Statistics for finance: mean, standard deviation and regression on the BA II Plus
One-variable and two-variable statistics done right: sample versus population dispersion, correlation, R², and forecasting in both directions with a verified worked example.
Risk is dispersion and relationships are regressions — the two statistical objects finance cannot live without. The BA II Plus statistics worksheet computes both families in seconds, and FinScope reproduces the same decimal-safe results in the browser. The discipline is knowing which statistic the question wants.
One-variable statistics: mean and the two sigmas
Enter data points in the 1-var list and the worksheet returns Σx, Σx², mean, and both standard deviations. The rule: sample standard deviation (s) divides by n − 1 and is used whenever your data is a sample from a larger population; population standard deviation (σ) divides by n and is used only when you literally have the whole population. For [3, 5, 7] the verified outputs are mean 5, s 2 and σ 1.6330 — the sample figure is always the larger one.
| Dataset | Mean | Sample s (n−1) | Population σ (n) |
|---|---|---|---|
| 3, 5, 7 | 5 | 2.0000 | 1.6330 |
| 10, 12, 14, 14, 18 | 13.6 | 2.9665 | 2.6533 |
The second row also shows the raw sums the worksheet tracks — Σx = 68 and Σx² = 960 — which exam questions occasionally ask for directly.
Two-variable statistics and regression
Switch to the 2-var list, put the independent variable in X and the dependent in Y, and select LIN mode. For the verified points (1,2), (2,5), (3,10) the worksheet reports slope b = 4 and intercept a = −2.3333, so the fitted line is Y = 4X − 2.3333. The correlation coefficient is r = 0.9897 and R² ≈ 0.9796 — the line explains about 98% of the variation in Y.
- 2ND DATA → choose 2-var, LIN mode.
- X01 = 1, Y01 = 2; X02 = 2, Y02 = 5; X03 = 3, Y03 = 10.
- 2ND STAT → read a = −2.3333, b = 4, r = 0.9897.
- Forecast: press X′ = 4 → the worksheet returns Y′ = 13.6667 (engine-verified).
- Inverse forecast: enter Y′ = 13.6667 → X′ returns exactly 4.
Where each statistic shows up in finance
- Mean and σ: expected return and volatility — the two coordinates of every risky asset.
- Sample s: the default for historical return series, because history is always a sample.
- r and R²: factor exposure checks, beta estimation, and judging whether a model line is worth trusting.
- Two-sided forecasts: the quick sensitivity question — 'what does the model say at X = 4, or X = 4.5?'
Frequently asked questions
When do I use sample standard deviation instead of population?
Whenever your data is a sample of a larger universe — which is the normal case in finance, since observed returns are a sample of all possible outcomes. Sample s divides by n − 1 to correct for that uncertainty: [3, 5, 7] gives s = 2 but σ = 1.6330. Use σ only when the dataset truly is the entire population.
What does the correlation coefficient r really tell me?
The strength and direction of the linear relationship between X and Y, from −1 to +1. In the verified example r = 0.9897, meaning the three points lie almost exactly on a line. Square it for R² ≈ 0.9796 — the proportion of Y's variance the line explains. r close to zero rules out a straight-line link, not all dependence.
How do the X′ and Y′ forecast keys work?
After fitting a regression, X′ solves the fitted line backwards: give it a Y and it returns the X that would produce it; Y′ works forward: give it an X and it returns the predicted Y. In the verified example Y′ at X = 4 is 13.6667 and feeding 13.6667 back as Y′ returns X′ = 4 exactly. Forecasts far outside the observed X range are extrapolation, not prediction.
Practice it live
Open the matching worksheet and reproduce every number in this article.
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