Percentiles & Quantiles

"Is 64 a good score for Virat?" The mean and max don't answer that. A percentile does: it tells you what fraction of the data lies below a value. Selectors use this all the time — "his 90th-percentile score is 88" means only 10% of his innings beat 88.

Definition

The p-th percentile of a dataset is the value below which p percent of the sorted data falls. A quantile is the same idea written as a fraction from 0 to 1 instead of a percentage: the 0.25 quantile = the 25th percentile.

• Sorted data — percentiles are about rank, so NumPy conceptually sorts the values first.
• Interpolation — when the exact position falls between two values, NumPy (by default) draws a straight line between them (method="linear").
• Quartiles — the 25th, 50th and 75th percentiles split the data into four equal parts: Q1, Q2 (= median), Q3.

Syntax

np.percentile(a, q, axis=None, method="linear", keepdims=False)   # q in 0..100
np.quantile(a, q, axis=None, method="linear", keepdims=False)     # q in 0..1

Note: like median, these are function-only — there is no a.percentile() method. In NumPy 2 the old interpolation= argument is replaced by method=.

Example — quartiles of Virat's season

import numpy as np

virat = np.array([72, 45, 88, 31, 64, 0, 103, 52, 19, 77])
print("sorted:", np.sort(virat))

q1, q2, q3 = np.percentile(virat, [25, 50, 75])
print("Q1 =", q1, " median =", q2, " Q3 =", q3)
print("same with quantile:", np.quantile(virat, [0.25, 0.5, 0.75]))
print("90th percentile:", np.percentile(virat, 90))
print("IQR (Q3 - Q1):", q3 - q1)

sorted: [  0  19  31  45  52  64  72  77  88 103]
Q1 = 34.5  median = 58.0  Q3 = 75.75
same with quantile: [34.5  58.   75.75]
90th percentile: 89.5
IQR (Q3 - Q1): 41.25

How it works: with 10 sorted values, the 25th percentile sits at position 0.25 × (10 − 1) = 2.25 → between 31 and 45, a quarter of the way: 31 + 0.25 × 14 = 34.5. The IQR (interquartile range) is the spread of the middle 50% of innings — a spread measure that ignores the extreme duck and century.

Example — the method parameter

import numpy as np

runs = np.array([10, 20, 30, 40])
for m in ["linear", "lower", "higher", "nearest", "midpoint"]:
    print(f"{m:<9}", np.percentile(runs, 40, method=m))

linear    22.0
lower     20
higher    30
nearest   20
midpoint  25.0

How it works: the 40th percentile position is 0.4 × 3 = 1.2 → between 20 (pos 1) and 30 (pos 2). linear gives 22.0, lower/higher pick a neighbour, nearest picks the closer one, midpoint averages the two.

Example — per-player percentiles and outlier detection

import numpy as np

scores = np.array([
    [72, 45, 88, 31, 64],
    [56,  0, 74, 88, 32],
    [45, 32,  0, 67, 18],
    [33, 61, 45,  0, 77],
])
print("75th percentile per player:", np.percentile(scores, 75, axis=1))

runs = np.array([12, 8, 15, 10, 14, 11, 150])
q1, q3 = np.percentile(runs, [25, 75])
iqr = q3 - q1
upper = q3 + 1.5 * iqr
print("upper fence:", upper, " outliers:", runs[runs > upper])

75th percentile per player: [72. 74. 45. 61.]
upper fence: 20.5  outliers: [150]

How it works: the 1.5 × IQR rule (used by box plots) flags any value above Q3 + 1.5×IQR (or below Q1 − 1.5×IQR) as an outlier. The 150 is clearly out of line with the rest.

Common mistakes

• ❌ np.quantile(a, 25) → ValueError: Quantiles must be in the range [0, 1]. ✅ np.quantile(a, 0.25) or np.percentile(a, 25).
• ❌ np.percentile(a, 0.25) — runs fine but gives the 0.25th percentile (almost the min). ✅ Use 25 with percentile.
• ❌ a.percentile(50) → AttributeError. ✅ It's function-only: np.percentile(a, 50).
• ❌ Expecting the result to be one of the data values. ✅ With method="linear" it can be in-between; use "nearest"/"lower" if you need an actual data point.

Comparison — percentile vs quantile vs median

Interview tip

Q: What is the IQR and why is it preferred over the range for skewed data? A: IQR = Q3 − Q1, the spread of the middle 50% of the data. The range (max − min) depends only on the two most extreme values, so one century or one duck changes it hugely; the IQR ignores the extremes, which makes it robust to outliers.