# How do I estimate centile values?

If all the values of a variable are ranked from the lowest value to the highest, the middle value (or if there are an even number of values the average of the two middle ones) is the median or 50th centile. Other centile values can separate the distribution into n% above the nth centile and (100-n) % below the nth centile.

• The centile, c, that corresponds to rank r can be calculated from the following formula for n observations:

Centile, c, of rth rank = r/(n+1) x 100,

For the rth observation from the smallest if all the observations are ranked from the lowest to the highest.

For example with 25 observations the one that is the 13th from the lowest end (13th rank) is equivalent to the following centile:

13/(25 + 1) x 100 = 13/26 x 100 = 50th

(The "n+1" in the formula above instead of "n" is to produce symmetrical centile estimates. For example in 25 observations the one that ranks 5th (4 observations below it) should be an equivalent centile position from the median as the one that ranks 21st (4 observations above it)).
So 5/26 x 100 = 19.2 centile, i.e. 30.8 centile points below the median and 21/26 x 100 = 80.8 centile, i.e. 30.8 centile points above the median

• The rank, r that corresponds to centile c can be calculated from the following formula for n observations:

Rank, r, of cth centile = c/100 x (n+1)

For example with 25 observations the rank that corresponds to the 10th centile = 10/100 x (25 + 1) = 2.6 (i.e., 10% of the distribution lies at or below the value corresponding to rank 2.6)
The value that corresponds to a rank of 2.6 is 60% of the distance between the second and third smallest values.

Similarly the rank that corresponds to the 90th centile = 90/100 x (25 +1) = 23.4 (not 22.5)
(i.e., 10% of the distribution lies at or above the value corresponding to rank 23.4).

If one wishes to know the observed rank that corresponds to an given centile round up for ranks below the middle one (so 2.6 above becomes 3) and round down for ranks above the middle one (so 23.4 becomes 23).