normal curve is always divided in the same respective manner. At the peak, in the center, stands the point of the mean of the distribution being graphed. On both the right and left sides each, the graph is divided into 3 equal parts, 1, 2, and 3 to the right and -1, -2, -3 to the left respectively. The important thing to remember is that at each of these standard deviation represents a fixed percentile. In other words, every standard deviation unit on the axis, including standard deviation units -3 to +3 have specific percentiles that are always paired with them, regardless the data or values in the distribution. So, what are the pairs of percentiles/standard deviation units? -2 = 2.5th percentile; -1 = 16th percentile; 0 = 50th percentile (also the mean of the distribution as previously stated); +1 = 84th percentile; +2 = 97.5th percentile; +3 = 99.8th percentile.

Percentage also becomes a factor in measuring a distribution graphically. On any normal curve, 99.7% of data lies between the -3 and +3 values, 95% between -2 and +2, 68% between -1 and +1, 34% between 0 and -1 or 0 and +1, 16% between -1 and -2 or +1 and +2 and 2.5% between -2 and -3 or +2 and +3. The remaining 0.3% of the data is between -3 and negative infinity or +3 and positive infinity.