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1.3: Table of Critical Values of F

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    \(F\) Distribution

    The tables following Figure \(\PageIndex{1}\) show the critical t-score. If the your calculated F-score is bigger (more extreme) than the critical F-score, then you reject the null hypothesis. Similarly, if the calculated F-score is to the right (in the shaded area), then the null hypothesis should be rejected. This means that there is a small probability (less than 5%, p<.05) that all of the means are similar (suggesting that at least one mean is different from one other mean). In contrast, if the calculated F-score is smaller than the critical value (to the right of the shaded area), then the null hypothesis is retained; there is a large probability (larger than 5%. p>.05) that the group means are similar.

    Sample F-distribution that is a line graph that leans to the left, with the extreme section to the right of an F-score at the right tail labeled "Probability p" shaded to show the Rejection Region.
    Figure \(\PageIndex{1}\)- Critical Values for F (\(F^*\)) with probability \(p\) to its right. (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax)
    Note

    Remember:

    (Critical \(<\) Calculated) \(=\) Reject null \(=\) At least one mean is different from at least one other mean. \(= p<.05\)

    (Critical \(>\) Calculated) \(=\) Retain null \(=\) All of the means are similar. \(= p>.05\)

    The information in the Note above reminds you how to format the statistical sentence, with underlining meaning to replace the letters with the answers for that specific scenario: \(t\)(dfBG, dfWG)=calculated F-score, \(p\)<.05 or \(t\)(dfBG, dfWG)=calculated F-score, \(p\)>.05 (depending on whether your calculated t-scores is more extreme than the critical F-score that you find in the following tables).

    Tables of Critical Values of F (ANOVA)

    The critical values table for ANOVA is a little different than other tables of critical values because ANOVA is a ratio of variances between groups and variances within each group. This means that there are two degrees of freedom, one for the numerator and one for the denominator. There are three sets of tables, one for each of three probabilities that the null hypothesis is true (1%, 5%, and 10%). This textbook will always use the 5% alpha (\(p = 0.05\), but your professor or your circumstances might indicate another alpha level is preferred. All of the tables show degrees of freedom of the numerator (which is based on the number of groups) in the columns; the options are from 1 to 5. If you have 6 or more groups (or combinations), use the column for 5. But honestly, if you have more than 5 groups, you should be using statistical software that will provide the probability so that you wouldn't have to use a table of critical values!

    To use each table, find the degrees of freedom for your denominator (sometimes Within Groups and sometimes called Error) on the first column, then the degrees of freedom of the numerator on the first row; use those to identify what the critical value is. If you're not sure how to calculate the degrees of freedom, the formulas for calculating the degrees of freedom for each (numerator and denominator) for each type of ANOVA are shown below the tables.

    Because tables are limited by size, not all critical t-scores are listed. For example, if you had F(2, 49), then the degrees of freedom of the numerator would be 2 (DFN = 2) and the degrees of freedom for the denominator would be 49 (\(df_D = 49\)). However, the DF for the denominator jumps from 40 to 50. There are a couple of options when your Degrees of Freedom is not listed on the table.

    • One option is to use the Degrees of Freedom that is closest to your sample's Degrees of Freedom. For our example of F(2, 49), that would mean that we would use the dfD of 50 because 50 is closer to 49 than 40 is. That would mean that the critical F-score for F(2, 49) would be 3.18.
    • Another option is to always we round down. For our example of F(2, 49), we use the dfD of 40 because it is the next lowest dfD listed. That would mean that the critical F-score for F(2, 49) would be 3.23. This option avoids inflating Type I Error (false positives).

    Ask your professor which option you should use! Whichever option you choose, your statistical sentence should include the actual degrees of freedom for the numerator and for the denominator, regardless of which number is listed in the table; the table is used to decide if the null hypothesis should be rejected or retained.

    Critical Values of F for p=1%

    The first table (Table \(\PageIndex{1}\)) shows the critical values when \(p\)=0.01. Use the degrees of freedom for the numerator and the denominator to find the critical value; if you're not sure how to calculate the degrees of freedom, the formulas for calculating the degrees of freedom for each (numerator and denominator) for each type of ANOVA are shown below the tables.

