Observed Values and Critical Values

Observed Value Meaning

In statistical analyses, the observed value is one of the most important figures as it allows researchers to identify if their data follows the trend predicted in their hypothesis.

The observed value is found when carrying out statistical tests called inferential tests.

Now that we've figured out the use of the observed value let's explore the observed value meaning.

The observed value differs depending on what inferential test is used. For instance, a correlation measures the association/ relationship between variables; using the figure r. In contrast, a t-test compares the mean of two groups; using the figure t.

Critical Values Psychology

But how do we know if the observed value we found is significant? And this is where the critical value comes in.

First, we need a significance level (p-value) to do this. Usually, the significance level is p = 0.05, although this value can change.

A p-score of .05 suggests there is a 5% probability the results we found are due to chance. And p = .01 means there is a 1% probability of the results being due to chance.

Hypothesis Testing Critical Values

There are different rules regarding the critical value in the tests we mentioned above, the chi-squared test, Mann Whitney U test, Wilcoxon, and Spearman's Rho test.

• Chi-squared test: significant if the observed value (χ²) is equal to or larger than the critical value

• Mann-Whitney U test: significant if the observed value (U) is equal to or smaller than the critical value

• Wilcoxon test: significant if the observed value (T) is equal to or smaller than the critical value

• Spearman's Rho test: significant if the observed value (r) is equal to or larger than the critical value

When using a critical values table, the observed value can be compared to the critical value to see if the results are statistically significant. Each statistical test will have its own critical values table. The critical value we need also depends on if our hypothesis is one or two-tailed.

We need to know two essential things for the critical values table: the 'N' number (number of participants) and the 'df' (degrees of freedom). Each critical table will have a column of N values or df values depending on what test it is.

• N = number of participants. In an independent group design, there will be different N numbers for each group of participants; this is written as Na (group A) and Nb (group B).
• df = degrees of freedom refers to the elements allowed to vary in statistical tests. It is used for tests where the number of categories is important, such as the chi-square test (which compares nominal data for different categories to see if there are any differences). The more degrees of freedom, the more categories there are.

We need to look at the N or df column in critical tables provided by statistical tests to find a comparable critical value. Then we can compare the observed value to the critical value and decide on significance based on the test parameters we covered above.

Let's look at an example to help you make sense of this.

Critical and Observed Value Psychology Example

The Mann-Whitney U test compares the scores between two groups (independent groups design), focusing on ranks and ordinal data. Let's look at the steps involved to see if our results are significant.

As we see in this table, ten participants are in each group.

We need to work out the observed value, 'U'. To do this, we need to calculate scores for the two groups (Ua and Ub). The U score will be the lower score of the two.

First, we need to rank each score; this is done for both groups. The highest score is rank 1, the one after that is rank 2, and so on.

Now let's work out at Ua. We need to know Na and Nb, the total number of scores in each group. There were 10 participants in each group, so a total of 10 scores for each group, Na = 10 and Nb = 10.

First, multiply Na and Nb (10 x 10 = 100)

Then multiply Na by (Na + 1) and then divide by 2 (10 x 11/2 = 110/2 = 55) and add the two scores together (100 + 55 = 155). Afterwards, Group A's ranks should be added together (18 + 16 + 13 + 9 + 19 + 12 + 10 + 7 + 8 +6 = 118). And finally, subtract this from the number in the last step (155 - 118 = 37).

Thus, Ua = 37

The U value is the lower of the two, so here, U = 37.

Next, we need to note if our hypothesis is one or two-tailed and the p-value. Let's suppose our hypothesis is one-tailed with a p-value of 0.05.

Then we must consult our critical values table for the Mann-Whitney U test.

Critical Values Table

The critical values table for the Mann-Whitney U test, p ≤ 0.05 (one-tailed), p ≤ 0.10 (two-tailed).

The values we need have been highlighted. First, we find Na, which in our case is 10. Then we find Nb, which is 10 too. We find the value where these two meet, which is the critical value. Here it is, 27.

Our observed value is 37, which is larger than the critical value of 27.

Therefore, our results are not significant, so we can accept the null hypothesis and reject the alternative hypothesis.