Measures of Central Tendency

A measure of central tendency is a single value that represents the centre or "typical" value of a dataset. The three classical measures are the mean, median and mode.

1. Arithmetic Mean

Ungrouped data:                Grouped data (frequency distribution):

        Σx                              Σ(f × x)
   x̄ = ────                        x̄ = ──────────
         n                                 Σf

   x̄ = sample mean,  μ = population mean
   x = individual value,  f = frequency,  n = number of observations

Worked example (ungrouped). Marks: 45, 52, 68, 74, 81

Σx = 45 + 52 + 68 + 74 + 81 = 320
n  = 5
x̄  = 320 / 5 = 64

Worked example (grouped).

x̄ = Σfx / Σf = 1020 / 40 = 25.5

Properties of the arithmetic mean:

1. The sum of deviations from the mean is always zero: Σ(x − x̄) = 0
2. The sum of squared deviations from the mean is minimum — Σ(x − x̄)² is smaller than for any other value
3. It uses every observation, so it is affected by every value — including outliers
4. Means of subgroups can be combined into a combined mean:

        n₁x̄₁ + n₂x̄₂
  x̄₁₂ = ──────────────
           n₁ + n₂

Example: Section A (n=40, mean=62), Section B (n=60, mean=70)
  x̄ = (40×62 + 60×70) / 100 = (2480 + 4200) / 100 = 66.8

Other Means

Relationship: for any positive dataset, AM ≥ GM ≥ HM (equality only when all values are identical).

# Weighted mean — CGPA calculation
subjects = [("DBMS", 8, 4), ("DA", 9, 3), ("OS", 7, 4), ("NLP", 10, 2)]
num = sum(grade * credits for _, grade, credits in subjects)
den = sum(credits for _, _, credits in subjects)
print("CGPA:", round(num / den, 2))    # CGPA: 8.15

# Geometric mean — average growth rate over 3 years: 10%, 25%, -5%
from statistics import geometric_mean
factors = [1.10, 1.25, 0.95]
g = geometric_mean(factors)
print("Average annual growth:", round((g - 1) * 100, 2), "%")   # 9.35 %
# (the arithmetic mean of 10, 25, -5 = 10% would OVERSTATE the true growth)

2. Median

The median is the middle value when data is arranged in ascending order — it divides the dataset into two equal halves.

Ungrouped data:
   n odd  ->  Median = value of the ((n+1)/2)th item
   n even ->  Median = mean of the (n/2)th and (n/2 + 1)th items

Grouped data:
                    (n/2 - cf)
   Median = L + ─────────────────  × h
                        f

   L  = lower boundary of the median class
   n  = Σf (total frequency)
   cf = cumulative frequency of the class BEFORE the median class
   f  = frequency of the median class
   h  = class width

Worked example (ungrouped, odd n). 12, 7, 3, 15, 9 → sorted: 3, 7, 9, 12, 15 → n = 5 (odd) → position (5+1)/2 = 3rd → Median = 9

Worked example (ungrouped, even n). 3, 7, 9, 12, 15, 20 → n = 6 → mean of 3rd and 4th = (9 + 12)/2 = 10.5

Worked example (grouped). Using the table above (Σf = 40, so n/2 = 20):

L = 20, cf = 13, f = 12, h = 10
Median = 20 + ((20 - 13) / 12) × 10
       = 20 + (7/12) × 10
       = 20 + 5.83
       = 25.83

3. Mode

The mode is the most frequently occurring value.

Grouped data:
                      (f₁ - f₀)
   Mode = L + ────────────────────────── × h
                (2f₁ - f₀ - f₂)

   L  = lower boundary of the modal class (the class with highest frequency)
   f₁ = frequency of the modal class
   f₀ = frequency of the class before it
   f₂ = frequency of the class after it
   h  = class width

Worked example (grouped). Modal class is 20–30 (highest f = 12):

L = 20, f₁ = 12, f₀ = 8, f₂ = 10, h = 10
Mode = 20 + ((12 - 8) / (24 - 8 - 10)) × 10
     = 20 + (4 / 6) × 10
     = 20 + 6.67
     = 26.67

A dataset may be unimodal (one mode), bimodal (two), multimodal (many), or have no mode at all.

Empirical Relationship

For a moderately skewed distribution:

   Mode ≈ 3 × Median − 2 × Mean

Check with our grouped data:
   3(25.83) - 2(25.5) = 77.49 - 51 = 26.49   ≈ 26.67 computed directly ✓

Effect of Skewness — Which Measure to Trust

The mean is dragged toward the long tail. This is why median income is always reported instead of mean income — a handful of billionaires would make the "average" income meaningless.

Comparison Table

import pandas as pd
import numpy as np

salaries = pd.Series([25000, 28000, 30000, 32000, 35000, 38000, 2500000])

print("Mean:  ", round(salaries.mean(), 2))      # Mean:   384000.0
print("Median:", salaries.median())              # Median: 32000.0
print("Mode:  ", salaries.mode().tolist())       # every value unique -> all listed
print("Skewness:", round(salaries.skew(), 3))    # 2.645  strongly right-skewed

# The mean says "average salary is 3.84 lakh" — but 6 of 7 employees earn under 40k.
# The MEDIAN of 32,000 is the honest summary here.

# Trimmed mean — a compromise: drop the extreme 10% from each end
from scipy import stats
print("Trimmed mean (10%):", round(stats.trim_mean(salaries, 0.1), 2))

# All three at once, on a full DataFrame
df = pd.DataFrame({
    "marks": [45, 52, 68, 74, 81, 68, 90, 68],
    "city":  ["Delhi", "Noida", "Delhi", "Gurgaon", "Delhi", "Noida", "Delhi", "Noida"],
})

print("Mean marks:  ", df["marks"].mean())          # 68.25
print("Median marks:", df["marks"].median())        # 68.0
print("Mode marks:  ", df["marks"].mode()[0])       # 68
print("Mode city:   ", df["city"].mode()[0])        # Delhi  (mean/median invalid here)

print(df["marks"].describe())

Central tendency tells you where the data sits. The next lesson answers the equally important question: how spread out is it?