Percentile

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 falls. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found. 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 falls. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found. 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 below 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. 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: so, 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. In finance, Value at Risk is a standard measure to assess (in a model dependent way) the quantity under which the value of the portfolio is not expected to sink within a given period of time and given a confidence value. 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 ( σ {displaystyle 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 σ {displaystyle sigma } to +3 σ {displaystyle sigma } range. For example, with human heights very few people are above the +3 σ {displaystyle 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 σ {displaystyle sigma } is the 0.13th percentile, −2 σ {displaystyle sigma } the 2.28th percentile, −1 σ {displaystyle sigma } the 15.87th percentile, 0 σ {displaystyle sigma } the 50th percentile (both the mean and median of the distribution), +1 σ {displaystyle sigma } the 84.13th percentile, +2 σ {displaystyle sigma } the 97.72nd percentile, and +3 σ {displaystyle sigma } the 99.87th percentile. This is related to the 68–95–99.7 rule or the three-sigma rule. Note that in theory the 0th percentile falls at negative infinity and the 100th percentile at positive infinity, although in many practical applications, such as test results, natural lower and/or upper limits are enforced. There is no standard definition of percentile,however all definitions yield similar results when the number of observations is very large and the probability distribution is continuous. In the limit, as the sample size approaches infinity, 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. This can be seen as a consequence of the Glivenko–Cantelli theorem. Some methods for calculating the percentiles are given below.