Measures of Dispersion

Abdullah Al Mahmud

2026-04-06

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 Variance (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)
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 Variance (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)
A	20	3.2
B	20	8.5
C	25	3.2

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