    Table \(\PageIndex{1}\)- Critical Values of F for \(p\)=0.01
    Degrees of Freedom Degrees of Freedom for Numerator
    Degrees of Freedom for Denominator (Error) 1 2 3 4 5
    5 16.26 13.27 12.06 11.39 10.97
    6 13.75 10.92 9.78 9.15 8.75
    7 12.25 9.55 8.45 7.85 7.46
    8 11.26 8.65 7.59 7.01 6.63
    9 10.56 8.02 6.99 6.42 6.06
    10 10.04 7.56 6.55 5.99 5.64
    11 9.65 7.21 6.22 5.67 5.32
    12 9.33 6.93 5.95 5.41 5.06
    13 9.07 6.70 5.74 5.21 4.86
    14 8.86 6.51 5.56 5.04 4.69
    15 8.68 6.36 5.42 4.89 4.56
    16 8.53 6.23 5.29 4.77 4.44
    17 8.40 6.11 5.19 4.67 4.34
    18 8.29 6.01 5.09 4.58 4.25
    19 10.56 8.02 6.99 6.42 6.06
    20 8.10 5.85 4.94 4.43 4.10
    21 8.02 5.78 4.87 4.37 4.04
    22 7.95 5.72 4.82 4.31 3.99
    23 7.88 5.66 4.76 4.26 3.94
    24 7.82 5.61 4.72 4.22 3.90
    25 7.77 5.57 4.68 4.18 3.85
    26 7.72 5.53 4.64 4.14 3.82
    27 7.68 5.49 4.60 4.11 3.78
    28 7.64 5.45 4.57 4.07 3.75
    29 7.60 5.42 4.54 4.04 3.73
    30 7.56 5.39 4.51 4.02 3.70
    40 7.31 5.18 4.31 3.83 3.51
    50 7.17 5.06 4.20 3.72 3.41
    60 7.08 4.98 4.13 3.65 3.34
    100 6.90 4.82 3.98 3.51 3.21
    200 6.76 4.71 3.88 3.41 3.11
    1000 6.66 4.63 3.80 3.34 3.04

    Critical Values of F for p=5%

    The second table (Table \(\PageIndex{2}\)) shows the critical values when \(p\)=0.05. Use the degrees of freedom for the numerator and the denominator to find the critical value; if you're not sure how to calculate the degrees of freedom, the formulas for calculating the degrees of freedom for each (numerator and denominator) for each type of ANOVA are shown below the tables.

    Table \(\PageIndex{2}\)- Critical Values of F for \(p\)=0.05
    Degrees of Freedom Degrees of Freedom for Numerator
    Degrees of Freedom for Denominator (Error) 1 2 3 4 5
    5 6.61 5.79 5.41 5.19 5.05
    6 5.99 5.14 4.76 4.53 4.39
    7 5.59 4.74 4.35 4.12 3.97
    8 5.32 4.46 4.07 3.84 3.69
    9 5.12 4.26 3.86 3.63 3.48
    10 4.96 4.10 3.71 3.48 3.33
    11 4.84 3.98 3.59 3.36 3.20
    12 4.75 3.89 3.49 3.26 3.11
    13 4.67 3.81 3.41 3.18 3.03
    14 4.60 3.74 3.34 3.11 2.96
    15 4.54 3.68 3.29 3.06 2.90
    16 4.49 3.63 3.24 3.01 2.85
    17 4.45 3.59 3.20 2.96 2.81
    18 4.41 3.55 3.16 2.93 2.77
    19 5.12 4.26 3.86 3.63 3.48
    20 4.35 3.49 3.10 2.87 2.71
    21 4.32 3.47 3.07 2.84 2.68
    22 4.30 3.44 3.05 2.82 2.66
    23 4.28 3.42 3.03 2.80 2.64
    24 4.26 3.40 3.01 2.78 2.62
    25 4.24 3.39 2.99 2.76 2.60
    26 4.23 3.37 2.98 2.74 2.59
    27 4.21 3.35 2.96 2.73 2.57
    28 4.20 3.34 2.95 2.71 2.56
    29 4.18 3.33 2.93 2.70 2.55
    30 4.17 3.32 2.92 2.69 2.53
    40 4.08 3.23 2.84 2.61 2.45
    50 4.03 3.18 2.79 2.56 2.40
    60 4.00 3.15 2.76 2.53 2.37
    100 3.94 3.09 2.70 2.46 2.31
    200 3.89 3.04 2.65 2.42 2.26
    1000 3.85 3.00 2.61 2.38 2.22

