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.