Normal Distribution

The normal (Gaussian) distribution is the symmetric bell-shaped distribution characterized by a mean and a standard deviation. Its importance in practice is less about data being normal than about methods assuming it, and the forecasting literature states that default plainly: “most time series models produce normally distributed forecasts — that is, we assume that the distribution of possible future values follows a normal distribution.”

Where the assumption is doing work

Several familiar constructions inherit it, usually without saying so at the point of use.

• Interval multipliers. The 1.96 in a 95% interval, and 1.645 at 90%, come from the normal distribution. A confidence interval for a mean takes the form \(\bar{Y} \pm z_{1-\alpha/2}\sigma/\sqrt{N}\), and a margin of error is that multiplier times a standard error.
• Forecast intervals. Built as a point forecast plus or minus a multiplier times a forecast standard deviation — see prediction intervals.
• Test statistics. z tests directly, and t tests as the version that accounts for an estimated standard deviation — the handbook notes that intervals are “by necessity wider if the standard deviation is estimated from limited data.” See z-test and t-test.

Two reasons the default is defensible. Under suitable central-limit conditions, such as iid contributions with finite variance, standardized sums approach a normal distribution. Independence alone is insufficient; infinite-variance contributions and dominant individual terms can defeat the approximation. And it is analytically convenient: closed forms exist for almost everything, which is why the multipliers are constants you can memorize.

Where it fails, and in which direction

The center is usually fine and the tails are usually not. That matters because the tail is what risk questions ask about — how bad can a bad week be, how often will we breach capacity, what is the worst case we should plan for. A distribution with heavier tails than normal produces extreme outcomes more often than the model says, and the model’s confidence about it is unchanged.

Three common reasons business data is not normal, all recognizable by eye.

• A hard lower bound. Durations, counts, revenue and latency cannot go below zero, so a symmetric interval can produce an impossible bound. The handbook’s own advice covers the symptom — “the lower confidence bound should be reported as zero instead” — and a symmetric interval hitting a natural limit is a sign the approximation is being pushed too far.
• Skew. Most values small, a few very large. Revenue per customer and session duration typically look like this, and then the mean and median differ materially.
• Mixtures. Two populations in one dataset — free and paid, mobile and desktop — produce a shape no single bell curve describes, so examine meaningful observed groups separately when available, or use a mixture model when the population structure is latent or the combined distribution is the target.

What to do about it

Check before relying on the tails: plot the distribution, and for a model plot the residuals rather than the raw data, since it is the residual assumption the intervals rest on. Where normality is unreasonable, residual bootstrapping can simulate forecasts without imposing a normal residual distribution. Ordinary independent residual resampling requires an appropriate iid or exchangeable residual model; zero correlation and constant variance alone do not establish that condition. Dependent residuals may require a block or another dependence-aware bootstrap — see bootstrap.

And where a specific quantile is what the decision needs, estimate that quantile directly instead of deriving it from an assumed shape — see quantile regression and quantile forecasts. The practical summary: normality is a reasonable working assumption for central estimates and a poor one for tail claims, so the level of scrutiny should follow which of the two you are making.

References

• Hyndman & Athanasopoulos, Forecasting: Principles and Practice (3rd ed.), §5.5 “Distributional forecasts and prediction intervals”
• NIST/SEMATECH e-Handbook of Statistical Methods, §7.1.4 “What are confidence intervals?”