How To Calculate Prediction Interval By Hand

Prediction intervals are crucial in statistics to estimate the range within which future observations will fall. Calculating them by hand is an essential skill for data analysts and statisticians.

1. Enter the sample size, mean, and standard deviation of your data.
2. Select the desired confidence level.
3. Click ‘Calculate’ to see the prediction interval and a visual representation.

The formula for calculating a prediction interval is:

x̄ ± t_(α/2, n-1) * s * √(1 + 1/n)

Where:

• x̄ is the sample mean,
• t_(α/2, n-1) is the critical value of the t-distribution with (n-1) degrees of freedom and confidence level (1 – α),
• s is the standard deviation of the sample,
• n is the sample size.

Comparison of Prediction Intervals at Different Confidence Levels

Confidence Level	Prediction Interval
90%	…
95%	…
99%	…

• Always use the t-distribution for small sample sizes (n < 30).
• For large sample sizes, the normal distribution (z-distribution) can be used as an approximation.
• Ensure your data is normally distributed before calculating prediction intervals.

What is the difference between a confidence interval and a prediction interval?

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For more information, see:

• NIST’s Engineering Statistics Handbook
• Penn State’s Statistics 200 Course