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T table (Student's t critical values)

--- --- Methodology

A t table lists the critical values of Student's t distribution: the cut-offs beyond which a test result is significant. Use the lookup box for any df, any α or an observed t, or read the printed two-tailed table below, from df 1 to ∞.

0.05 is the usual choice.

n − 1 for a one-sample test, n₁ + n₂ − 2 for a pooled two-sample test.

Two-tailed critical t at α = 0.05, df 18 ±2.101
Confidence level0.95
Rejection region|t| > 2.101
z value at the same α±1.960
Show the working, step by step

  1. A two-tailed test splits α between the two tails, so each tail holds α/2 and the cut-off has 1 − α/2 of the distribution below it.

  2. Read it from the t table (row df = 18, column α = 0.05) or compute it:

    t* = ±2.101

  3. The t value is always a little further out than the z value, because estimating the standard deviation costs precision. As df grows the two converge:

    t* 2.101 against z* 1.960

The printed two-tailed table below lists the usual α levels; this tool handles any α and one-tailed lookups.

How to read a t table

The rows are the degrees of freedom and the columns are the significance levels. The table below is two-tailed: the number in a cell is the value t* such that α of the t distribution's area sits in both tails combined.

Worked lookup for df = 18, α = 0.05, two-tailed:

  1. Find the row df = 18.
  2. Find the column 0.05.
  3. The cell reads 2.101 (highlighted below).
  4. A two-tailed test rejects H₀ when |t| > 2.101; a confidence interval multiplies the standard error by 2.101.

The one-tailed value at α = 0.05 is the 0.10 column of this table (1.734 at df = 18), because a one-tailed test puts all of α in one end instead of splitting it.

Student's t distribution, two-tailed critical values: area α in the two tails combined
df 0.10.050.020.010.005
1 6.31412.70631.82163.657127.321
2 2.9204.3036.9659.92514.089
3 2.3533.1824.5415.8417.453
4 2.1322.7763.7474.6045.598
5 2.0152.5713.3654.0324.773
6 1.9432.4473.1433.7074.317
7 1.8952.3652.9983.4994.029
8 1.8602.3062.8963.3553.833
9 1.8332.2622.8213.2503.690
10 1.8122.2282.7643.1693.581
15 1.7532.1312.6022.9473.286
18 1.7342.1012.5522.8783.197
20 1.7252.0862.5282.8453.153
25 1.7082.0602.4852.7873.078
30 1.6972.0422.4572.7503.030
40 1.6842.0212.4232.7042.971
50 1.6762.0092.4032.6782.937
60 1.6712.0002.3902.6602.915
100 1.6601.9842.3642.6262.871
∞ 1.6451.9602.3262.5762.807

One-tailed values from the two-tailed table

Printed two-tailed tables double up as one-tailed ones: the one-tailed α is the two-tailed 2α. The most-used pairs, at df = 18:

One-tailed critical values, read off the two-tailed table
One-tailed αTwo-tailed αCritical t, df 18Typical use
0.100.201.330Directional test at 10%
0.050.101.734Directional test at 5%
0.0250.052.101Two-tailed test at 5%, or 95% CI
0.010.022.552Two-tailed test at 2%, or 98% CI

A worked significance test

A sample of n = 19 readings gives t = 2.4 against H₀: μ = 100. The df are n − 1 = 18, so the two-tailed 5% cut-off is 2.101 from the table above. Because 2.4 > 2.101, the result is significant at the 5% level. The lookup box above gives the exact p-value for t = 2.4 at df = 18; the t-test calculator runs the whole test from raw data.

Why the t table is not just the z table

When the standard deviation is estimated from the sample instead of known, the test statistic carries that extra uncertainty and its distribution has heavier tails: a 2.0 is a normal two-sided 0.046 event but a 0.061 event at df = 18. Every row of the t table is therefore further from zero than the ∞ (z) row, and the gap closes as the sample grows — df 100 needs only 1.984 where z needs 1.960. That is why the t table exists at all: the normal table is wrong by exactly this amount for small samples, and that amount is not small when n is 5.

The values are computed by the same inverse-CDF routine as the critical value calculator, not transcribed from a printed table, and rounded to 3 decimal places — the standard for t.

T table (Student’s t critical values): the worked example on this page, with its result and chart
T table (Student’s t critical values): the worked example above, at a glance.

Common questions

How do you read a t table?

Find your degrees of freedom down the left column and your significance level across the top. The cell where they meet is the critical value t*. For a two-tailed test at α = 0.05 with df = 18 that cell reads 2.101: reject the null when |t| exceeds 2.101. Most printed t tables are two-tailed; the lookup box above handles either convention.

How do I find the degrees of freedom?

They come from the test, not the table. A one-sample or paired t-test has df = n − 1; a pooled two-sample t-test has df = n₁ + n₂ − 2. A confidence interval for a mean uses the same df as the test would. The degrees of freedom calculator works it out from your sample sizes.

What is the critical t value at 95% with 18 degrees of freedom?

±2.101 for a two-tailed test. That is the row df = 18, column 0.05 cell of the table above. It is the multiplier in a 95% confidence interval for a mean with n = 19, and the cut-off for a two-tailed test at the 5% level. Compare z: with known σ the same interval would use 1.960; the extra 0.141 is the price of estimating σ.

How is a one-tailed t table different?

The same numbers, read with the α doubled. A one-tailed test at α = 0.05 has the same cut-off as a two-tailed test at α = 0.10, so it is the 0.10 column, not the 0.05 column: with df = 18 that is 1.734 against 2.101. One-tailed tests are easier to pass, which is why the direction must be fixed before the data are seen.

Why does the t table have a row for infinite df?

The ∞ row is the normal (z) limit: as the sample grows, estimating σ costs less, the t distribution becomes the normal distribution, and the 0.05 two-tailed value settles at 1.960. That is the last row of the table above and why textbooks note "use z when n is large". For df = 100 the value is 1.984, already close.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.

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