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.

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

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

The sample mean has no impact on the width of a confidence interval!

The z-score for a two-sided 99% confidence interval is 2.807, which is the 99.5-th quantile of the standard normal distribution N(0,1).