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8.3.1: Table of Critical t-scores

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    17362
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    Student's \(t\) Distribution

    Table \(\PageIndex{1}\), which follows Figure \(\PageIndex{1}\), shows the critical t-score.  If the absolute value of your calculated t-score is bigger (more extreme) than the critical t-score, then you reject the null hypothesis.  In Figure \(\PageIndex{1}\), the critical t-score is represented by the line; if the absolute value of the calculated t-score is to the left (where the alpha sign is, \(α\) ), then the null hypothesis should be rejected.  

    Standard normal curve with line representing the critical t-score on the extreme right side showing the Alpha.

    Figure \(\PageIndex{1}\)- Upper critical values of Student's t Distribution (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax)

    Note

    Remember

    Critical < |Calculated| = Reject null = means are different = p<.05

    Critical > |Calculated| = Retain null = means are similar = p>.05

     

    Table of Critical Values for Student's \(t\)

    Table \(\PageIndex{1}\) shows the critical t-scores for different probabilities (p-values) that represent how likely it would be to get a calculated t-scores this big if the sample was really from the population, by the Degrees of Freedom (df, to represent the size of the sample). More information Degrees of Freedom is below the table.  If we think that the sample is not from the population, we would expect a larger t-score and want a small p-value.  

    Table \(\PageIndex{1}\)- Table of Critical t-values
    Degrees of Freedom (df) p = 0.10 p = 0.05 p = 0.025 p = 0.01
    1 3.078 6.314 12.706 31.821
    2 1.886 2.920 4.303 6.965
    3 1.638 2.353 3.182 4.541
    4 1.533 2.132 2.776 3.747
    5 1.476 2.015 2.571 3.365
    6 1.440 1.943 2.447 3.143
    7 1.415 1.895 2.365 2.998
    8 1.397 1.860 2.306 2.896
    9 1.383 1.833 2.262 2.821
    10 1.372 1.812 2.228 2.764
    11 1.363 1.796 2.201 2.718
    12 1.356 1.782 2.179 2.681
    13 1.350 1.771 2.160 2.650
    14 1.345 1.761 2.145 2.624
    15 1.341 1.753 2.131 2.602
    16 1.337 1.746 2.120 2.583
    17 1.333 1.740 2.110 2.567
    18 1.330 1.734 2.101 2.552
    19 1.328 1.729 2.093 2.539
    20 1.325 1.725 2.086 2.528
    21 1.323 1.721 2.080 2.518
    22 1.321 1.717 2.074 2.508
    23 1.319 1.714 2.069 2.500
    24 1.318 1.711 2.064 2.492
    25 1.316 1.708 2.060 2.485
    26 1.315 1.706* 2.056 2.479
    27 1.314 1.703 2.052 2.473
    28 1.313 1.701 2.048 2.467
    29 1.311 1.699 2.045 2.462
    30 1.310 1.697 2.042 2.457
    40 1.303 1.684 2.021 2.423
    60 1.296 1.671 2.000 2.390
    100 1.290 1.660 1.984 2.364
    \(\infty\) 1.282 1.645 1.960 2.326

    Degrees of Freedom

    • One-Sample t-test:  N-1
    • Independent Sample t-test:  N1 + N2 - 2
    • Dependent Sample t-test:  N-1 (in which N is the number of pairs)

    Because tables are limited by size, not all critical t-scores are listed.  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 example, if your df = 49. you would use the df row for 40, (so a p=0.05 would have a critical value of 1.684).  If your df=55, you would use the df row for 60.  It's sorta silly, but, mathematically, any score is closer to df=100 than infinity (\(\infty\)), so if your sample is more than 100 scores, use df=100.  
    • Another option is to always we round down.  For our example of df=49, we would still use the df row for 40.  If your df=55, you would still use the df row for 40.  And if your sample is more than 100 scores, use df=100.  This option avoids inflating Type I Error (false positives).

    Ask your professor which option you should use!

    Contributors and Attributions


    8.3.1: Table of Critical t-scores is shared under a CC BY license and was authored, remixed, and/or curated by Michelle Oja.