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Appendix 3: Critical Values of t

  • Page ID
    81434
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    The table below gives values of \(t(α,ν)\) where \(α\) defines the confidence level and \(ν\) defines the degrees of freedom. Values for \(α\) are defined as follows

    \[α = 1 - \textrm{confidence level (as fraction)}\]

    For example, for a 95% confidence level, \(α = 1 - 0.95 = 0.05\). The degrees of freedom is the number of independent measurements given any constraints that we place on the measurements. For example, if we have n measurements and we calculate their mean, \(\bar{x}\), then we have n - 1 degrees of freedom because the mean, \(\bar{x}\), and the values for the first four measurements, x1, x2, x3, and x4, removes the independence of the fifth measurement, x5, whose value is defined exactly as

    \[x_5=\bar{x} -x_1-x_2-x_3-x_4\]

    The values of t in this table are two-tailed in that they define a confidence interval that is symmetrical around the mean. For example, for a 95% confidence interval (\(α=0.05\)), half of the area not included within the confidence interval is at the far right of the distribution and half is at the far left of the distribution. For a one-tailed confidence interval, in which the excluded area is on one side of the distribution, divide the values of \(α\) in half; thus, for a one-tailed 95% confidence interval, we use values of t from the column where \(α=0.10\).

    \(ν\)

    \(α=0.10\)

    \(α=0.05\)

    \(α=0.02\)

    \(α=0.01\)

    1

    6.314

    12.706

    31.821

    63.657

    2

    2.920

    4.303

    6.965

    9.925

    3

    2.353

    3.182

    4.541

    5.841

    4

    2.132

    2.776

    3.747

    4.604

    5

    2.015

    2.571

    3.365

    4.032

    6

    1.943

    2.447

    3.143

    3.707

    7

    1.895

    2.365

    2.998

    3.499

    8

    1.860

    2.306

    2.896

    3.255

    9

    1.833

    2.262

    2.821

    3.250

    10

    1.812

    2.228

    2.764

    3.169

    12

    1.782

    2.179

    2.681

    3.055

    14

    1.761

    2.145

    2.624

    2.977

    16

    1.746

    2.120

    2.583

    2.921

    18

    1.734

    2.101

    2.552

    2.878

    20

    1.725

    2.086

    2.528

    2.845

    30

    1.697

    2.042

    2.457

    2.750

    50

    1.676

    2.009

    2.311

    2.678

    \(\infty\)

    1.645

    1.960

    2.326

    2.576


    This page titled Appendix 3: Critical Values of t is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Contributor.

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