Editing Student's t-distribution (section)

===Table of selected values===
The following table lists values for {{mvar|t}}&nbsp;distributions with {{mvar|ν}} degrees of freedom for a range of one-sided or two-sided critical regions. The first column is {{mvar|ν}}, the percentages along the top are confidence levels <math>\ \alpha\ ,</math> and the numbers in the body of the table are the <math>t_{\alpha,n-1}</math> factors described in the section on [[#Confidence intervals|confidence intervals]].

The last row with infinite {{mvar|ν}} gives critical points for a normal distribution since a {{mvar|t}}&nbsp;distribution with infinitely many degrees of freedom is a normal distribution. (See [[#Related distributions|Related distributions]] above).

{| class="wikitable"
|-
! ''One-sided''
! 75%
! 80%
! 85%
! 90%
! 95%
! 97.5%
! 99%
! 99.5%
! 99.75%
! 99.9%
! 99.95%
|-
! ''Two-sided''
! 50%
! 60%
! 70%
! 80%
! 90%
! 95%
! 98%
! 99%
! 99.5%
! 99.8%
! 99.9%
|-
!1
|1.000
|1.376
|1.963
|3.078
|6.314
|12.706
|31.821
|63.657
|127.321
|318.309
|636.619
|-
!2
|0.816
|1.061
|1.386
|1.886
|2.920
|4.303
|6.965
|9.925
|14.089
|22.327
|31.599
|-
!3
|0.765
|0.978
|1.250
|1.638
|2.353
|3.182
|4.541
|5.841
|7.453
|10.215
|12.924
|-
!4
|0.741
|0.941
|1.190
|1.533
|2.132
|2.776
|3.747
|4.604
|5.598
|7.173
|8.610
|-
!5
|0.727
|0.920
|1.156
|1.476
|2.015
|2.571
|3.365
|4.032
|4.773
|5.893
|6.869
|-
!6
|0.718
|0.906
|1.134
|1.440
|1.943
|2.447
|3.143
|3.707
|4.317
|5.208
|5.959
|-
!7
|0.711
|0.896
|1.119
|1.415
|1.895
|2.365
|2.998
|3.499
|4.029
|4.785
|5.408
|-
!8
|0.706
|0.889
|1.108
|1.397
|1.860
|2.306
|2.896
|3.355
|3.833
|4.501
|5.041
|-
!9
|0.703
|0.883
|1.100
|1.383
|1.833
|2.262
|2.821
|3.250
|3.690
|4.297
|4.781
|-
!10
|0.700
|0.879
|1.093
|1.372
|1.812
|2.228
|2.764
|3.169
|3.581
|4.144
|4.587
|-
!11
|0.697
|0.876
|1.088
|1.363
|1.796
|2.201
|2.718
|3.106
|3.497
|4.025
|4.437
|-
!12
|0.695
|0.873
|1.083
|1.356
|1.782
|2.179
|2.681
|3.055
|3.428
|3.930
|4.318
|-
!13
|0.694
|0.870
|1.079
|1.350
|1.771
|2.160
|2.650
|3.012
|3.372
|3.852
|4.221
|-
!14
|0.692
|0.868
|1.076
|1.345
|1.761
|2.145
|2.624
|2.977
|3.326
|3.787
|4.140
|-
!15
|0.691
|0.866
|1.074
|1.341
|1.753
|2.131
|2.602
|2.947
|3.286
|3.733
|4.073
|-
!16
|0.690
|0.865
|1.071
|1.337
|1.746
|2.120
|2.583
|2.921
|3.252
|3.686
|4.015
|-
!17
|0.689
|0.863
|1.069
|1.333
|1.740
|2.110
|2.567
|2.898
|3.222
|3.646
|3.965
|-
!18
|0.688
|0.862
|1.067
|1.330
|1.734
|2.101
|2.552
|2.878
|3.197
|3.610
|3.922
|-
!19
|0.688
|0.861
|1.066
|1.328
|1.729
|2.093
|2.539
|2.861
|3.174
|3.579
|3.883
|-
!20
|0.687
|0.860
|1.064
|1.325
|1.725
|2.086
|2.528
|2.845
|3.153
|3.552
|3.850
|-
!21
|0.686
|0.859
|1.063
|1.323
|1.721
|2.080
|2.518
|2.831
|3.135
|3.527
|3.819
|-
!22
|0.686
|0.858
|1.061
|1.321
|1.717
|2.074
|2.508
|2.819
|3.119
|3.505
|3.792
|-
!23
|0.685
|0.858
|1.060
|1.319
|1.714
|2.069
|2.500
|2.807
|3.104
|3.485
|3.767
|-
!24
|0.685
|0.857
|1.059
|1.318
|1.711
|2.064
|2.492
|2.797
|3.091
|3.467
|3.745
|-
!25
|0.684
|0.856
|1.058
|1.316
|1.708
|2.060
|2.485
|2.787
|3.078
|3.450
|3.725
|-
!26
|0.684
|0.856
|1.058
|1.315
|1.706
|2.056
|2.479
|2.779
|3.067
|3.435
|3.707
|-
!27
|0.684
|0.855
|1.057
|1.314
|1.703
|2.052
|2.473
|2.771
|3.057
|3.421
|3.690
|-
!28
|0.683
|0.855
|1.056
|1.313
|1.701
|2.048
|2.467
|2.763
|3.047
|3.408
|3.674
|-
!29
|0.683
|0.854
|1.055
|1.311
|1.699
|2.045
|2.462
|2.756
|3.038
|3.396
|3.659
|-
!30
|0.683
|0.854
|1.055
|1.310
|1.697
|2.042
|2.457
|2.750
|3.030
|3.385
|3.646
|-
!40
|0.681
|0.851
|1.050
|1.303
|1.684
|2.021
|2.423
|2.704
|2.971
|3.307
|3.551
|-
!50
|0.679
|0.849
|1.047
|1.299
|1.676
|2.009
|2.403
|2.678
|2.937
|3.261
|3.496
|-
!60
|0.679
|0.848
|1.045
|1.296
|1.671
|2.000
|2.390
|2.660
|2.915
|3.232
|3.460
|-
!80
|0.678
|0.846
|1.043
|1.292
|1.664
|1.990
|2.374
|2.639
|2.887
|3.195
|3.416
|-
!100
|0.677
|0.845
|1.042
|1.290
|1.660
|1.984
|2.364
|2.626
|2.871
|3.174
|3.390
|-
!120
|0.677
|0.845
|1.041
|1.289
|1.658
|1.980
|2.358
|2.617
|2.860
|3.160
|3.373
|-
!∞
|0.674
|0.842
|1.036
|1.282
|1.645
|1.960
|2.326
|2.576
|2.807
|3.090
|3.291
|-
! ''One-sided''
! 75%
! 80%
! 85%
! 90%
! 95%
! 97.5%
! 99%
! 99.5%
! 99.75%
! 99.9%
! 99.95%
|-
! ''Two-sided''
! 50%
! 60%
! 70%
! 80%
! 90%
! 95%
! 98%
! 99%
! 99.5%
! 99.8%
! 99.9%
|}

