Percentiles and Spread

Five-Number Summary

Describe a dataset's spread with five landmarks: minimum, Q1, median, Q3, and maximum. With n=9, percentile positions (n−1)×p/100 for p∈{0,25,50,75, 100} land on exact integers (0, 2, 4, 6, 8) — no interpolation. Loop over the five percents, compute each position, read sv[pos]. With numpy, np.percentile(x, [0,25,50,75,100]) gives the same five values via linear interpolation (exact at integer positions).

By hand

With the library

np.percentile(x, [0,25,50,75,100]) extends the same linear-interpolation method to all five percentiles. Percentile 0 returns the minimum and 100 returns the maximum — no separate min()/max() calls needed. Values are float; cast with int() to match.

naive.py 
values = [4, 7, 2, 9, 1, 8, 5, 6, 3]
sv = sorted(values)
n = len(sv)
percents = [0, 25, 50, 75, 100]
summary = []
for p in percents:
    pos = int((n - 1) * p / 100)
    summary.append(sv[pos])
lo, q1, median, q3, hi = summary
print('RESULT:', (lo, q1, median, q3, hi))
library.py 
import numpy as np
from dalib.display import set_display
set_display()

x = [4, 7, 2, 9, 1, 8, 5, 6, 3]
lo, q1, median, q3, hi = np.percentile(x, [0, 25, 50, 75, 100])
print('min:', int(lo))
print('Q1:', int(q1))
print('median:', int(median))
print('Q3:', int(q3))
print('max:', int(hi))
print('RESULT:', (int(lo), int(q1), int(median), int(q3), int(hi)))

min: 1
Q1: 3
median: 5
Q3: 7
max: 9
RESULT: (1, 3, 5, 7, 9)

Implementation notes

  • The five-number summary is the data behind a box-and-whisker plot: the box spans Q1–Q3, the line inside is the median, and the whiskers extend to min and max (before outlier clipping).
  • np.percentile with p=0 and p=100 is equivalent to min() and max(); calling them through percentile keeps the loop uniform and avoids two extra passes.
  • Cross-reference: median-and-quartiles (this chapter) for the quartile position derivation; interquartile-range (this chapter) for the Q3−Q1 spread derived from this summary.