Statistical Tests Calculator, Critical Values Tables and Formulae
On This Page:
- Chi-Squared Test
- SignTest
- Mann-Whitney U Test
- Wilcoxon's T Test
- Spearman's Rho Test
Critical Values Tables and Formulae
On this page you will find the critical values tables for the tests listed to the right.Further down the page are the formulae and step-by-step instructions on how to calculate each test.
If you would prefer to have these tests calculated for you, then use the Statistics Calculator instead.
Formulae
Chi-Squared Test
Χ2 = Σ ((O-E)2 / E)
E= Expected frequencies for each cell.
- Add up the total for each row.
- Add up the total for each column.
- Add the row totals and column totals together.
- Calculate the expected frequency for each cell: E=(Row Total x Column Total) / Total
- Calculate chi-squared for each cell: (O-E)2 / E.
- Add up the values of chi-squared for all cells.
- Calculate the degrees of freedom (df): df=(No. Rows - 1) x (No. Columns - 1).
- Look up value of Χ2 in table.
Sign Test
This isn't a formula as such, but...
S = the sum of the negative differences OR the sum of the positive differences, depending on which is smaller.
S = the sum of the negative differences OR the sum of the positive differences, depending on which is smaller.
- For each row, identify whether the change from condition 1 to condition 2 is positive (+), negative (-) or no change (=).
- Add up the total for the positive and negative changes.
- Ignore cases where there has been no change - this means your sample size for the critical value will be smaller.
- Look up value of S in table, based on your new sample size (see above).
Mann-Whitney U Test
NB= number of scores in condition two
- Rank all of the scores in both conditions together, from 1st place
to lowest value. Make sure that if there is more than one value that is
the same that they all share the mid rank position:
- eg. values of 4 5 5 6 6 6 6 7 8 9 9 9 10 10 11 12 13 15 18 19
- would be ranked 1 2.5 2.5 5.5 5.5 5.5 5.5 8 9 11 11 11 13.5 13.5 15 16 17 18 19 20
- Sigma RA = the total of the ranks for the scores from condition one.
- Substitute values into formula
- Look up value of U in table
Wilcoxon's T Test
This isn't a formula as such, but...
T = the sum of the ranks for the negative differences OR the sum of the ranks for the positive differences, depending on which is smaller.
T = the sum of the ranks for the negative differences OR the sum of the ranks for the positive differences, depending on which is smaller.
- Subtract the values in column 2 from their matched values in column 1 (D=A-B)
- Rank the differences obtained in step one, lowest difference = 1st place (ignoring whether these are positive or negative - based only on the number and discarding any differences of 0)
- Add together the ranks for all of the positive differences
- Add together the ranks for all the negative differences
- Whichever of these totals is smallest = T
- Look up the value of T in the table
Spearman's Rho
D = difference between the ranks of IV1 and IV2
- Rank the values for IV1
- Rank the values for IV2
- Subtract the ranks for IV2 from IV1 (D=RA-RB)
- Square the differences from step 3
- Total the values obtained in step 4.
- Substitute values into formula
- Look up value of rs in table
Critical Values Tables
Critical Values for Chi-Squared Test
USE THIS TABLE TO FIND OUT IF YOUR VALUE FOR (Chi-Squared) Χ2 IS SIGNIFICANT.The value you obtained must be greater than or equal to the value given in the table for your degrees of freedom (df) in order to be significant.
