Confidence intervals for a mean return

From a point estimate to an interval

The sample mean is a point estimate : one best guess at the population mean. An interval estimate wraps a range around it. A confidence interval is an interval estimate built to contain the true mean with a stated probability, the confidence level , 1 − α.

Why it works: by the central limit theorem, x̄ is approximately normal around μ with a standard deviation of σ ÷ √n, so 95% of sample means land within 1.96 standard errors of μ. Turned around, x̄ ± 1.96 standard errors covers μ in 95% of samples.

A 95% interval is a statement about the method: build intervals this way from many samples and about 95 in 100 contain μ. Each tail outside holds α ÷ 2, which is 2.5% at 95%.

In practice σ is unknown, so the sample standard deviation s replaces it. The calculation below is the large-sample version.

The formula and the margin of error

Memorize

The interval is centered on the sample mean. How far it reaches on each side is set by a critical Z-value and the standard error.

Confidence interval for a meanx̄ ± Z × s ÷ √n

• Point estimate Use it the sample mean x̄, the center of the interval
• Margin of error Memorize Z × s ÷ √n, the half-width
  • Critical Z-value Memorize 1.96 at 95%, 2.576 at 99%; α ÷ 2 in each tail
  • Standard error Memorize s ÷ √n, how precise x̄ is

$$\bar{x} \pm Z_{\alpha/2} \times \frac{s}{\sqrt{n}}$$

The lower bound is $\bar{x} - Z_{\alpha/2}\, s/\sqrt{n}$ and the upper bound is $\bar{x} + Z_{\alpha/2}\, s/\sqrt{n}$. The distance from the mean to either bound is the margin of error , the half-width of the interval.

$$E = Z_{\alpha/2} \times \frac{s}{\sqrt{n}}$$

It is the sampling error you allow for on each side of the estimate. The critical values:

Confidence level	Z	Probability in each tail
90%	1.645	5%
95%	1.96	2.5%
99%	2.576	0.5%

The 95% and 99% values are the ones to know.

Worked example: three steps

Memorize

Questions also turn the formula around: they give the interval and ask for the margin, the mean or the sample size.

Move the confidence level and the sample size, then switch to the second tab and draw twenty intervals from a population with a known mean. About one in twenty misses at 95%.

What makes an interval wider or narrower?

Change	Effect on the interval	Why
Higher confidence level	wider	a bigger Z
Larger n	narrower	√n is in the denominator; four times the data halves the width
More volatile returns (larger s)	wider	the standard error rises with s

The same fund with 576 months (48 years) and the same s: standard error 4.20 ÷ 24 = 0.175%, and the 95% margin is 0.343%, half the 144-month margin of 0.686%.

Exam tip Multiply the data by four and the width halves. If an option shows a quarter, it is the trap.

Go deeper How many observations for a target margin?

Turn the margin formula around: n = (Z × s ÷ E)². For s = 4.20%, a 95% interval and a target margin of 0.35%, n = (1.96 × 4.20 ÷ 0.35)² = 23.52² = 553.2, so you would need 554 monthly returns, about 46 years. Halving the target margin always means four times the data.

When does an interval mislead?

Use it

The interval leans on the central limit theorem, which assumes a finite variance and data drawn from one stable distribution. Price levels that trend for years break that: their histogram can have two humps, the long-run average price says little about today, and an interval around it is not meaningful. These are non-stationary data . Work with percentage changes (returns), which are far more stable, before you estimate.

A commodity that traded near 300 for a decade and near 1,500 for the next has an “average price” of about 900 that it hardly ever traded at. An interval around that average describes a price that never occurred.