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Standard Deviation Formula & How to Calculate It
Standard deviation measures how spread out values are from the mean. A low SD means data is tightly clustered (consistent). A high SD means data is widely spread (variable).
Population: σ = √( Σ(x−μ)² / N )
Sample: s = √( Σ(x−x̄)² / (n−1) )
Variance: σ² = Σ(x−μ)² / N | s² = Σ(x−x̄)² / (n−1)
Sample: s = √( Σ(x−x̄)² / (n−1) )
Variance: σ² = Σ(x−μ)² / N | s² = Σ(x−x̄)² / (n−1)
Population vs Sample Standard Deviation
- Population (σ, divides by N): Use when you have data for the ENTIRE group. Example: all students in one class, all products from one batch.
- Sample (s, divides by n−1): Use when your data is a SAMPLE from a larger population. The n−1 (Bessel's correction) corrects for bias. Example: surveying 100 people from a city of millions.
The 68-95-99.7 Rule (Empirical Rule)
For normally distributed data, the standard deviation tells you exactly what percentage of values fall within each range:
- μ ± 1σ contains approximately 68.27% of all values
- μ ± 2σ contains approximately 95.45% of all values
- μ ± 3σ contains approximately 99.73% of all values
Coefficient of Variation (CV)
CV = (Standard Deviation / Mean) × 100%
CV < 15% = Low variability | CV 15–30% = Moderate | CV > 30% = High
CV < 15% = Low variability | CV 15–30% = Moderate | CV > 30% = High
Real-World Uses of Standard Deviation
- Finance: Measures investment risk — higher SD = more volatile
- Quality control: Detects manufacturing consistency — Six Sigma uses 6σ
- Education: Grades distribution — tight SD means class performed similarly
- Medicine: Clinical trial variability — consistency of drug effects
- Weather: Temperature variation — high SD means unpredictable weather
Standard Deviation Examples
Data {2,4,4,4,5,5,7,9}: Mean=5, Pop SD=2, Samp SD≈2.14
Data {5,5,5,5,5}: Mean=5, SD=0 (no spread)
Data {1,100}: Mean=50.5, SD=49.5 (extreme spread)
Data {5,5,5,5,5}: Mean=5, SD=0 (no spread)
Data {1,100}: Mean=50.5, SD=49.5 (extreme spread)
Interpreting Standard Deviation
- SD = 0: All values are identical — no variability
- Low SD: Values clustered near the mean — very consistent
- High SD: Values spread far from the mean — high variability
- Compare datasets: Always compare SD relative to the mean using CV%
Z-Score Formula
Z = (x − μ) / σ
Z = 0 → value equals mean
Z = +1 → value is 1 SD above mean
Z = −2 → value is 2 SDs below mean
Z = 0 → value equals mean
Z = +1 → value is 1 SD above mean
Z = −2 → value is 2 SDs below mean
Standard Deviation in Different Fields
- Finance (Volatility): Higher SD = higher investment risk. Stock A SD=2% is less volatile than Stock B SD=8%
- Six Sigma Quality: Process must be within 6 standard deviations of the target — less than 3.4 defects per million
- IQ Scores: Mean=100, SD=15. Z-score=2 means IQ=130 (top 2.5%)
- Weather: Low SD temperature = predictable climate; High SD = unpredictable
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