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Can you provide a formula with an example for solving with Anova?

Question

Can you provide a formula with an example for solving with Anova?

check_circleAnswer
Step 1

Let\'s take a dataset of 3 groups with 12 observations. Hypotheses can be set as shown below. 

A
B
1 G1
G2
G3
2
2
2
3
2
4
1
1
5
4
2
H All means are not equal
st
st
st
LC
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A B 1 G1 G2 G3 2 2 2 3 2 4 1 1 5 4 2 H All means are not equal st st st LC

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Step 2

Let us then determine the degrees of freedom within group and between groups to determine the critical F value.

Use significance level alpha =0.05 

"Benween = k -1 =3 -1 2
dF
dF
within = N - k = 12 3 =9
dF
Total drBetween +dFwithin =2+9=11
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"Benween = k -1 =3 -1 2 dF dF within = N - k = 12 3 =9 dF Total drBetween +dFwithin =2+9=11

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Step 3

Degrees of freedom Numerator = Degrees of freedom between =2

Degrees of freedom Denominator = Degrees of freedom within = 9

F cr...

Denominator
Degrees of
Freedom (k)
Numerator Degrees of Freedom (k)
2
6
7
9
10
12
15
20
24
30
40
60
120
161 4 199.5 215.7
224.6
230.2 234.0
250.1
2368 238.9
19.35 19.37
240.5
241.9 243.9 245.9
1940 19.4
8,74
248.0 249.1
251.1 252 2 253.3 254.3
18.51
10.13
7.71
19.00
19.16
19.25
9.12
6.39
19.30 19.33
9.0
8.94
6.26
19.38
8.81
6.00
19.43
8.70
5.86
19.45
1945
19.46
19.47
19.48
19.49
19.50
8.53
9.55
9.28
8.89
6.09
8.85
6.04
8.79
8.66
8.64
8.62
8.59
8.57
5.69
8.55
566
4
6.94
6.59
6.16
5.96
5.77
5.91
572
3.80
5.75
5.63
5.79
S.14
661
S99
5.59
S.32
5.41
S.19
4.53
4.12
S.0s
4.39
3.97
3.69
3.48
4.95
4.28
4.88
4.82
4,15
3.73
4.77
4.10
3.68
4.74
4,65
4.00
3.57
3.28
3.07
4.62
3.94
3.51
3.22
4.56
3.87
3.44
3.15
4.53
3.84
3.4
4.50
3.81
3.38
446
3.77
4.36
4.43
3.74
4.40
3.70
3.27
2.97
2.75
4.76
4.35
4.07
4.21
3.79
3.50
4.06
3.67
3.23
2.93
2.71
4.74
3.87
3.64
3.35
3.14
3.34
3.04
283
3.30
3.01
2.79
4.46
4.26
3.84
3.63
3.58
3.44
3.39
3.18
3.12
2.90
3.08
2.86
S
5.12
3.86
3.37
3.29
3.23
3.01
2.94
10
11
4.96
3.71
4.10
3.48
3.36
3.33
3.20
3.22
3.14
3.07
3.02
2.98
2.85
2.75
2.91
2.79
2.85
2.72
2.62
2.53
2.77
2.74
2.61
2.51
2.70
2.57
2,47
2.66
2.62
2.49
2.58
2.54
2.40
484
475
467
3.98
3.89
3.59
3.09
3.01
2.9
2.83
295
285
2.90
2.65
2.54
2.46
2.53
2.45
12
13
3.49
3.26
3.11
3.03
3.00
2.80
2.69
2.43
2.34
727
2.38
2.30
2.34
2.30
3.81
3.74
3.18
3.41
3.34
2.92
2.77
2.70
2.71
2.67
2.60
2.42
2.35
2.38
2.25
2.21
2.13
14
4.60
3.11
2.96
285
2.76
2.65
2.60
2.53
2.46
2.39
2.31
2.22
2.18
15
16
3.68
3.63
3.59
3.55
3.52
3.29
3.24
3.20
3.16
4.54
4.49
4.45
4.41
4.38
3.06
3.01
2.96
2.93
2.90
2.90
2.85
281
2.77
2.74
2.79
2.74
2.70
2.71
2.66
2,61
2.58
2.54
2.64
2.59
2.55
2.59
2.54
2.49
2.54
2.48
2.42
2.38
2. 34
2.31
240
2.35
2.31
2.27
223
2.33
2.28
2.23
2.19
2.16
2.29
2.24
2.19
2.15
2.11
2.25
2,19
2.15
2.11
2.07
2.20
2.16
2.11
206
202
1.98
2.11
2.06
201
2.07
2.49
245
2.4
2.38
2.15
2.01
1.96
1.92
188
17
2.10