    Critical Values of F for p=10%

    The third table (Table \(\PageIndex{3}\)) shows the critical values when \(p\)=0.10. Use the degrees of freedom for the numerator and the denominator to find the critical value; if you're not sure how to calculate the degrees of freedom, the formulas for calculating the degrees of freedom for each (numerator and denominator) for each type of ANOVA are shown below the tables.

    Table \(\PageIndex{2}\)- Critical Values of F for \(p\)=0.05
    Degrees of Freedom Degrees of Freedom for Numerator
    Degrees of Freedom for Denominator (Error) 1 2 3 4 5
    5 4.06 3.78 3.62 3.52 3.45
    6 3.78 3.46 3.29 3.18 3.11
    7 3.59 3.26 3.07 2.96 2.88
    8 3.46 3.11 2.92 2.81 2.73
    9 3.36 3.01 2.81 2.69 2.61
    10 3.29 2.92 2.73 2.61 2.52
    11 3.23 2.86 2.66 2.54 2.45
    12 3.18 2.81 2.61 2.48 2.39
    13 3.14 2.76 2.56 2.43 2.35
    14 3.10 2.73 2.52 2.39 2.31
    15 3.07 2.70 2.49 2.36 2.27
    16 3.05 2.67 2.46 2.33 2.24
    17 3.03 2.64 2.44 2.31 2.22
    18 3.01 2.62 2.42 2.29 2.20
    19 3.36 3.01 2.81 2.69 2.61
    20 2.97 2.59 2.38 2.25 2.16
    21 2.96 2.57 2.36 2.23 2.14
    22 2.95 2.56 2.35 2.22 2.13
    23 2.94 2.55 2.34 2.21 2.11
    24 2.93 2.54 2.33 2.19 2.10
    25 2.92 2.53 2.32 2.18 2.09
    26 2.91 2.52 2.31 2.17 2.08
    27 2.90 2.51 2.30 2.17 2.07
    28 2.89 2.50 2.29 2.16 2.06
    29 2.89 2.50 2.28 2.15 2.06
    30 2.88 2.49 2.28 2.14 2.05
    40 2.84 2.44 2.23 2.09 2.00
    50 2.81 2.41 2.20 2.06 1.97
    60 2.79 2.39 2.18 2.04 1.95
    100 2.76 2.36 2.14 2.00 1.91
    200 2.73 2.33 2.11 1.97 1.88
    1000 2.71 2.31 2.09 1.95 1.85

    Degrees of Freedom

    Between Groups ANOVA (independent groups)

    • Numerator (effect of IV): k-1
    • Denominator (error): N-k
    • Total: N-1

    Repeated Measures ANOVA (dependent groups)

    • Numerator (effect of IV Between Groups): k-1
    • Denominator (Within Groups or Error): \((k-1) \times (P-1)\)
    • Participants: P-1
    • Total: N-1

    Factorial ANOVA (2+ IVs)

    1. Cells: \((k_1 \times k_2)\; – 1\)
      1. Remembering that “k” is the number of groups, k1 is the number of levels of the first IV and , k2 is the number of levels of the other IV.
    2. Between group for one variable (IV11): \(k_1 – 1\)
    3. Between group for the other variable (IV2): \(k_2 – 1\)
    4. Interaction: \( df_1\times df_2 \)
    5. Within group: \( df_{Total}– df_{Cells} \)
    6. Total: N - 1
      1. With N being the number of scores.

    Regression

    • Model (numerator): 1
    • Error (denominator): N-2
    • Total: N-1

    Table of q-values

    The table of q-values needed to complete Tukey's HSD pairwise comparison by hand can be found at this Real-Statistics.com webpage. There is not a separate, dedicated page in this textbook.

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