One-tailed
0.1
0.05
0.025
0.01
0.005
0.0025
Two-tailed
0.2
0.1
0.05
0.02
0.01
0.001
DF
1
1.64
2.71
3.84
5.41
6.64
10.83
2
3.22
4.60
5.99
7.82
9.21
13.82
3
4.64
6.25
7.82
9.84
11.34
16.27
4
5.99
7.78
9.49
11.67
13.28
18.46
5
7.29
9.24
11.07
13.39
15.09
20.52
6
8.56
10.64
12.59
15.03
16.81
22.46
7
9.80
12.02
14.07
16.62
18.48
24.32
8
11.03
13.36
15.51
18.17
20.09
26.12
9
12.24
14.68
16.92
19.68
21.67
27.88
10
13.44
15.99
18.31
21.16
23.21
29.59
11
14.63
17.28
19.68
22.62
24.72
31.26
12
15.81
18.55
21.03
24.05
26.22
32.91
13
16.98
19.81
22.36
25.47
27.69
34.53
14
18.15
19.81
23.68
26.87
29.14
36.12
15
19.31
22.31
25.00
28.26
30.58
37.70
16
20.46
23.54
26.30
29.63
32.00
39.29
17
21.62
24.77
27.59
31.00
33.41
40.75
18
22.76
25.99
28.87
32.35
34.80
42.31
19
23.90
27.20
30.14
33.69
36.19
43.82
20
25.04
28.41
31.41
35.02
37.57
45.32
21
26.17
29.62
32.67
36.34
38.93
46.80
22
27.30
30.81
33.92
37.66
40.29
48.27
23
28.43
32.01
35.17
38.97
41.64
49.73
24
29.55
33.20
36.42
40.27
42.98
51.18
25
30.68
34.38
37.65
41.57
44.31
52.62
26
31.80
35.56
38.88
42.86
45.64
54.05
27
32.91
36.74
40.11
44.14
46.96
55.48
28
34.03
37.92
41.34
45.42
48.28
56.89
29
35.14
39.09
42.69
49.69
49.59
58.30
30
36.25
40.26
43.77
47.96
50.89
59.70
32
38.47
42.59
46.19
50.49
53.49
62.49
34
40.68
44.90
48.60
53.00
56.06
65.25
36
42.88
47.21
51.00
55.49
58.62
67.99
38
45.08
49.51
53.38
57.97
61.16
70.70
40
47.27
51.81
55.76
60.44
63.69
73.40
44
51.64
56.37
60.48
65.34
68.71
78.75
48
55.99
60.91
65.17
70.20
73.68
84.04
52
60.33
65.42
69.83
75.02
78.62
89.27
56
64.66
69.92
74.47
79.82
83.51
94.46
60
68.97
74.40
79.08
84.58
88.38
99.61
Critical Values for Sign Test
USE THIS TABLE TO FIND OUT IF YOUR VALUE FOR S IS SIGNIFICANT.
The value you obtained must be less than or equal to the value given in the table for your sample size (n) in order to be significant.
The value you obtained must be less than or equal to the value given in the table for your sample size (n) in order to be significant.
One Tailed
0.05
0.025
0.01
0.005
0.0005
Two Tailed
0.10
0.005
0.02
0.001
0.0005
N
5
0
6
0
0
7
0
0
0
8
1
0
0
0
9
1
1
0
0
10
1
1
0
0
11
2
1
1
0
0
12
2
2
1
1
0
13
3
2
1
1
0
14
3
2
2
1
0
15
3
3
2
2
1
16
4
3
2
2
1
17
4
4
3
2
1
18
5
4
3
3
1
19
5
4
4
3
2
20
5
5
4
3
2
25
7
7
6
5
4
30
10
9
8
7
5
35
12
11
10
9
7
Critical Values for Mann-Whitney U Test
(from Billet, P. (2004) 'The Open Door Website: Critical Values for the Mann-Whitney U-Test' http://www.saburchill.com/IBbiology/stats/003.html (online) [accessed 06/05/05])
USE THIS TABLE TO FIND OUT IF YOUR VALUE FOR (Mann-Whitney) U IS SIGNIFICANT.
The value you obtained must be less than or equal to the value given in the table for your sample sizes (N) in order to be significant.