; Calculating the confidence interval :

Let's say we have a sample with size&nbsp;11, sample mean&nbsp;10, and sample variance&nbsp;2. For 90% confidence with 10&nbsp;degrees of freedom, the one-sided {{mvar|t}}&nbsp;value from the table is 1.372&nbsp;. Then with confidence interval calculated from

:<math>\ \overline{X}_n \pm t_{\alpha,\nu}\ \frac{S_n}{\ \sqrt{n\ }\ }\ ,</math>

we determine that with 90% confidence we have a true mean lying below

:<math>\ 10 + 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ } = 10.585 ~.</math>

In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean.

And with 90% confidence we have a true mean lying above

:<math>\ 10 - 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ } = 9.414 ~.</math>

In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean.

So that at 80% confidence (calculated from 100%&nbsp;−&nbsp;2&nbsp;×&nbsp;(1&nbsp;−&nbsp;90%) = 80%), we have a true mean lying within the interval

:<math>\left(\ 10 - 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ },\ 10 + 1.372\ \frac{ \sqrt{2\ } }{\ \sqrt{11\ }\ }\ \right) = (\ 9.414,\ 10.585\ ) ~.</math>

Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method; see [[confidence interval]] and [[prosecutor's fallacy]].

Nowadays, statistical software, such as the [[R (programming language)|R programming language]], and functions available in many [[Spreadsheet|spreadsheet programs]] compute values of the {{mvar|t}}&nbsp;distribution and its inverse without tables.