Measures of Dispersion Notes

Measures of Dispersion

Introduction

Dispersion is described as a single value that describes the spread of a distribution.
The common measures of dispersion are:
- Range
- Interquartile Range
- Variance
- Standard Deviation

Range

The range (R) is the difference between the highest value (HV) and the lowest value (LV) in a data set.
It is considered the weakest measure of dispersion.
Formula: $R = HV - LV$

Interquartile Range

The interquartile range is the middle 50% of a data set.
It is the difference between the upper quartile (UQ) and the lower quartile (LQ).
Formula: $Interquartile Range = UQ - LQ$

Example Calculation

Given Data Set: 21, 24, 25, 25, 28, 29, 30, 31, 32, 33, 39, 42, 48
Calculate the range, interquartile range, variance, and standard deviation.

Data and Calculations Table

x	x̄ (mean)	(x – x̄)	(x – x̄)²
21	31.31	-10.31	106.2961
24	31.31	-7.31	53.4361
25	31.31	-6.31	39.8161
25	31.31	-6.31	39.8161
28	31.31	-3.31	10.9561
29	31.31	-2.31	5.3361
30	31.31	-1.31	1.7161
31	31.31	-0.31	0.0961
32	31.31	0.69	0.4761
33	31.31	1.69	2.8561
39	31.31	7.69	59.1361
42	31.31	10.69	114.2761
48	31.31	16.69	278.5561
Σx = 407			Σ(x – x̄)² = 712.7693

Mean (x̄) Calculation: x̄ = $\\\Sigma x / n = 407 / 13 = 31.3076 ≈ 31.31
Sum of Squared Differences: $\\\Sigma(x – x̄)² = 712.7693$

1. Range Calculation:

Highest Value (HV) = 48
Lowest Value (LV) = 21
$R = 48 - 21 = 27$

Interquartile Range Calculation:

Median = 30
Lower Quartile (LQ) = 25 (average of 25 and 25)
Upper Quartile (UQ) = 36 (average of 33 and 39)
$Interquartile Range = 36 - 25 = 11$

Variance Calculation:

Formula: $S² = \\\Sigma(x – x̄)² / (n - 1)$
$S² = 712.7693 / (13 - 1) = 712.7693 / 12$
$S² = 59.3974 ≈ 59.40$

Standard Deviation Calculation:

Formula: $Sd = \\\sqrt{S²}$
$Sd = \\\sqrt{59.40} = 7.707 ≈ 7.71$

Problem Analysis 1-2

Given the results of 6 students in a 50-item multiple-choice exam: 47, 36, 42, 35, 23, 27
Task: Rearrange the data, construct a table for x, x̄, (x-x̄), and (x-x̄)², and compute the range, interquartile range, first quartile, third quartile, mean, variance, and standard deviation.