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# Observed Values and Critical Values

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When we conduct psychological research, how do we know if the results we have found are significant? We can compare observed values with critical ones procured using statistical tests. Let us take a look at what these are.

Analysis of data helps researchers identify if their results are significant, freepik.com/storyset

## Observed and critical values

After we have conducted our study and collected our data, we can run some tests on the data to see if it supports or rejects our hypothesis. These tests are inferential statistics tests. Some tests you may have come across in your studies are:

The chi-squared test, Mann Whitney U test, Wilcoxon, and Spearman’s Rho test

When run with data, each of these statistical tests will produce a value, and this value is the observed value.

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

The critical value is a set value that we look at to see if what we have found is due to the variables we are investigating or chance. We compare the observed value to the critical value provided by the statistical test we decide to use (this is why it's essential to make sure you're using the proper test).

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

The 'p' stands for probability.

Here we are saying there is a 5% probability the results we found are due to chance. If p = 0.01, there would only be a 1% probability of the results being due to chance. In the tests we have mentioned above, the chi-squared test, Mann Whitney U test, Wilcoxon, and Spearman’s Rho test, there are different rules when it comes to the critical value.

• 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

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.

• One-tailed: particular direction of findings, such as getting more sleep, will lead to better exam grades.
• Two-tailed: not sure about the direction of findings, just that there will be some effect that can go either direction. Sleep affects exam grades (not specified good or bad effect, just a general effect of some sort).

We need to know two important things for the critical values table: the 'N' number (number of participants) and the 'df' (degrees of freedom). Each 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 our table provided by statistical tests until we find a comparable critical value. Then we compare the observed value to the critical value and decide on significance based on the test parameters we covered above.

Let us look at how observed and critical values work with an example. We will use the example of the Mann-Whitney U test.

## Observed and critical value example

The Mann-Whitney U test compares the different 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 can see in this table, there are ten participants in each group.

 Group A scores Group B scores 3 24 5 6 8 4 12 22 2 10 9 18 11 20 15 1 14 7 17 19

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

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

 Group A scores Rank Group B scores Rank 3 18 24 1 5 16 6 15 8 13 4 17 12 9 22 2 2 19 10 11 9 12 18 5 11 10 20 3 15 7 1 20 14 8 7 14 17 6 19 4

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

Add the two scores together (100 + 55 = 155)

Add together all the ranks for Group A (18 + 16 + 13 + 9 + 19 + 12 + 10 + 7 + 8 +6 = 118)

Subtract this from the number in the last step (155 - 118 = 37)

Ua = 37

3. Repeat for Ub; we won't go over the steps again. In this case Ub = 63.

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

5. Next is our hypothesis one or two-tailed, and what is the p-value? Let's suppose our hypothesis is one-tailed with a p-value of 0.05.

6. Now, we need to consult our critical values table for the Mann-Whitney U test. A table is shown below:

Critical values for Mann-Whitney U test, p ≤ 0.05 (one-tailed), p ≤ 0.10 (two-tailed)

 Nb 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Na 5 4 5 6 8 9 11 12 13 15 16 18 19 20 22 23 25 6 5 7 8 10 12 14 16 17 19 21 23 25 26 28 30 32 7 6 8 11 13 15 17 19 21 24 26 28 30 33 35 37 39 8 8 10 13 15 18 20 23 26 28 31 33 36 39 41 44 47 9 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 10 11 14 17 20 24 27 31 34 37 41 44 48 51 55 58 62 11 12 16 19 23 27 31 34 38 42 46 50 54 57 61 65 69 12 13 17 21 26 30 34 38 42 47 51 55 60 64 68 72 77 13 15 19 24 28 33 37 42 47 51 56 61 65 70 75 82 84 14 16 21 26 31 36 41 46 51 56 61 66 71 77 82 87 92 15 18 23 28 33 39 44 50 55 61 66 72 77 83 88 94 100 16 19 25 30 36 42 48 54 60 65 71 77 83 89 95 101 107 17 20 26 33 39 45 51 57 64 70 77 83 89 96 102 109 115 18 22 28 35 41 48 55 61 68 75 82 88 95 102 109 116 123 19 23 30 37 44 51 58 65 72 80 87 94 101 109 116 123 130 20 25 32 39 47 54 62 69 77 84 92 100 107 115 123 130 138

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. Our results are not significant, so we can retain the null hypothesis and reject the alternative hypothesis.

## Observed Values and Critical Values - Key takeaways

• An observed value is a result we get when we run a statistical test.
• The critical value is a set value that we look at to see if what we have found is due to the variables we are investigating or chance.
• The observed value can be compared to the critical value to see if it is significant or not.
• For some tests, the observed value needs to be the same or lower than the critical value to be significant. It needs to be the same or higher than the critical value for other tests to be significant.

Critical values are values that tell us whether our statistical test results are significant or not.

The observed value is the result obtained from a statistical test. We can compare the observed value to the critical value to see if it is significant or not.

Finding the observed value depends on what statistical test is being used.

The critical value is a set value that we look at to see if what we have found is due to the variables we are investigating or chance.  For some tests, the observed value needs to be the same or lower than the critical value to be significant. It needs to be the same or higher than the critical value for other tests to be significant.

## Final Observed Values and Critical Values Quiz

Question

What is an observed value?

An observed value is the result we get when we run a statistical test.

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Question

What is a critical value?

The critical value is a set value that we look at to see if what we have found is due to the variables we are investigating or due to chance.

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Question

A one tailed hypothesis is:

A very specific direction of findings

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Question

What does N stand for?

Number of participants

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Question

What does df stand for?

Degrees of freedom

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Question

Fill in the blank: A chi-squared test is significant if the observed value is ___ than the critical value

equal to or larger

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Question

Fill in the blank: A Mann-Whitney U test is significant if the observed value is ___ than the critical value

equal to or smaller

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Question

For a Wilcoxon test, the observed value T = 15. The critical value is 12. Are the results significant?

No, for significance the observed value needs to be equal to or smaller than the critical value.

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Question

For a Spearman’s Rho test, the observed value r = 0.7, the critical value is 0.4. Are the results significant?

Yes, for significance the observed value needs to be equal to or larger than the critical value.

Show question

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