# Statistical power calculation

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

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Confidence intervals

##### Simple linear regression, fundamentals

##### Two-sample inference

##### ANOVA & the F-distribution

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