2.06
2.03
18
251
2.48
266
2.46
1,97
19
3.13
263
2.42
1.93
20
2.45
2.42
4.35
432
4.30
4.28
4.26
3.49
2.39
2.28
2.25
2.23
3,10
3.07
3.05
2.87
2.84
2.82
2.71
2.68
2.66
260
257
2.51
2.49
2.35
2.32
2.20
2.12
2.08
2.04
1.99
1.96
1.95
1.92
1.89
1.90
187
184
L81
1.84
L8
1.78
21
3.47
2.37
2.18
2.15
213
2.10
2.07
2.05
2.03
2.01
1.98
22
3.44
2.55
2.46
2.40
2.34
1.94
1.91
1.89
2.30
2.27
23
3.42
3.40
3.03
2.80
2.64
253
244
2.37
2.36
2.32
2.20
205
203
201
1.98
1.96
1.94
1.86
1.76
24
3.01
2.78
2.62
2.51
2.42
2.30
2.25
2.18
2.11
184
1.79
1,73
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Denominator Degrees of Freedom (k) Numerator Degrees of Freedom (k) 2 6 7 9 10 12 15 20 24 30 40 60 120 161 4 199.5 215.7 224.6 230.2 234.0 250.1 2368 238.9 19.35 19.37 240.5 241.9 243.9 245.9 1940 19.4 8,74 248.0 249.1 251.1 252 2 253.3 254.3 18.51 10.13 7.71 19.00 19.16 19.25 9.12 6.39 19.30 19.33 9.0 8.94 6.26 19.38 8.81 6.00 19.43 8.70 5.86 19.45 1945 19.46 19.47 19.48 19.49 19.50 8.53 9.55 9.28 8.89 6.09 8.85 6.04 8.79 8.66 8.64 8.62 8.59 8.57 5.69 8.55 566 4 6.94 6.59 6.16 5.96 5.77 5.91 572 3.80 5.75 5.63 5.79 S.14 661 S99 5.59 S.32 5.41 S.19 4.53 4.12 S.0s 4.39 3.97 3.69 3.48 4.95 4.28 4.88 4.82 4,15 3.73 4.77 4.10 3.68 4.74 4,65 4.00 3.57 3.28 3.07 4.62 3.94 3.51 3.22 4.56 3.87 3.44 3.15 4.53 3.84 3.4 4.50 3.81 3.38 446 3.77 4.36 4.43 3.74 4.40 3.70 3.27 2.97 2.75 4.76 4.35 4.07 4.21 3.79 3.50 4.06 3.67 3.23 2.93 2.71 4.74 3.87 3.64 3.35 3.14 3.34 3.04 283 3.30 3.01 2.79 4.46 4.26 3.84 3.63 3.58 3.44 3.39 3.18 3.12 2.90 3.08 2.86 S 5.12 3.86 3.37 3.29 3.23 3.01 2.94 10 11 4.96 3.71 4.10 3.48 3.36 3.33 3.20 3.22 3.14 3.07 3.02 2.98 2.85 2.75 2.91 2.79 2.85 2.72 2.62 2.53 2.77 2.74 2.61 2.51 2.70 2.57 2,47 2.66 2.62 2.49 2.58 2.54 2.40 484 475 467 3.98 3.89 3.59 3.09 3.01 2.9 2.83 295 285 2.90 2.65 2.54 2.46 2.53 2.45 12 13 3.49 3.26 3.11 3.03 3.00 2.80 2.69 2.43 2.34 727 2.38 2.30 2.34 2.30 3.81 3.74 3.18 3.41 3.34 2.92 2.77 2.70 2.71 2.67 2.60 2.42 2.35 2.38 2.25 2.21 2.13 14 4.60 3.11 2.96 285 2.76 2.65 2.60 2.53 2.46 2.39 2.31 2.22 2.18 15 16 3.68 3.63 3.59 3.55 3.52 3.29 3.24 3.20 3.16 4.54 4.49 4.45 4.41 4.38 3.06 3.01 2.96 2.93 2.90 2.90 2.85 281 2.77 2.74 2.79 2.74 2.70 2.71 2.66 2,61 2.58 2.54 2.64 2.59 2.55 2.59 2.54 2.49 2.54 2.48 2.42 2.38 2. 34 2.31 240 2.35 2.31 2.27 223 2.33 2.28 2.23 2.19 2.16 2.29 2.24 2.19 2.15 2.11 2.25 2,19 2.15 2.11 2.07 2.20 2.16 2.11 206 202 1.98 2.11 2.06 201 2.07 2.49 245 2.4 2.38 2.15 2.01 1.96 1.92 188 17 2.10 2.06 2.03 18 251 2.48 266 2.46 1,97 19 3.13 263 2.42 1.93 20 2.45 2.42 4.35 432 4.30 4.28 4.26 3.49 2.39 2.28 2.25 2.23 3,10 3.07 3.05 2.87 2.84 2.82 2.71 2.68 2.66 260 257 2.51 2.49 2.35 2.32 2.20 2.12 2.08 2.04 1.99 1.96 1.95 1.92 1.89 1.90 187 184 L81 1.84 L8 1.78 21 3.47 2.37 2.18 2.15 213 2.10 2.07 2.05 2.03 2.01 1.98 22 3.44 2.55 2.46 2.40 2.34 1.94 1.91 1.89 2.30 2.27 23 3.42 3.40 3.03 2.80 2.64 253 244 2.37 2.36 2.32 2.20 205 203 201 1.98 1.96 1.94 1.86 1.76 24 3.01 2.78 2.62 2.51 2.42 2.30 2.25 2.18 2.11 184 1.79 1,73

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