Standard Error of the Mean Calculator

How precisely does a history of returns pin down the true average? Change any number in the problem and the bell curve redraws.

1. Standard error problem

Enter the average and standard deviation of a return history. Switching the data frequency rescales the figures you entered.

Data frequency
Statistic to estimate
98 years (1,176 observations)

A portfolio has a mean monthly return of 0.75%.

Its returns have a monthly standard deviation of 4%, measured over 98 years of monthly data (1,176 observations).

Question: How precisely does this sample estimate the true mean return, at 95% confidence? Show your work.

2. Sampling distribution

0.40%−3 SE0.52%−2 SE0.63%−1 SE0.75%mean0.87%+1 SE0.98%+2 SE1.10%+3 SE0.52%0.98%0.63%0.87%Sample mean 0.75%
Mean ± 1 SE (about 68%)Mean ± 1.96 SE (95%)

The bell curve shows where the mean monthly return could plausibly lie given 1,176 observations. Its width is set by the standard error (4.00% ÷ √1,176); the darker band is ±1 standard error and the full shaded band is the 95% confidence interval. Add years of data and the curve narrows, but only with the square root of n.

3. Show your work

  1. Count the observations.n = 98 years × 12 per year = 1,176
  2. Take the square root of n.√1,176 = 34.29
  3. Divide the standard deviation by √n.Standard error of the mean = 4.00% ÷ 34.29 = 0.117%
  4. Add and subtract one standard error.0.75% ± 0.117% = 0.63% to 0.87%About 68% confidence under a normal distribution.
  5. Multiply by the 95% critical value.Margin = 1.960 × 0.117% = 0.229%
  6. Build the confidence interval around the mean.0.75% ± 0.229% = 0.52% to 0.98%
  7. Compare the signal to the noise.t = 0.75% ÷ 0.117% = 6.43
  8. For contrast, add and subtract one standard deviation for a single month.0.75% ± 4.00% = −3.25% to 4.75%About 68% of single-month outcomes under a normal distribution.
  9. Widen that to the 95% range for one month.0.75% ± 1.960 × 4.00% × √(1 + 1/1,176) = −7.09% to 8.59%
Answer: With 1,176 observations, a 95% confidence procedure places the underlying mean monthly return between 0.52% and 0.98%. The interval excludes zero, so at 95% confidence the sample distinguishes the mean from zero. One standard error either side gives 0.63% to 0.87%. A single month's return, by contrast, falls within one standard deviation, −3.25% to 4.75%, only about two-thirds of the time, and its 95% range runs from −7.09% to 8.59%. Quadrupling the data to 4,704 observations would only halve the standard error.

The default inputs are the hypothetical illustration from IFA's article Inferential Statistics: What 98 Years of Monthly Returns Can—and Cannot—Tell Investors. They are not the performance of any IFA Index, IFA Index Portfolio or client account. All figures on this page are calculated from the values you enter; the source of any statistic shown is the user's input.

This calculator is for education only. When you change the data frequency, the average is scaled by the number of periods and the standard deviation by its square root; this simple scaling assumes independent returns and ignores compounding. It assumes independent observations drawn from a stable, normal distribution, uses the large-sample approximation √(π/2) × standard deviation ÷ √n for the standard error of the median, and uses normal (z) critical values of 1.645, 1.960 and 2.576. Actual market returns can cluster in volatility, change over time and produce more extreme outcomes than a normal curve suggests, so real-world uncertainty can be greater than shown. Investing involves risk, including the potential loss of principal. Past performance does not guarantee future results, and nothing here is a prediction of future returns.

Index Fund Advisors, Inc. is a Registered Investment Adviser with the U.S. Securities and Exchange Commission; however, such registration does not imply a certain level of skill or training and no inference to the contrary should be made.