Standard Deviation – Computation & Importance 📘📈
Standard Deviation (S.D.) is the most scientific and widely used measure of dispersion. It tells us how widely data values are spread around the mean.
It is based on all observations and uses square deviations, making it mathematically powerful.
1. Definition
Standard Deviation is the square root of the average of squared deviations from the mean.
It is derived from Variance, which is:
Variance (σ²) = Σ (X – Mean)² / N
And Standard Deviation:
σ = √Variance
Key IdeaStandard Deviation measures consistency: lower S.D. = more consistency, higher S.D. = more variability.
2. Formulas
For Ungrouped Data
Variance
σ² = Σ (X – X̄)² / N
Standard Deviation
σ = √[ Σ (X – X̄)² / N ]
For Grouped Data
If midpoints (X) and frequencies (f) are given:
Variance
σ² = Σ f (X – X̄)² / Σf
Standard Deviation
σ = √[ Σ f (X – X̄)² / Σf ]
Shortcut FormulaFor grouped data,
Variance can also be computed using:
σ² = [ΣfX² / Σf] – (Mean)²
This significantly reduces calculation time in exams.
3. Solved Examples
Example 1 — Ungrouped Data
Data: 5, 7, 9, 12, 17
Step 1: Mean
Mean = (5 + 7 + 9 + 12 + 17) / 5 = 50 / 5 = 10
Step 2: Compute (X – Mean)²
| X | X – 10 | (X – 10)² |
|---|---|---|
| 5 | –5 | 25 |
| 7 | –3 | 9 |
| 9 | –1 | 1 |
| 12 | 2 | 4 |
| 17 | 7 | 49 |
Σ (X – Mean)² = 25 + 9 + 1 + 4 + 49 = 88
Step 3: Variance
σ² = 88 / 5 = 17.6
Step 4: Standard Deviation
σ = √17.6 ≈ 4.20
Example 2 — Grouped Data
Class intervals & frequency:
| Class | Midpoint X | f |
|---|---|---|
| 0–10 | 5 | 3 |
| 10–20 | 15 | 5 |
| 20–30 | 25 | 7 |
Step 1: Compute Mean
ΣfX = (3×5) + (5×15) + (7×25)
= 15 + 75 + 175 = 265
Σf = 3 + 5 + 7 = 15
Mean = 265 / 15 ≈ 17.67
Step 2: Compute f(X – Mean)²
| X | X–17.67 | (X–17.67)² | f | f(X–Mean)² |
|---|---|---|---|---|
| 5 | –12.67 | 160.56 | 3 | 481.68 |
| 15 | –2.67 | 7.12 | 5 | 35.60 |
| 25 | 7.33 | 53.72 | 7 | 376.04 |
Σ f(X–Mean)² = 481.68 + 35.60 + 376.04 = 893.32
Step 3: Variance
σ² = 893.32 / 15 ≈ 59.55
Step 4: Standard Deviation
σ = √59.55 ≈ 7.72
4. Merits of Standard Deviation ✔️
- Based on all observations
- Uses square deviations → mathematically sound
- Most reliable measure of dispersion
- Suitable for advanced statistics
- Used in correlation, regression, probability, finance, economics
- Strong theoretical foundation
5. Limitations ❌
- More difficult to compute
- Affected by extreme values
- Harder to explain to non-technical users
Exam ReminderIf the question says "Calculate dispersion using the best measure",
always choose Standard Deviation unless specified otherwise.
6. Applications in Business & Economics
- Measuring risk in investments
- Analysing variability in production, wages, demand
- Quality control checks
- Comparing stability of sales or revenue
- Forecasting and prediction models
7. Summary ✨
- S.D. tells how much data varies from the mean
- Variance is the square of Standard Deviation
- Most scientific and widely used measure
- Essential for advanced analysis
Quiz Time 🎯
Test Your Knowledge
Question 1 of 5
1. Standard Deviation is the square root of: