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M09.02.04 Measures of variability
Variability, or dispersion, in a dataset, indicates how spread out the values are. The range and standard deviation are commonly used measures of variability, each with distinct characteristics and applications in statistical analysis.
Types of Measures of Variability
1. Range
- Definition: The range is the simplest measure of variability, calculated as the difference between the highest and lowest values in the dataset.
- Limitations: It is highly sensitive to extreme values, making it less stable than other measures like standard deviation.
2. Standard Deviation (SD)
- Definition: Standard deviation is a more reliable measure of variability that describes the average distance of each data point from the mean.
- Formula: The standard deviation, sss, is calculated as:
- X represents each value,
,
- ‘n’ is the total number of values.
3. Variance (s²)
- Definition: Variance is the square of the standard deviation, representing the average squared deviations from the mean.
- Formula:
Calculation of Standard Deviation: Step-by-Step Process
- Calculate Deviations: Subtract the mean from each score to find deviations from the mean.
- Square Deviations: Square each deviation to make them all positive.
- The sum of Squares: Add all squared deviations.
- Average of Squares: Divide by the number of values minus one (for sample SD).
- Square Root: Take the square root of the result to obtain the standard deviation.
Example Calculation
| Score (X) | Mean (X‾\overline{X}X) | Deviation (X−X‾X – \overline{X}X−X) | Squared Deviation ((X−X‾)2(X – \overline{X})^2(X−X)2) |
|---|---|---|---|
| 5 | 4 | 1 | 1 |
| 3 | 4 | -1 | 1 |
| 6 | 4 | 2 | 4 |
| 2 | 4 | -2 | 4 |
| 4 | 4 | 0 | 0 |
- Sum of Squared Deviations = 1 + 1 + 4 + 4 + 0 = 10
- Variance (s²) =
- Standard Deviation (s)
Standard Deviation and Normal Distribution
In a normal distribution:
- 68% of values fall within ±1 SD of the mean.
- 95.5% of values fall within ±2 SDs of the mean.
- 99.7% of values fall within ±3 SDs of the mean.
Visual Representation
| Standard Deviations | Proportion of Data Covered |
|---|---|
| ±1 SD | 68% |
| ±2 SDs | 95.5% |
| ±3 SDs | 99.7% |
Points to Remember
- The Range is quick to calculate but highly sensitive to outliers.
- Standard Deviation is more stable and reflects the typical deviation of scores from the mean.
- In normal distributions, SDs allow predictions about the proportion of cases within certain ranges.
Normal Distribution Bell Curve with Standard Deviations
Bibliography and References
- CDC. (2023). “Statistics in Epidemiology and Biostatistics”. CDC.
- Porta, M. (2014). “A Dictionary of Epidemiology.”