Confidence Intervals

A practical guide to understanding and calculating the range where a true population mean likely lies.

What is a Confidence Interval?

It's often impossible to survey an entire population (like every stock in the market). Instead, we take a smaller sample (like the S&P 500) and calculate its mean (average) return.

A confidence interval uses this sample mean to construct a range of values and says, "We are X% confident that the true average of the entire population falls within this range." It's a way of putting boundaries on uncertainty.

A Common Misconception

A 95% confidence interval does **not** mean there is a 95% probability that the true population mean falls within that specific interval. The true mean is fixed. It either is or isn't in our calculated interval. Instead, it means that if we were to repeat our sampling process many times, 95% of the confidence intervals we construct would contain the true population mean.

The Formula

CI = \bar{x} \pm Z \cdot \frac{\sigma}{\sqrt{n}}

\text{Standard Error} = \frac{\sigma}{\sqrt{n}}

Interactive Calculator

Adjust the parameters to see how they affect the confidence interval in real-time.

Sample Mean (x̄)

Population Standard Deviation (σ)

Sample Size (n)

Confidence Level: 95%

Calculated 95% Confidence Interval

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