Observed Values and Critical Values

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When conducting statistical analyses, many numbers, charts and graphs are outputted. These can be confusing, and it can be challenging to determine what exactly they mean and how we can identify if these figures support or disprove our hypothesis. The two crucial statistical values to determine this are observed values and critical values.

First, we will delve into the observed value meaning and then move on to understand critical values psychology.
Then we will learn how hypothesis testing critical values work.
To help with your learning, we will look at a critical and observed value psychology example and provide you with a critical values table for the Mann-Whitney U test.

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.

A hypothesis is a predictive statement that states what the researcher expects to find in their study.

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

Some examples of inferential tests are the chi-squared test, Mann-Whitney U test, Wilcoxon, and Spearman's Rho test.

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

The observed value meaning is the statistical figure that calculates/ measures the trend/ pattern investigated in the research.

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.

The critical value is a set value that we look at to see if what we have found (the observed value) is due to the variables investigated or chance.

The process involves comparing the observed value to the critical value provided by the statistical test we decide to use (this is why it's essential to ensure you're using the proper test).

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

The 'p' stands for probability.

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.

The 0 before the decimal point in critical values is not reported per APA standards.

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.

A one-tailed hypothesis is when the researcher predicts that the findings will go in a particular direction, e.g. getting more sleep will lead to better exam grades.

A two-tailed hypothesis is a predictive statement that suggests there will be an effect, but the direction is unsure, e.g. if there will be an increase or decrease.

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.

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, '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.

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 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

Then, repeat for Ub; we won't go over the steps again. In this case Ub = 63.

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.

How we do calculations depends on the test used because the various inferential tests measure/ calculate different things.

Critical Values Table

The critical values table for the 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.

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

From this critical and observed psychology example, we can infer there is no statistical difference between the scores of the two groups. i.e. the null hypothesis is accepted.

Observed Values and Critical Values - Key takeaways

The observed value meaning is the statistical figure that calculates/ measures the trend/ pattern investigated in the research.
Critical values psychology is a statistical figure used to identify if the results from inferential tests are significant or if they occurred due to chance.
The critical and observed value psychology research is compared to identify whether the null or alternative hypothesis should be accepted/ rejected.
The critical values table allows the comparison of observed and critical values to identify if the results are statistically significant.
How researchers calculate and make inferences based on the observed and critical values depends on the inferential test the researcher uses.

Frequently Asked Questions about Observed Values and Critical Values

Critical values psychology is a statistical figure used to identify if the results from inferential tests are significant or if they occurred due to chance.

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 the statistical test used and can be found using calculations and the corresponding critical values table.

Hypothesis testing critical values is when we look to see if what we have found is due to the variables proposed in the hypothesis 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.

The observed value meaning is the statistical figure that calculates/ measures the trend/ pattern investigated in the research.

Critical values psychology is a statistical figure used to identify if the results from inferential tests are significant or if they occurred due to chance.

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What is an observed value?

The observed value meaning is the statistical figure that calculates/ measures the trend/ pattern investigated in the research.

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.

What is a one-tailed hypothesis?

The researcher states the direction of the results.

What does N stand for?

The number of participants.

What does df stand for?

Degrees of freedom.

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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Observed Values and Critical Values

Observed Value Meaning

Critical Values Psychology

Hypothesis Testing Critical Values

Critical and Observed Value Psychology Example

Critical Values Table

Observed Values and Critical Values - Key takeaways

Frequently Asked Questions about Observed Values and Critical Values

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	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

	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

Observed Values and Critical Values

Observed Value Meaning

Critical Values Psychology

Hypothesis Testing Critical Values

Critical and Observed Value Psychology Example

Critical Values Table

Observed Values and Critical Values - Key takeaways

Frequently Asked Questions about Observed Values and Critical Values

What do critical values mean?

What is the difference between critical and observed value?

How do we find the observed value in psychology?

What are critical values for a test of hypothesis?

What are critical values and observed values in research?

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	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