Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Observed Values and Critical Values

Save Save
Print Print
Edit Edit
Sign up to use all features for free. Sign up now
Observed Values and Critical Values

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.

Observed Values and Critical Values, Data analysis of different charts, StudySmarterAnalysis 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 scoresGroup B scores
324
56
84
1222
210
918
1120
151
147
1719

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 scoresRankGroup B scoresRank
318241
516615
813417
129222
2191011
912185
1110203
157120
148714
176194

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.

Frequently Asked Questions about Observed Values and Critical Values

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?

Show answer

Answer

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

Show question

Question

What is a critical value?

Show answer

Answer

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. 

Show question

Question

A one tailed hypothesis is:


Show answer

Answer

A very specific direction of findings

Show question

Question

What does N stand for?


Show answer

Answer

Number of participants

Show question

Question

What does df stand for?

Show answer

Answer

Degrees of freedom

Show question

Question

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


Show answer

Answer

equal to or larger

Show question

Question

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


Show answer

Answer

equal to or smaller

Show question

Question

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


Show answer

Answer

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

Show question

Question

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


Show answer

Answer

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

Show question

More about Observed Values and Critical Values
60%

of the users don't pass the Observed Values and Critical Values quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Just Signed up?

Yes
No, I'll do it now

Sign up to highlight and take notes. It’s 100% free.