statistics

Percentile

The k-th percentile is the value below which k% of observations fall. The 50th percentile is the median; the 25th and 75th are the quartiles.

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The k-th percentile PkP_k is the value below which k%k\% of observations in a dataset fall. Common percentiles:

  • 25th percentile (Q1): lower quartile.
  • 50th percentile (Q2): median.
  • 75th percentile (Q3): upper quartile.
  • 90th, 95th, 99th: tails — used for extremes.

Interquartile range (IQR) = Q3−Q1Q3 - Q1, a robust measure of spread (insensitive to outliers, unlike standard deviation).

Computation: sort data, compute rank r=(k/100)(n+1)r = (k/100)(n+1), take the rr-th observation (interpolating between integer ranks). Multiple slightly-different definitions exist — Excel, R, NumPy use different conventions.

Applications: standardised test scores, paediatric growth charts, P99 latency in engineering, box plots. Percentiles are non-parametric — they make no assumption about distribution shape.