The statistical power calculation is the calculation of the probability that we reject a false null hypothesis. So, this calculation returns a numbered probability. Statistical power is also known via the term ‘Power of Test’.

If we test a H0 for µ ≥ 60 in a world where the true mean is 53 and our statistical power calculation = 0.70, we can say that there is approximately a 70% probability that we will reject H0, in case the true mean is 53.

The procedure

The statistical power calculation, or calculation of Power of Test, can be done by following these three steps:

• Calculate the critical value (z, when normally distributed)
• Calculate the critical value of the sample mean (x̄)
• Calculate the Power through the Type II Error

Statistical power calculation example

Let’s run through an example where we have a one sample and one-tailed z-test around a mean. The same procedure goes for other tests.

Say we are going to test the following hypotheses:

H0: µ ≥ 60

Ha: µ < 60

Significance level (α) = 0.10

We are about to run a simple random sample of 38 and assume that our population is normally distributed with a known standard deviation (σ) = 24 and with an unknown mean (µ).

Step 1: Critical value for z

The critical z for an alpha level of 0.10 is -1.28. This can be looked up in the normal distribution table and is also embedded in statistical software.

Step 2: Critical value for the sample mean (x̄):

We apply the z-statistic formula plugging in our existing values in order to calculate the critical value:

So, our critical value for x̄ is 55.01 meaning that we will reject for any sample mean equal to or less than 55.01.

Step 3: The statistical power calculation

What is the statistical power in case the true H0 is 53? We can calculate this via the z formula finding first the Type II Error:

The probability of z > 0.52 is looked up in the normal distribution table and returns a value of 0.30. This is our Type II Error (β). The power calculation is 1-β, so we get a Power of Test, or a statistical power, of 0.70 in case the true mean should be 53.

As also described in Statistical power, we can see the relation between alpha, beta and power in a power curve and in the bell curves:

Statistical power calculation in Excel

Statistical power calculation in Excel can be calculated with the help of the =NORM.S.INV and the =NORM.DIST functions:

Left tailed and right tailed examples:

Learning statistics

Some of my preferred materials for learnings on statistical power calculation:

• JBstatistics (video: 11:31): Calculating Power and the Probability of a Type II Error (A One-Tailed Example)
• JBstatistics (video 13:39): Calculating Power and the Probability of a Type II Error (A Two-Tailed Example)
• com (an online calculator for power): Power calculator
• Khan Academy (video 5:03): Introduction to Type I and II Errors

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