# Finding a balance between Type I and Type II errors

The main advantage of looking at statistics sideways is that it makes it easier to understand the relationship between two probability curves. Those are the population probability curve (on the rigt), and the sample probability curve (on the left). The population probability curve shows what alpha is for the statistical experiment; the sample probability curve shows what beta is.

### Alpha

Figure 01 has three curves in it, showing three different values for alpha. The three probability curves are all identical, but the value of the sample correlation (which is the line on the right side of the graph that bisects the population probability curve) increases between them. That increase shifts the curve upward (in our sideways view) and decreases the area under (where “under” means between the curve and the vertical baseline) the probability curve that is bounded by the null line. That decreasing area is equal to alpha.   This is what you would expect to happen. Remember that alpha (a.k.a., the “confidence level”, or “p value”) indicates the probability that the population correlation is not null. Which is another way of saying that it measures the chances that the correlation is “real”, and not a result of random error. More importantly alpha is a measure of the chance of making a “Type I” error.

So, the larger the sample correlation is, the less likely it is that no such correlation exists in the population, and that all you have found is random error. If the sample correlation is quite small, it's easy to believe that the correlation is the result of sampling errors, because it wouldn't take much error to create the illusion of a small correlation. But it would take a very large and therefore unlikely amount of sampling error to create the illusion of a large correlation.

### Beta

Figure 02 also has three graphs in it, with the statistical power increasing between them, while its inverse, beta, is decreasing.   Alpha is the same for all three graphs, and therefore the minimum sample correlation that is statistically significant is also the same. As the population correlation (which is the line on the left side of the graph that bisects the sample probability curve) gets higher in the sideways graph, it gets farther away from the minimum statistically significant sample correlation, and the proportion of the area under the sample probability curve that's above the minimum correlation gets larger. That area is the statistical power.

The inverse of statistical power is beta; meaning it's the area that's below the minimum correlation. So, as the population correlation increases, beta gets smaller. And beta is the chance of making a Type II error. 