Mean Confidence Interval

Question, try, then observe

A 2% mean: how precise is the estimate?

Keep the sample mean monthly return at 2%, with population SD unknown. Adjust n, sample SD, and confidence. Compare the current Student's t interval with a fixed baseline.

Illustrative data, not live market data. A monthly return of 2 = 2%; dispersion and return differences use percentage points, with no annualization.

01 · Make a predictionWith other settings held constant, how does a larger sample change the interval?

Sample size n 24

Sample standard deviation s 4 pp

Solid line: current interval. Dashed line: fixed baseline n=24, s=4 pp, 95%. Both share one scale, which adjusts to fit the wider interval.

Standard error of the mean (pp)

SE = \frac{4}{\sqrt{24}} \approx 0.8165

The t critical value uses current confidence and n−1 degrees of freedom

ME = 2.0687 \times 0.8165 \approx 1.6891

Mean ± margin of error; endpoints are monthly return %

2 \pm 1.6891 \Rightarrow [0.3109, 3.6891]

Lower / upper bound0.31% – 3.69%

Standard error SE0.82 pp

Margin of error1.69 pp

t critical value / df2.0687 / 23

The current interval is 3.38 pp wide. More observations narrow it; more dispersion or higher confidence widen it. This estimates the mean, not the return range for individual months.

95% means that, under the assumptions, repeated sampling with this method produces intervals that cover the fixed population mean about 95% of the time. It is not a 95% probability assigned to this realized interval. Observations must be independent; small samples need approximate normality. Autocorrelation, heavy tails, or dependent financial observations may invalidate these assumptions.

First calculate SE=s/√n. Multiply by the t critical value for n−1 degrees of freedom to obtain the margin of error, then add and subtract it from the mean.

What did you just see?

Think of these data as one outcome of repeated sampling: more independent observations generally reduce uncertainty in the , while greater dispersion needs a wider interval.

Assumptions of the t method

Observations should form an independent, representative sample. Approximate population normality is especially important with small samples. Degrees of freedom are n−1 and at least two observations are required. A larger sample does not automatically fix serial dependence or sampling bias.

SD and SE are different

Sample SD describes dispersion of individual observations. SE, s/√n, describes sampling variation of the under the assumptions. The interval estimates a population mean, not the range of individual monthly returns.

Another example

Enter a 2% mean, a four-percentage-point sample SD and n=12, or supply sample observations.
Choose 95% confidence; with unknown population SD use t with 11 degrees of freedom.
Read the SE, margin and bounds, and check whether independence and distribution assumptions are reasonable.

Show formal definition and formula

A mean confidence interval estimates a population mean using a specified sampling procedure. The t method substitutes sample SD for unknown population SD.

Two-sided t interval for the mean

CI = x̄ \pm t_{1 - α / 2, n - 1} \frac{s}{\sqrt{n}}

Add and subtract the t critical value at n−1 degrees of freedom times the sample standard error; population SD is unknown.

Standard error and margin of error

SE = \frac{s}{\sqrt{n}}; ME = t* SE

Standard error is s divided by √n. Margin of error is the t critical value times standard error.

n / x̄ / s: n is sample size, x̄ the sample mean, and s the sample SD using n−1 in the variance denominator.
1−α / df: Confidence is 1−α, and df=n−1. This tool uses two-sided Student's t.
t* / SE / ME: t* is the quantile for the degrees of freedom and confidence level; SE=s/√n and ME=t*×SE.
Units: Mean return and bounds use %. SD, standard error, and margins use percentage points; variance uses squared percentage points. No annualization.

Common mistakes

A mean interval is not a prediction range for the next return. Do not describe 95% as the probability that the fixed true mean falls within this already computed interval.

Frequently asked questions

What does a 95% confidence interval mean?

Across repeated independent samples and intervals built by the same method, about 95% of intervals cover the fixed population mean when the assumptions hold. It is not a 95% probability statement about this one computed interval.

Why not always use 1.96?

When population SD is unknown and estimated with sample SD, the t critical value depends on degrees of freedom and confidence. Small-sample t critical values generally exceed their normal counterparts.

Can I directly use daily stock returns?

You can compute it, but must check assumptions. Serial dependence, volatility clustering, heavy tails or unrepresentative samples can invalidate the nominal coverage of a simple mean t interval.

You can now explore…

How n, spread and confidence change an interval
Interpret 95% through repeated sampling

Term reference3 terms

Open a term when you need a reminder. Underlined words in the article work too.