Standard Deviation

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Standard Deviation

What it is

Standard deviation (SD) is a measure of how spread out the values in a dataset are around the mean. It quantifies the typical distance of data points from the average and is widely used to describe variability and risk.

Key idea

  • A small SD: data are clustered near the mean.
  • A large SD: data are widely dispersed from the mean.
  • In finance, a higher SD indicates greater volatility (risk).

Formula and calculation

Two common forms:
– Population standard deviation: SD = sqrt( (1/N) * Σ(xi − μ)² )
– Sample standard deviation: SD = sqrt( (1/(n−1)) * Σ(xi − x̄)² )

Where:
– xi = each data point
– μ = population mean
– x̄ = sample mean
– N = population size
– n = sample size

Step-by-step (sample SD):
1. Compute the mean x̄.
2. Subtract the mean from each data point (deviations).
3. Square each deviation.
4. Sum the squared deviations.
5. Divide by n − 1 (for a sample) to get the variance.
6. Take the square root to get the standard deviation.

Quick tools: Excel functions STDEV.S (sample) and STDEV.P (population).

Example

Data: 5, 7, 3, 7
1. Mean = (5 + 7 + 3 + 7) / 4 = 5.5
2. Squared deviations: 0.25, 2.25, 6.25, 2.25 (sum = 11)
3. Sample variance = 11 / (4 − 1) = 3.6667
4. Sample SD = sqrt(3.6667) ≈ 1.915

Apple stock example (summary):
– Five annual returns: 88.97%, 82.31%, 34.65%, −26.41%, 28.32%
– Mean ≈ 41.57%; computed sample SD ≈ 46.90% (high volatility across those years).

Relationship to variance

  • Variance = average of squared deviations from the mean.
  • Standard deviation = square root of variance.
  • SD has the same units as the original data; variance does not (it is in squared units), which often makes SD easier to interpret.

For normally distributed data:
– ≈68% of observations lie within ±1 SD of the mean
– ≈95% within ±2 SD
– ≈99.7% within ±3 SD

Key properties (corrected)

  • Non-negative: SD ≥ 0.
  • Same units as the data.
  • Scale behavior: multiplying all data by a constant c multiplies the SD by |c|.
  • Not additive in general: variances add for independent variables (Var(X+Y) = Var(X)+Var(Y)), but SDs do not simply add.
  • Sensitive to outliers: because deviations are squared, extreme values have a large effect.

Uses in business and finance

  • Risk management: quantify variability of returns, sales, or other outcomes.
  • Financial analysis: compare volatility across assets and funds.
  • Forecasting: assess uncertainty in sales and demand; help plan inventory and cash flow.
  • Quality control: measure process variability (Six Sigma, control charts).
  • Project management: evaluate schedule or cost variability (e.g., earned value analysis).

Strengths and limitations

Strengths
– Uses all data points, not just extremes.
– Intuitive and widely understood.
– Useful in many algebraic and probabilistic calculations.
– Expressed in original units, aiding interpretation.

Limitations
– Sensitive to outliers and skewed data.
– Can be harder to compute manually than simple measures like range.
– Treats all deviation as “risk” even when outcomes above the mean are favorable.
– SD of small samples can be an imprecise estimate of population variability.

Practical tips

  • For skewed data or when robustness is needed, consider median absolute deviation (MAD) or trimmed statistics.
  • Use software (Excel, statistical packages) to compute SD accurately.
  • Choose sample vs. population SD appropriately: use n−1 (sample) when estimating population variability from a sample.

What a high or low SD means for investors

  • High SD: greater historical volatility — potential for larger gains and losses.
  • Low SD: more stable returns — lower volatility, typically lower expected surprise.
  • Neither is inherently “better”; suitability depends on an investor’s risk tolerance and objectives.

Bottom line

Standard deviation is a fundamental measure of dispersion that helps summarize how spread out data are around their mean. It is essential for assessing variability and risk in finance, operations, quality control, forecasting, and many other business applications. Use it with awareness of its sensitivity to outliers and the difference between sample and population calculations.