# Analysing a single ordinal variable

## 2b: Effect size

If a sample size is very large, almost everything will be significant, but perhaps not relevant anymore. It is therefore recommended to also add a so-called effect size measure.

Unfortunately, there is not much written about measures of effect size that would be suitable for this test, but one that is sometimes recommended (Mangiafico, 2016; Simone, 2017) is a generic one.

It simply divides the standardized test statistics, by the square root of the sample size. The formula can be found in Rosenthal book (1991, p. 19) , so I will refer to as the Rosenthal correlation coefficient (as to differentiate it with other correlation coefficients).

In the example the standardized test statistic was 11.932 and the sample size was 954 (see output from software used in the previous part). We can then quickly calculate the Rosenthal correlation coefficient: *r* = 11.932 / SQRT(954) = 0.39. As for the interpretation various rules of thumb exist. One such rule of thumb is from Bartz (1999, p. 184) shown in Table 1.

Rosenthal Correlation | Interpretation |
---|---|

0.00 < 0.20 |
very low |

0.20 < 0.40 |
low |

0.40 < 0.60 |
moderate |

0.60 < 0.80 |
strong |

0.80 < 1.00 |
very strong |

The 0.39 from the example would then indicate a moderate effect size.

In the report the results of the test we could add this for example like:

A one-sample Wilcoxon signed-rank test indicated that the median was significantly different from 2.5, *Z *= 11.93, *p* < .001, with a moderate effect size (*r *= .39).

**Click here to see how you can obtain this effect size with Python, an Online calculator, or Manually.**

**with Python**

**Online calculator**

Enter the standardized test statistic, and the sample size:

**Manually (Formula and Example)**

**Formula**

The formula to determine the Rosenthal Correlation Coefficient is:

In this formula *Z* is the Z-statistic, which in the case of a Wilcoxon one-sample test is the adjusted W statistic, and *n* the sample size.

**Example**

Note this is a different example than used in this section.

We are given six scores from an ordinal scale and like to test if the median is significantly different from 3. The six scores are:

In the previous section we already calculated the adjusted W statistic, so we can set:

Since there are six scores we also know:

Filling out the formula for the Rosenthal Correlation, we then get:

Let's finish by combining all the reports parts now in the next part.

**Single ordinal variable**

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