Measures of Dispersion
Measures of dispersion help us understand how spread out or scattered the data is. They tell us how much the values in a dataset differ from the average (mean).
Types of Measures of Dispersion
- Range: Difference between the highest and lowest value.
- Mean Deviation: Average of the absolute differences from the mean.
- Variance: Average of the squared differences from the mean.
- Standard Deviation (SD): Square root of the variance. It is the most commonly used measure of dispersion.
Given Data
Values: 10, 12, 14, 18, 25, 30, 35, 40
Step 1: Calculate the Mean (Average)
Formula:
Mean (𝑥̄) = (Sum of all values) ÷ (Total number of values)
Mean (𝑥̄) = (Sum of all values) ÷ (Total number of values)
Mean = (10 + 12 + 14 + 18 + 25 + 30 + 35 + 40) ÷ 8 = 184 ÷ 8 = 23
Step 2: Prepare a Table for Standard Deviation
| Value (x) | x - Mean (x - 23) | (x - 23)² |
|---|---|---|
| 10 | -13 | 169 |
| 12 | -11 | 121 |
| 14 | -9 | 81 |
| 18 | -5 | 25 |
| 25 | 2 | 4 |
| 30 | 7 | 49 |
| 35 | 12 | 144 |
| 40 | 17 | 289 |
| Total | — | 882 |
Step 3: Calculate Standard Deviation
Formula:
SD = √(Σ(x - x̄)² / n)
SD = √(Σ(x - x̄)² / n)
SD = √(882 / 8) = √110.25 = 10.5
Final Result
- Mean: 23
- Standard Deviation: 10.5
- Range: 40 - 10 = 30
Conclusion: The mean of the data is 23, and the standard deviation is 10.5, which shows the average spread of the values around the mean.
