> ## Documentation Index
> Fetch the complete documentation index at: https://pyfia.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Understanding variance

> How pyFIA computes standard errors using FIA's stratified, design-based variance — and how it's validated against EVALIDator.

Every pyFIA estimate comes with a measure of uncertainty. This page explains how that standard error is calculated, why FIA's formula looks different from a textbook one, and how closely pyFIA matches the official EVALIDator tool.

## EXPNS is not a survey weight

The key to FIA variance is the expansion factor **EXPNS**:

```
EXPNS = total acres in stratum / number of plots in stratum
```

Unlike a standard survey weight, EXPNS already contains the inverse of the sample size (`1/n_h`). That single fact is why FIA's variance formula differs from the stratified-sampling formulas in most textbooks.

## The domain-total variance formula

For tree-based estimates (volume, biomass, tree count, growth, mortality, removals), pyFIA uses the stratified **domain total** variance:

$$
V(\hat{Y}) = \sum_h w_h^2 \, s^2_{yh} \, n_h
$$

where for each stratum $h$:

* $w_h$ = `EXPNS` (acres per plot)
* $s^2_{yh}$ = sample variance of the plot-level values (with `ddof=1`)
* $n_h$ = number of plots

<Note>
  Multiplying by $n_h$ looks backwards compared to a textbook mean-variance
  formula that divides by it. It's correct here because `EXPNS` already carries
  $1/n_h$, and we're estimating a **total**, not a mean. The per-acre standard
  error is then derived as `SE_total / total_area`.
</Note>

## Two principles that keep it unbiased

1. **Include every plot, with zeros.** Plots without any trees in your domain still contribute — as zeros — to the stratum variance. Dropping them biases the result. pyFIA joins all evaluation plots and fills missing values with zero before computing variance.

2. **Single-plot strata contribute no variance.** Variance is undefined for $n_h = 1$, so those strata are set to zero variance rather than dropped.

## Reading the output

Every estimate includes standard-error columns (suffix `_SE`) — for example `VOLCFNET_TOTAL_SE` for volume or `AREA_SE` for area. In addition, `area()` and `volume()` emit variance columns (`AREA_VARIANCE`; `VOLUME_ACRE_VARIANCE` and `VOLUME_TOTAL_VARIANCE`), each equal to the corresponding standard error squared. The other estimators currently return standard errors only:

```python theme={null}
volume(db, totals=True)
# ...VOLCFNET_TOTAL, VOLCFNET_TOTAL_SE, VOLUME_TOTAL_VARIANCE
biomass(db, totals=True)
# ...BIO_TOTAL, BIO_TOTAL_SE   (standard errors only)
```

<Note>
  Avoid the `variance=True` parameter — it does **not** reliably switch the
  output from standard errors to variances. For most estimators it has no
  effect, and for `tpa()` it currently drops the standard-error columns
  entirely. Use the `_SE` columns (and the `*_VARIANCE` columns where present)
  directly. Tracked in [#109](https://github.com/mihiarc/pyfia/issues/109).
</Note>

There are no separate confidence-interval columns — construct one from the estimate and its standard error (for a 95% interval, ±1.96 × SE).

## Validation against EVALIDator

[EVALIDator](https://apps.fs.usda.gov/Evalidator/evalidator.jsp) is the official USFS estimation tool and pyFIA's reference for correctness. Validated against Georgia (EVALID 132301 / 132303):

| Estimator              | pyFIA SE    | EVALIDator SE | Difference |
| ---------------------- | ----------- | ------------- | ---------- |
| Forest area            | 138,928     | 136,048       | 2.1%       |
| Volume (growing stock) | 545,617,192 | 549,272,904   | 0.67%      |
| Biomass (aboveground)  | 14,204,093  | 14,256,973    | 0.37%      |
| TPA (live)             | 204,196,873 | 199,352,867   | 2.4%       |
| Growth                 | 35,406,501  | 35,445,004    | 0.11%      |
| Mortality              | 18,549,971  | 18,517,419    | 0.18%      |
| Removals               | 58,202,325  | 58,906,897    | 1.2%       |

Standard errors typically land within 1-3% of EVALIDator. Differences come from rounding, finite-population corrections, and edge cases like single-plot strata.

You can run this comparison yourself — see [`validate_pyfia_estimate()`](/api/pyfia-evalidator-validation).

## A note on temporal methods

FIA defines several temporal estimators. pyFIA implements **TI (Temporally Indifferent)** — using all available data in an evaluation — which is EVALIDator's default. Annual and moving-average methods (SMA, LMA, EMA) are not currently implemented.

## References

* **Bechtold, W\.A. & Patterson, P.L. (2005).** *The Enhanced FIA Program.* Gen. Tech. Rep. SRS-80, Chapter 4. [DOI: 10.2737/SRS-GTR-80](https://doi.org/10.2737/SRS-GTR-80)
* **Westfall, J.A.; Patterson, P.L.; Coulston, J.W. (2011).** Post-stratified estimation: within-strata and total sample size recommendations. *Can. J. For. Res.* 41(5): 1130-1139.
* **USDA Forest Service (2018).** [Population Estimation User Guide](https://research.fs.usda.gov/sites/default/files/2024-05/wo-nov2018_ug_population_estimation.pdf).