To calculate a confidence interval (two-sided), you need to follow these steps:

1. Let's say the sample size is 100.
2. Find the mean value of your sample. Assume it's 3.
3. Determine the standard deviation of the sample. Let's say it's 0.5.
4. Choose the confidence level. The most common confidence level is 95%.
5. In the statistical table find the Z(0.95)-score, i.e., the 97.5th quantile of N(0,1) – in our case, it's 1.959.
6. Compute the standard error as σ/√n = 0.5/√100 = 0.05.
7. Multiply this value by the z-score to obtain the margin of error: 0.05 × 1.959 = 0.098.
8. Add and subtract the margin of error from the mean value to obtain the confidence interval. In our case, the confidence interval is between 2.902 and 3.098.

If you repeatedly draw samples and use each of them to find a bunch of 95% confidence intervals for the population mean, then the true population mean will be contained in about 95% of these confidence intervals. The remaining 5% of intervals will not contain the true population mean.

The width of a confidence interval increases when the margin of error increases, which happens when the:

• Significance level increases;
• Sample size decreases; or
• Sample variance increases.