Measures of Dispersion

Dispersion (variability/spread) measures how far the data values are scattered from the centre. Two datasets can share an identical mean yet be completely different:

Set A: 48, 49, 50, 51, 52      mean = 50, very tight
Set B:  10, 30, 50, 70, 90     mean = 50, hugely spread

Reporting only the mean would hide the entire difference — which is why every summary must pair a measure of centre with a measure of spread.

1. Range

   Range = Maximum value − Minimum value

   Coefficient of Range = (Max − Min) / (Max + Min)

Simplest measure, but uses only two values and is completely determined by the extremes.

Example. Marks 45, 52, 68, 74, 81 → Range = 81 − 45 = 36

2. Quartile Deviation (Semi-Interquartile Range)

   Q1 = lower quartile (25th percentile)
   Q3 = upper quartile (75th percentile)

   IQR = Q3 − Q1
   Quartile Deviation (QD) = (Q3 − Q1) / 2
   Coefficient of QD = (Q3 − Q1) / (Q3 + Q1)

The IQR covers the middle 50% of the data and completely ignores extreme values — making it the natural companion to the median, and the basis of the box plot and outlier fences.

3. Mean Deviation (Average Deviation)

                Σ|x − x̄|
   MD (mean) = ──────────
                   n

                  Σ|x − Median|
   MD (median) = ───────────────      (this version is the minimum possible MD)
                        n

Worked example. Data: 2, 4, 6, 8, 10 → x̄ = 6

|2-6| + |4-6| + |6-6| + |8-6| + |10-6| = 4 + 2 + 0 + 2 + 4 = 12
MD = 12 / 5 = 2.4

4. Variance and Standard Deviation

The most important measures in all of statistics.

POPULATION                              SAMPLE

        Σ(x − μ)²                              Σ(x − x̄)²
  σ² = ───────────                       s² = ───────────
            N                                     n − 1

  σ = √σ²                                 s = √s²

  Divisor n−1 (Bessel's correction) makes the sample variance an
  UNBIASED estimator of the population variance.

Shortcut / computational formula:

        Σx²      ( Σx )²
  σ² = ─────  −  ───────
         N          N

Worked example. Data: 2, 4, 6, 8, 10 (treat as population)

μ  = 30 / 5 = 6
σ² = 40 / 5 = 8
σ  = √8 = 2.828

As a SAMPLE instead:
s² = 40 / (5 − 1) = 10
s  = √10 = 3.162

Verify with the shortcut formula:

Σx² = 4 + 16 + 36 + 64 + 100 = 220
σ² = 220/5 − (30/5)² = 44 − 36 = 8  ✓

Properties of standard deviation:

1. σ ≥ 0 always; σ = 0 only when every value is identical
2. Same unit as the original data (variance is in squared units — its main drawback)
3. Adding a constant to every value leaves σ unchanged
4. Multiplying every value by a constant c multiplies σ by |c|
5. It is the measure minimised by the mean, and it underpins the normal distribution, z-scores, correlation and regression

5. Coefficient of Variation (CV)

        σ
  CV = ─── × 100 %
        x̄

CV is unit-free, so it is the correct tool for comparing variability between datasets with different units or very different means.

Worked example. Which is more consistent?

Batsman B has the smaller standard deviation but the larger CV — relative to his own average he is less consistent. Lower CV = more consistent/stable.

The Empirical Rule (68–95–99.7)

For an approximately normal distribution:

This is exactly why |z| > 3 is the standard outlier threshold — such values occur in under 0.3% of normal data.

Skewness and Kurtosis — Shape Measures

Karl Pearson's coefficient of skewness:

        Mean − Mode              3(Mean − Median)
  Sk = ─────────────    or  Sk = ─────────────────
             σ                           σ

Full Python Walkthrough

import pandas as pd
import numpy as np

data = pd.Series([2, 4, 6, 8, 10])

print("Range:            ", data.max() - data.min())          # 8
print("Variance (sample):", data.var())                       # 10.0   (ddof=1 default)
print("Variance (pop):   ", data.var(ddof=0))                 # 8.0
print("Std dev (sample): ", round(data.std(), 3))             # 3.162
print("Std dev (pop):    ", round(data.std(ddof=0), 3))       # 2.828

Q1, Q3 = data.quantile(0.25), data.quantile(0.75)
print("Q1:", Q1, "Q3:", Q3, "IQR:", Q3 - Q1)                  # Q1: 4.0 Q3: 8.0 IQR: 4.0
print("Quartile Deviation:", (Q3 - Q1) / 2)                   # 2.0

print("Mean Absolute Deviation:", (data - data.mean()).abs().mean())   # 2.4
print("CV:", round(data.std() / data.mean() * 100, 2), "%")   # 52.7 %
print("Skewness:", round(data.skew(), 3))                     # 0.0 (perfectly symmetric)
print("Kurtosis (excess):", round(data.kurtosis(), 3))        # -1.2

# Same mean, very different spread — the whole point of dispersion
A = pd.Series([48, 49, 50, 51, 52])
B = pd.Series([10, 30, 50, 70, 90])

summary = pd.DataFrame({
    "Set A": [A.mean(), A.std(), A.max() - A.min(), A.std() / A.mean() * 100],
    "Set B": [B.mean(), B.std(), B.max() - B.min(), B.std() / B.mean() * 100],
}, index=["Mean", "Std Dev", "Range", "CV %"]).round(2)
print(summary)
#          Set A  Set B
# Mean     50.00  50.00     <- identical
# Std Dev   1.58  31.62     <- 20x difference
# Range     4.00  80.00
# CV %      3.16  63.25

# describe() gives centre + spread + quartiles in one call
df = pd.DataFrame({"marks": [45, 52, 68, 74, 81, 68, 90, 33]})
print(df.describe().round(2))
#         marks
# count    8.00
# mean    63.88
# std     18.83
# min     33.00
# 25%     50.25
# 50%     68.00
# 75%     75.75
# max     90.00

Absolute vs Relative Dispersion

Centre plus spread describes one variable. The next lesson moves to two variables at once — correlation.