What is Standard Deviation?

Standard Deviation (σ) is a statistical measure that quantifies the dispersion of data points around the mean in a dataset. There are two methods to calculate it:

Population vs Sample Standard Deviation

The key difference lies in the denominator:

• Population Standard Deviation: Use when you have all possible values in your dataset
• Sample Standard Deviation: Use when your dataset is a subset of all possible values

The choice between population and sample calculation depends on your dataset's completeness. Population calculation produces smaller σ values, while sample calculation accounts for potential missing values.

Understanding the Basics

Standard deviation:

• Represents dispersion around the mean
• Acts as a proxy for volatility
• Uses the same units as the original dataset
• Can be calculated as the square root of variance

Example

If you measure people's heights with a mean of 155cm and σ = 11.5:

• One standard deviation (1σ) = ±11.5cm from the mean
• This means most heights fall within 143.5cm to 166.5cm

Z-Score: Standardizing Comparisons

When comparing datasets with different units or scales, standard deviation alone isn't enough. This is where Z-scores become valuable.

Z-score:

• Normalizes σ values into standard units
• Enables comparison between different datasets
• Measures how many standard deviations an observation is from the mean

Interpreting Z-Scores

• Within ±1σ: Contains 68% of observations (common values)
• Within ±2σ: Contains 95% of observations
• Within ±3σ: Contains 99.7% of observations (rare values)

Applications in Finance

Investment Strategies

1. Mean-Reversion Strategy:

   • High Z-Score → Short position (expect price decrease)
   • Low Z-Score → Long position (expect price increase)

2. Momentum Strategy:

   • High Z-Score → Long position (expect continued rise)
   • Low Z-Score → Short position (expect continued fall)

Fund Comparison Example

Consider two funds with 10% average return:

• Fund A: σ = 7% (Returns: 10%, 2%, 19%)
• Fund B: σ = 27% (Returns: 50%, -15%, -5%)

Fund A offers more stable returns despite the same average return.

Important Considerations

Benefits

• Helps quantify risk
• Enables comparison across different metrics
• Simple to understand and interpret
• Widely applicable across fields

Limitations

1. Sensitive to outliers
2. Assumes normal distribution
3. Accuracy depends on data quality

Further Reading

• Non-Linear Returns
• Volatility and Standard Deviation
• Standard Deviation in Stock Returns