In statistics, a percentile (or centile) is the value of a variable below which a certain percent of observations fall. For example, 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 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), and the 75th percentile as the third quartile (Q3).
Percentile can be estimated from a sample. Illustration with a example.
Assume a measurement of 55 samples
v1<-rnom(55)*100 N=55
eg
[1] -1.3924374 214.1235590 -55.8068048 -52.6550701 65.9187795
[6] -127.0083785 -55.9475734 -88.6760815 -152.1034023 54.7370840
[11] 3.2038870 -131.0856343 -11.2843457 32.3273583 -159.9716957
[16] -47.9830684 -60.8969309 -73.4535934 77.5085243 -145.1564030
[21] -85.8970074 63.3539220 -0.8484222 44.4983017 54.2943575
[26] 65.1081746 -80.8126492 95.5413286 -29.1480091 171.5289645
[31] 17.2395755 43.3895267 -158.8449973 -24.7021580 111.4312038
[36] -139.8413503 -90.3177177 -36.0836404 78.2439702 167.3131312
[41] 78.7074156 125.9342059 -75.5310449 162.5520463 2.2618479
[46] 5.1322097 -165.9821912 33.8848565 -141.1951021 147.5859301
[51] -33.8523485 -52.7056264 211.1667364 124.2635856 58.3251450
Rank or sort y as
v<-sort (v1)
[1] -165.9821912 -159.9716957 -158.8449973 -152.1034023 -145.1564030
[6] -141.1951021 -139.8413503 -131.0856343 -127.0083785 -90.3177177
[11] -88.6760815 -85.8970074 -80.8126492 -75.5310449 -73.4535934
[16] -60.8969309 -55.9475734 -55.8068048 -52.7056264 -52.6550701
[21] -47.9830684 -36.0836404 -33.8523485 -29.1480091 -24.7021580
[26] -11.2843457 -1.3924374 -0.8484222 2.2618479 3.2038870
[31] 5.1322097 17.2395755 32.3273583 33.8848565 43.3895267
[36] 44.4983017 54.2943575 54.7370840 58.3251450 63.3539220
[41] 65.1081746 65.9187795 77.5085243 78.2439702 78.7074156
[46] 95.5413286 111.4312038 124.2635856 125.9342059 147.5859301
[51] 162.5520463 167.3131312 171.5289645 211.1667364 214.1235590
The value, vP, of the P-th percentile of an ascending ordered dataset containing N elements with values 
.
The rank is calculated: n=P/100*(N+1), eg for 90% pencentile, the rank n = 0.9 * (55+1) , is 50.4
and then the n is split into its integer component k and decimal component d, such that n = k + d, k=50, d=0.4. Then vP is calculated as:
- v(p) = v(1) for n=1
- v(p) = v(N) for n=N
- v(p) = v(k) + d * [v(k+1) – v(k)] for 1<n<N
v(90%) = v[50] + 0.4 * (v[50+1]- v [50] ) = 147.5859301 + 0.4 * (162.5520463 – 147.5859301) = 153.5724
the 90th percentile in this sample is 153.5724
Microsoft Excel uses this method to calculate percentile too.
This chart demonstrates Percentile ranks (or percentiles) and Normal curve equivalents.
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A percentile rank of 80 indicates that 20 of the total number of cases scored above and 80 scored below in whatever characteristics were being studied. percentile per-sentl any one of the 99 values that divide the of a or sample into 100 intervals of equal probability or frequency for example 45 per cent of a population scores below the 45th percentile.