Measures of Dispersion

Abdullah Al Mahmud

2026-04-05

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 = \(\sqrt{Variance}\)

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, \(Me = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ term} + \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ term}}{2}\)
• Median score of 50 students is 75; what does it mean?
• Quartiles divide the data into 4 equal parts

Quartiles for Grouped Data

Recall Median

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

Where

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