how to calculate percentile

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 v_1 \le v_2 \le \dots \le v_N.

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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2 responses to “how to calculate percentile

  1. 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.

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