Measures of Dispersion

Abdullah Al Mahmud

2026-09-28

Why Dispersion?

Which batter is better?

  • Batter A: 0, 100, 10, 10

  • Batter B: 20, 30, 25, 35

  • Dispersion measures variation

  • Needed alongside central tendency for comparison

Is Dispersion Alone good?

  • Batter A: 0, 5, 10, 20, 4
  • Batter B: 10, 15, 12, 22, 16

Graphically



  • Which plot has the largest dispersion?
  • Which plot has the smallest dispersion?

Which is more scattered?

  • X: 10, 12, 15, 20
  • Y: 10, 50, 170, 470

Absolute Measures

  • Range
  • Mean Deviation
  • Quartile Deviation
  • Variance/Standard Deviation (Best)

Relative Measures

  • Coefficient of Range
  • Coefficient of Mean Deviation
  • Coefficient of Quartile Deviation
  • Coefficient of variation (Best)

Estimation

Data: 10, 12, 23, 16, 17, 20, 15

Range

  • Ungrouped Data: \(R = X_H - X_L\)
  • Grouped Data: \(R = L_n - L_1\)

Mean Deviation (MD)

Around mean, \(\displaystyle MD(\bar x)=\frac{1}{n}{\sum_{i=1}^{n} |x_i - \bar{x}|}\)

  • For grouped data, \(\displaystyle MD(\bar x) = \frac{1}{N}{\sum_{i=1}^{n} f_i|x_i - \bar{x}|}\)
  • Around Median, \(\rightarrow MD (Me) = ?\)
  • Around Mode, \(\rightarrow MD (Mo) = ?\)

Think!

What if?

\(\displaystyle MD(\bar x)=\frac{1}{n}{\sum_{i=1}^{n} (x_i - \bar{x})}\)

Grouped Data

“Classified” is not a proper term

Class Interval Frequency
10–19 5
20–29 12
30–39 18
40–49 10
50–59 6

Variance and Standard Deviation

Variance, \(\displaystyle \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2\)

  • Also, \(\displaystyle \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2\)
  • Mean of square - square of mean
  • \(\bar {x^2} - (\bar x)^2\)
  • SD = \(\sqrt{Variance}\)
  • Why squared in the first place?

Variance for Grouped Data

??

  • \(\displaystyle \sigma^2 = \frac1n{\sum_{i=1}^{n} f_i (x_i - \bar x)^2}\) or
  • \[\displaystyle \sigma^2 = \frac{\sum f_i x_i^2}{n} - \left(\frac{\sum f_i x_i}{n}\right)^2\]
  • Create just two columns for calculation (\(f_ix_i\) and \(f_ix_i^2)\)

Find Variance

Class Interval Mid Value (xᵢ) Frequency (fᵢ) \(f_ix_i\) \(f_ix_i^2\)
10–20 15 5
20–30 25 8
30–40 35 7

Coefficients

  • Coefficient of Range, \(CR=\frac{X_H-X_L}{X_H+X_L}\times 100\)
  • Coefficient of Mean Deviation, \(CMD(\bar x) = \frac{MD(\bar x)}{\bar x}\)
  • Coefficient of Quartile Deviation, \(QD = \frac{Q_3-Q_1}{Q_3+Q_1} \times 100\)
  • Coefficient of variation (Best), \(CV = \frac{\sigma}{\bar x} \times 100\)

Quartiles

Recall Median Formula

  • When \(n\) is odd, \(Me = \frac{n+1}{2}th\) term
  • When \(n\) is even, \(Me = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ term} + \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ term}}{2}\)
  • Find median: 10, 20, 13
  • What now: 10, 20, 13, 11
  • Median score of 50 students is 75; what does it mean?
  • Quartiles divide the data into 4 equal parts

Median and Quartiles Formula for Grouped Data

\(\text{Median} = L + \left( \frac{\frac{N}{2} - F_c}{f_m} \right) \times h\)

  • \(\text{Q_3} = L + \left( \frac{\frac{3N}{4} - F_c}{f_m} \right) \times h\)

Where

  • \(L=\) Lower boundary of the median class
  • \(N=\) Total frequency
  • \(F_c=\) Cumulative frequency before the median class
  • \(f_m=\) Frequency of the median class
  • \(h=\) Class width

Median and Quartiles for Grouped Data

Class Interval Frequency Cumulative Freq
10–20 5
20–30 11
30–40 18
40–50 10
50–60 6

\(\boxed {N = 50, L = ?}\)

  • \(Me \rightarrow \frac{N}{2}th\)
  • \(Q_1 \rightarrow \frac{N}{4}th\)
  • \(Q_3 \rightarrow \frac{3N}{4}th\)
  • \(F_c=?, f_m = ?,h = ?\)

Estimate CV

Given \(\sigma = 4, \bar x = 10\)

  • \(CV = \frac{\sigma}{\bar x}\times 100\)
  • \(\frac{4}{10}\times 100\)

Who is better?

Marks in 5 subjects out of 30

Student Mean (\(\bar x\)) Standard Deviation (σ) CV
A 20 3.2
B 20 8.5
C 25 3.2

\(CV = \frac{\sigma}{\bar x}\times 100\)