Standard Deviation is the most widely used and reliable measure of dispersion in statistics. It quantifies the average distance of each data point from the mean, giving a clear picture of data variability.
Definition
Standard Deviation measures the spread of a data set by calculating the square root of the average of squared deviations from the mean.
Formulas
For Ungrouped Data
Standard Deviation (σ):
σ = √[Σ(X − 𝑋̄)² / N]
Where:
X̄ = Mean
N = Number of observations
For Grouped Data
σ = √[Σf(X − 𝑋̄)² / Σf]
or, using the shortcut method:
σ = √[(ΣfX² / Σf) − (𝑋̄)²]
Coefficient of Standard Deviation:
Coefficient = σ / 𝑋̄
Used to compare variability of two or more datasets.
Example: Ungrouped Data
Data: 5, 10, 15, 20, 25
- Mean (𝑋̄) = (5 + 10 + 15 + 20 + 25) / 5 = 15
- Deviations: -10, -5, 0, 5, 10
- Squares: 100, 25, 0, 25, 100
- Σ(X − 𝑋̄)² = 250
- Standard Deviation = √(250 / 5) = √50 ≈ 7.07
Example: Grouped Data
| Class Interval | Frequency (f) | Midpoint (X) | fX | X² | fX² |
|---|---|---|---|---|---|
| 0–10 | 4 | 5 | 20 | 25 | 100 |
| 10–20 | 6 | 15 | 90 | 225 | 1350 |
| 20–30 | 10 | 25 | 250 | 625 | 6250 |
Σf = 20, ΣfX = 360, ΣfX² = 7700
Mean = 360 / 20 = 18
Standard Deviation = √[(7700 / 20) − 18²] = √(385 − 324) = √61 ≈ 7.81
Merits of Standard Deviation
- Considers all data values and their deviations
- Mathematically suitable for further statistical analysis
- Reliable and accurate for measuring consistency
- Applicable to both theoretical and applied research
Demerits of Standard Deviation
- Complex calculation for large data sets
- Heavily affected by extreme values
- Not easily interpretable without proper context
Comparison Of Key Measures of Dispersion
| Measure | Formula Base | Considers All Data? | Suitable for Further Analysis |
|---|---|---|---|
| Range | Max − Min | No | No |
| Quartile Deviation | (Q3 − Q1)/2 | Middle 50% | Limited |
| Mean Deviation | Σ|X − A| / N | Yes | Limited |
| Standard Deviation | √(Σ(X−X̄)² / N) | Yes | Yes |
Applications of Standard Deviation
- Risk assessment in finance and investment
- Quality control in manufacturing
- Statistical testing in research studies
- Analyzing consistency in performance or scores
Conclusion
Standard Deviation is the most comprehensive and mathematically sound measure of dispersion. It incorporates every value in the dataset and provides a clear understanding of how much values vary around the mean. For UGC NET Commerce aspirants, mastering this concept is essential for solving questions on statistical consistency, probability, and inferential statistics.