Appendix 3: Critical Values of t
- Page ID
- 81434
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 |