T Table (Student's t Distribution)

What a t table is for

A t table answers one narrow question fast: for a given number of degrees of freedom and a given tail area, where does the t distribution cross that threshold? Every cell in the grid is a critical value. If a test statistic lands beyond the cell you looked up, the result falls in the tail you chose.

The table exists because samples rarely arrive with a known population standard deviation. The normal curve assumes you know sigma, which almost never happens outside a textbook problem. Swap sigma for the sample standard deviation s and the ratio (mean - mu) / (s / sqrt(n)) stops following a normal curve. It follows a t distribution, which has heavier tails because s is itself an estimate. William Sealy Gosset worked this out at the Guinness brewery and published it in 1908 under the pen name Student, which is why the tables still say Student's t.

The NIST Reference Tables (Table 9.3, 2019 revision) and the Devore Table A.5 reproduced by Purdue both use the same layout: rows of degrees of freedom, columns of tail area. Their cells agree to three decimals, and every value on this page comes from those two tables. The widget below carries df 1 through 30 plus the rows for 40, 60, 120 and infinity, at 90, 95 and 99 percent two-tailed.

Degrees of freedom choose the row

Degrees of freedom count the pieces of information left after you spend some on estimates. A one-sample t test spends one on the sample mean, so df = n - 1. A two-sample pooled test spends one on each mean, so df = n1 + n2 - 2. A paired test works on the differences, so df = pairs - 1.

The reason is arithmetic. The deviations x - xbar always add to zero, so once you know n - 1 of them the last one is fixed. Only n - 1 deviations carry new information, and the t distribution widens or narrows to match that count.

Small df means fat tails. At df = 1 the 95 percent two-tailed critical value is 12.706. At df = 30 it is 2.042. At df = 120 it is 1.980, and at infinite df it is 1.960, the familiar z. The table makes the shrink visible: it lists every df from 1 to 30, then jumps to 40, 60, 120 and infinity because the gaps stop mattering. A sample of 42 gives df = 41, which is not a row on any printed table.

When df falls between listed rows, round down. Using the df = 40 row for a sample with df = 41 gives a slightly larger critical value, which makes the interval wider and the test harder to pass. That is the safe direction to be wrong in, and it is why printed tables jump in steps instead of interpolating.

One tail or two tails, and which column that means

Before you touch the grid, decide how alpha gets spent. A two-tailed test at 95 percent confidence leaves 5 percent to split, 2.5 percent in each tail, so you read the 0.025 column. A one-tailed test puts the whole 5 percent in one tail and reads the 0.05 column.

Columns are usually labeled by tail area, not by confidence level. That is the most common place to lose the answer: the 0.05 column of a two-tailed table is the 90 percent confidence column, not the 95 percent one. At df = 10 the 0.05 column reads 1.812 while the 0.025 column reads 2.228. Pick the wrong one and a confidence interval comes out roughly 19 percent too narrow.

Direction only matters for one-tailed tests. The t distribution is symmetric, so the left critical value is the negative of the right one and printed tables show positive values only. If the alternative says the mean is lower, reject when t falls below minus the cell value.

Three lookups you can follow

Reading the grid takes two moves: find the row, then the column. These three lookups cover most of the requests this page gets.

df = 11 at 95 percent. Row 11, column 0.025, cell 2.201. That is the multiplier for every 12-person sample building a 95 percent interval, and it is the default in the widget below.

df = 24 at 99 percent. Row 24, column 0.005, cell 2.797. The same row at 95 percent reads 2.064, so buying the extra confidence costs about 35 percent more width.

df = 5 at 90 percent. Row 5, column 0.05, cell 2.015. With only six observations the 90 percent multiplier is already larger than the 95 percent z value of 1.960, which shows how much the small sample is charging you.

Each of these is a boundary, not a probability. The cell tells you where 2.5 percent of the distribution sits beyond it. Turning that into a p-value takes software or a second lookup between two columns, which is the job of a p-value calculator.

t against z: what the extra width costs

At infinite df the t distribution becomes the normal distribution, so the last row of any t table reproduces the z values: 1.645, 1.960, 2.576. The useful part is how quickly the rows get there.

