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1.4: Table of Critical Values of r

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    When using this table, we're following the general pattern of rejecting the null hypothesis when the calculated value is larger than the critical value. The only difference from what we've been doing is the null hypothesis is not about whether group means are similar or not.

    Note

    (Critical \(<\) Calculated) \(=\) Reject null \(=\) There is a linear relationship. \(= p<.05 \)

    (Critical \(>\) Calculated) \(=\) Retain null \(=\) There is not a linear relationship. \(= p>.05\)

    Degrees of Freedom

    For Pearson's r, the Degrees of Freedom are N-2 in which N is the number of pairs of data. This is usually the case because you get two different pieces of information from each participant. For example, if you had 10 people who all provided their height and weight (two different quantitative variables), then N=10, so df=8. If you were comparing weight before and after an exercise program, then you would not use a Pearson's correlation since you have one quantitative variable (weight) in two groups.

    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:r(df)=calculated r, \(p\)<.05 or r(df)=calculated r, \(p\)>.05 (depending on whether your calculated correlation is more extreme than the critical score that you find in Table \(\PageIndex{1}\)).

    Table of Critical Values of r

    Table \(\PageIndex{1}\) is a simplified and accessible version of the table in Real Statistics Using Excel by Dr. Charles Zaiontz. Table \(\PageIndex{1}\) shows the critical scores of Pearson's r for different probabilities (p-values) that represent how likely it would be to get a calculated correlation this extreme if the two variables were unrelated in the population, by the Degrees of Freedom (df) to represent the size of the sample.

    Table \(\PageIndex{1}\)- Critical Values for Pearson's r
    Degrees of Freedom (df) p = 0.1 p = 0.05 p = 0.01
    1 0.988 0.997 1.000
    2 0.900 0.950 0.990
    3 0.805 0.878 0.959
    4 0.729 0.811 0.917
    5 0.669 0.754 0.875
    6 0.621 0.707 0.834
    7 0.582 0.666 0.798
    8 0.549 0.632 0.765
    9 0.521 0.602 0.735
    10 0.497 0.576 0.708
    11 0.476 0.553 0.684
    12 0.458 0.532 0.661
    13 0.441 0.514 0.641
    14 0.426 0.497 0.623
    15 0.412 0.482 0.606
    16 0.400 0.468 0.590
    17 0.389 0.456 0.575
    18 0.378 0.444 0.561
    19 0.369 0.433 0.549
    20 0.360 0.423 0.537
    21 0.352 0.413 0.526
    22 0.344 0.404 0.515
    23 0.337 0.396 0.505
    24 0.330 0.388 0.496
    25 0.323 0.381 0.487
    26 0.317 0.374 0.479
    27 0.311 0.367 0.471
    28 0.306 0.361 0.463
    29 0.301 0.355 0.456
    30 0.296 0.349 0.449
    35 0.275 0.325 0.418
    40 0.257 0.304 0.393
    45 0.243 0.288 0.372
    50 0.231 0.273 0.354
    60 0.211 0.250 0.325
    70 0.195 0.232 0.302
    80 0.183 0.217 0.283
    90 0.173 0.205 0.267
    100 0.164 0.195 0.254
    150 0.134 0.159 0.208
    200 0.116 0.138 0.181
    300 0.095 0.113 0.148
    400 0.082 0.098 0.128
    500 0.073 0.088 0.115
    700 0.062 0.074 0.097
    1000 0.052 0.062 0.081
    5000 0.023 0.028 0.036

    Because tables are limited by size, not all critical values are listed. For example, if you had 100 participants, your Degrees of Freedom would be 98 (df=N-2=100-2=98=100). However, the table provides df=90 or df=100. 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 r (98), that would mean that we would use the Degrees of Freedom of 100 because 98 is closer to 100 than to 90. That would mean that the critical r-value for r(98) would be 0.194604 for a p-value of 0.05.
    • Another option is to always we round down. For our example of N=100, we use the Degrees of Freedom of 90 because it is the next lowest df listed. That would mean that the critical r-value for r(98) would be 0.204968 for a p-value of 0.05. 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 , regardless of which number is listed in the table; the table is used to decide if the null hypothesis should be rejected or retained.

    Contributors and Attributions

    Dr. MO (Taft College)

    & Real Statistics Using Excel by Dr. Charles Zaiontz