Reading about statistical power they draw this:

enter image description here

and say:

Illustration of the power and the significance level of a statistical test, given the null hypothesis (sampling distribution 1) and the alternative hypothesis (sampling distribution 2).

I don't understand how $\alpha$ and $\beta$ relate to the plots. Can you give me some hints please?

  • Are those Gaussian probability distribution curves (could be any other.) each for a single group of people -say for height, or are they a distribution of the means of many groups of people ? (just taking the mean as an example.)

1 Answer 1


The graph is correct, but probably more confusing than it is helpful.

The 2 gaussians are meant to represent 2 different scenarios: the first, when $H_0$ is true (sampling distribution 1), the other, when $H_a$ is true (sampling distribution 2).

The term "sampling distribution" is properly used, referring for example to a single mean (for a 1-sample t test), a difference of means (for a 2-sample t-test), or to some other sampling distribution, for any given statistic $t$, which estimates a parameter $\theta$.

In the case of the figure, the null hypothesis is that the parameter in question is $0$ (a common, but not universal situation).

A better figure would be to stack the 2 scenarios as in the figures below:

alpha and beta errors

How alpha and beta relate to the plots should be clearer:

  • Alpha errors can only occur when $H_0$ is true, while beta errors can only occur when $H_a$ is true
  • Alpha errors are when you incorrectly reject $H_0$ when it is in fact true (scenario 1), beta errors when you fail to reject $H_0$ when it is in fact false (scenario 2)
  • If one increases alpha (at the cost of more chances of alpha errors), this will decreases beta (i.e. increase your power)
  • Similarly, if one decreases alpha, this will increase beta, thus decreasing your power
  • $\begingroup$ You risk being misunderstood by referring to the parameter $\theta$ as a "statistic." I believe this is certain to confuse anyone who isn't already deeply familiar with estimation and hypothesis testing. $\endgroup$
    – whuber
    Commented May 20 at 2:32
  • 1
    $\begingroup$ I apologize in advance if my suspicions are misplaced: Did you make these plots? Or are you borrowing the plots from someone? If the latter, please add a reference to the source. $\endgroup$
    – dipetkov
    Commented May 20 at 8:41
  • $\begingroup$ ok, i got it finally, at least in broad terms, thank you, that's a master explanation ! $\endgroup$
    – Mah Neh
    Commented May 20 at 14:49
  • 1
    $\begingroup$ dipetkov, this is indeed my graph. @whuber, yes indeed, I misused statistic vs. parameter.; Post has been edited accordingly. $\endgroup$
    – jginestet
    Commented May 20 at 16:39

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