At df = 5 the 95 percent critical value is 31 percent wider than z. At df = 11 the gap is 12.3 percent. At df = 24 it is down to 5.3 percent. The folk rule that 30 observations are enough to use z is a rough version of this: at df = 29 the gap is 4.3 percent, with 2.045 against 1.960.

Modern practice skips the rule and uses t whenever sigma is estimated, because software makes the lookup free. The choice still matters in a hand calculation, in an exam, and in any report where somebody multiplied by 1.96 on 8 subjects and called it a 95 percent interval. On 8 subjects the correct multiplier is 2.365, so the quoted interval is about 17 percent too tight.

Hand-building a 95 percent interval, step by step

The formula is one line: sample mean plus or minus t times s divided by the square root of n. The t comes from df = n - 1 and the confidence level you want.

Take 12 measurements with a mean of 42.3 and a standard deviation of 8.5. df = 12 - 1 = 11, and row 11 at 95 percent gives 2.201. The standard error is 8.5 / sqrt(12) = 8.5 / 3.4641 = 2.4537. The margin is 2.201 x 2.4537 = 5.4007. The interval runs from 42.3 - 5.4007 = 36.90 to 42.3 + 5.4007 = 47.70.

A second sample has 25 readings, a mean of 118.4 and s = 15.2. At 99 percent: df = 24, t = 2.797, standard error = 15.2 / 5 = 3.04, margin = 2.797 x 3.04 = 8.5029, interval 109.90 to 126.90. The same data at 95 percent uses 2.064 and gives 6.2746, for an interval of 112.13 to 124.67. Buying the extra confidence costs 2.23 units of width here.

NIST's page on confidence limits for the mean sets out the same formula and adds a caution worth repeating. A 95 percent interval does not mean the true mean has a 95 percent chance of sitting inside this particular interval. The mean is fixed and the interval is what moves. The 95 percent describes the method: intervals built this way cover the truth about 95 times out of 100.

Where the t table stops being the right tool

The table assumes independent observations and a population that is roughly normal, or a sample large enough for the central limit theorem to carry the mean. Neither assumption is checked by the lookup. A heavily skewed sample of 6 with one extreme value gets a wide interval that still may not cover the truth, and no critical value repairs that.

The one-sample case is the easy one. Two-sample tests come in two flavors: the pooled version uses df = n1 + n2 - 2 and assumes equal variances, while the Welch version computes a fractional df such as 17.4 that no printed table contains. Software handles Welch; a printed table forces you to round down to 17, which is again the conservative direction.

Proportions are a separate family. A confidence interval for a share uses z, not t, because its standard error comes from the estimated proportion rather than from a measured s.

The table also gives a boundary, not an exact p-value. If t = 2.5 at df = 11, it sits between the 2.201 cell (0.025) and the 2.718 cell (0.01), so the two-tailed p is somewhere between 0.02 and 0.05. Getting the exact number needs the incomplete beta function, which is what a p-value calculator does. For the interval itself, though, the table is all you need, and the confidence interval calculator checks the arithmetic if you want a second opinion.

The rows worth remembering

Three numbers cover a large share of everyday work: 2.201 for df = 11 at 95 percent, 2.042 for df = 30 at 95 percent, and 1.960 for the normal limit. The full slice below is the one to bookmark.

Source: NIST Reference Tables, Table 9.3 (Student t-Variate, 2019 revision) and Devore's Table A.5 as reproduced by the Purdue statistics department. The NIST PDF is at nist.gov. Read down the 0.025 column and you can watch 12.706 fall to 1.960 in nine rows.

The widget repeats this grid for df 1 through 30 plus 40, 60, 120 and infinity, and it does the arithmetic that usually follows a lookup. Enter df and a confidence level to get the critical value, then add a sample mean, standard deviation and size to get the standard error, the margin, the interval, and a one-sample t statistic against any null value you type. If the df you enter does not equal n - 1, the output says so rather than hiding the mismatch.

One reminder about precision: the table is rounded to three decimals, which is plenty for reporting an interval but not for reproducing a p-value to four places. Use the table for the decision and software for the decimal.