Level of significance: 5% (P = 0.05 two-tailed, and p=0.025 one-tailed test)
nb
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
na
3
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
13
13
4
1
2
3
4
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
17
18
19
20
21
22
23
5
2
3
5
6
7
8
9
11
12
13
14
15
17
18
19
20
22
23
24
25
27
28
29
30
32
33
6
5
6
8
10
11
13
14
16
17
19
21
22
24
25
27
29
30
32
33
35
37
38
40
42
43
7
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
8
13
15
17
19
22
24
26
29
31
34
36
38
41
43
45
48
50
53
55
57
60
62
65
9
17
20
23
26
28
31
34
37
39
42
45
48
50
53
56
59
62
64
67
70
73
76
10
23
26
29
33
36
39
42
45
48
52
55
58
61
64
67
71
74
77
80
83
87
11
30
33
37
40
44
47
51
55
58
62
65
69
73
76
80
83
87
90
94
98
12
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
101
105
109
13
45
50
54
59
63
67
72
76
80
85
89
94
98
102
107
111
116
120
14
55
59
64
67
74
78
83
88
93
98
102
107
112
118
122
127
131
15
64
70
75
80
85
90
96
101
106
111
117
122
125
132
138
143
16
75
81
86
92
98
103
109
115
120
126
132
138
143
149
154
17
87
93
99
105
111
117
123
129
135
141
147
154
160
166
18
99
106
112
119
125
132
138
145
151
158
164
171
177
19
113
119
126
133
140
147
154
161
168
175
182
189
20
127
134
141
149
156
163
171
178
186
193
200
21
142
150
157
165
173
181
188
196
204
212
22
158
166
174
182
191
199
207
215
223
23
175
183
192
200
209
218
226
235
24
192
201
210
219
228
238
247
25
211
220
230
239
249
258
26
230
240
250
260
270
27
250
261
271
282
28
272
282
293
29
294
305
30
317
Critical Values for Wilcoxon's T Test
USE THIS TABLE TO FIND OUT IF YOUR VALUE FOR (Wilcoxon) T IS SIGNIFICANT.
The value you obtained must be less than or equal to the value given in the table for your sample sizes (N) in order to be significant.
p is the probability of your results occurring by chance (as stated by the null hypothesis), so the lower the value of p, the more certain you can be that your result is significant.
eg. p=0.10 means a 1 in 10 probability the result occurred by chance, whereas p=0.025 mean a 25 in a 1000 (1 in 40) probability the result occurred by chance.
The value you obtained must be less than or equal to the value given in the table for your sample sizes (N) in order to be significant.
p is the probability of your results occurring by chance (as stated by the null hypothesis), so the lower the value of p, the more certain you can be that your result is significant.
eg. p=0.10 means a 1 in 10 probability the result occurred by chance, whereas p=0.025 mean a 25 in a 1000 (1 in 40) probability the result occurred by chance.
1 tailed
0.05
0.025
0.01
0.001
2 tailed
0.10
0.05
0.02
0.002
6
2
0
7
3
2
0
8
5
3
1
9
8
5
3
10
10
8
5
0
11
13
10
7
1
12
17
13
9
2
13
21
17
12
4
14
25
21
15
6
15
30
25
19
8
16
35
29
23
11
17
41
34
27
14
18
47
40
32
18
19
53
46
37
21
1 tailed
0.05
0.025
0.01
0.001
2 tailed
0.10
0.05
0.02
0.002
20
60
52
43
26
21
67
58
49
30
22
75
65
55
35
23
83
73
62
40
24
91
81
69
45
25
100
89
76
51
26
110
98
84
58
27
119
107
92
64
28
130
116
101
71
29
141
125
111
78
30
151
137
120
86
31
163
147
130
94
32
175
159
140
103
33
187
170
151
112
Critical Values for Spearman's Rho
USE THIS TABLE TO FIND OUT IF YOUR VALUE FOR (Spearman's Rho) rs IS SIGNIFICANT.
The value you obtained must be greater than or equal to the value given in the table for your sample size (N) in order to be significant.
If your N is not in the table, use the next one down - e.g., for an N of 17, use the table values for 16.
The value you obtained must be greater than or equal to the value given in the table for your sample size (N) in order to be significant.
If your N is not in the table, use the next one down - e.g., for an N of 17, use the table values for 16.
N
0.05
0.02
0.01
5
1
1
6
0.886
0.943
1
7
0.786
0.893
0.929
8
0.738
0.833
0.881
9
0.683
0.783
0.833
10
0.648
0.746
0.794
12
0.591
0.712
0.777
14
0.544
0.645
0.715
16
0.506
0.601
0.665
18
0.475
0.564
0.625
20
0.45
0.534
0.591
22
0.428
0.508
0.562
24
0.409
0.485
0.537
26
0.392
0.465
0.515
28
0.377
0.448
0.496
30
0.364
0.432
0.478