T vs Z Critical Values Chart

Quick answer: the critical t value (t*) is always larger than the critical z value (z*), and it shrinks toward z* as degrees of freedom grow. For a two-tailed test at α = 0.05, t* = 12.706 at df = 1, 2.228 at df = 10, 2.042 at df = 30, and approaches z* = 1.960 (the t-table’s ∞ row).

Pick one- or two-tailed and α, then slide df to watch the gap close. The table under the chart lists t*, z*, and the gap for the df you are most likely to need.

t* vs z* chart

Two-tailed α = 0.05 (95% confidence, df = 10): t* = 2.228 vs z* = 1.960. t* is 0.268 larger (13.7% above z*).

For these settings t* is within 5% of z* from df = 26 and within 1% from df = 123.

Two-tailed α = 0.05: t* vs z* by degrees of freedom

dft*z*t* − z*% above z*
112.7061.96010.746548.3%
24.3031.9602.343119.5%
33.1821.9601.22262.3%
42.7761.9600.81641.6%
52.5711.9600.61131.2%
62.4471.9600.48724.8%
72.3651.9600.40520.7%
82.3061.9600.34617.7%
92.2621.9600.30215.4%
102.2281.9600.26813.7%
112.2011.9600.24112.3%
122.1791.9600.21911.2%
132.1601.9600.20010.2%
142.1451.9600.1859.4%
152.1311.9600.1718.7%
162.1201.9600.1608.2%
172.1101.9600.1507.7%
182.1011.9600.1417.2%
192.0931.9600.1336.8%
202.0861.9600.1266.4%
212.0801.9600.1206.1%
222.0741.9600.1145.8%
232.0691.9600.1095.6%
242.0641.9600.1045.3%
252.0601.9600.1005.1%
262.0561.9600.0964.9%
272.0521.9600.0924.7%
282.0481.9600.0884.5%
292.0451.9600.0854.3%
302.0421.9600.0824.2%
402.0211.9600.0613.1%
502.0091.9600.0492.5%
602.0001.9600.0402.0%
801.9901.9600.0301.5%
1001.9841.9600.0241.2%
1201.9801.9600.0201.0%
2001.9721.9600.0120.6%
5001.9651.9600.0050.3%
∞ (z)1.9601.9600.0000.0%

Critical values computed for Student’s t and the standard normal distribution, rounded to 3 decimals; they match the t-table on our homepage wherever both list the same df.

Worked example

Question: you build a 95% confidence interval from n = 11 observations. How much wider is it with t than it would be with z?

Step 1. df = n − 1 = 10. A 95% interval is two-tailed, α = 0.05.

Step 2. From the chart (or the two-tailed t-table): t* = 2.228. The z critical value is z* = 1.960.

Answer: the margin of error is t*/z* times as wide, about 13.7% wider with t. At df = 5 the gap is 31.2%; at df = 30 it is only 4.2%.

FAQ

Why is the critical t value bigger than the critical z value?

Student’s t has heavier tails than the normal curve because the standard deviation is estimated from the sample. To leave the same α in the tail, the cutoff has to sit further out, so t* > z* for every finite df.

When is t* close enough to z*?

It depends on α. For a two-tailed test at α = 0.05, t* is within 5% of z* = 1.960 from df = 26 and within 1% from df = 123. Smaller α needs more df: choose your α on the chart to see the exact df.

What is the ∞ row in the t-table?

It is the limit as df grows without bound, which is the standard normal z* value. For two-tailed α = 0.05 the ∞ row is 1.960, the same as the z critical value on the z-table.

Do one-tailed and two-tailed tests use different values?

Yes. A two-tailed test splits α across both tails, so two-tailed α = 0.05 uses z* = 1.960 (t* = 2.228 at df = 10), while one-tailed α = 0.05 uses z* = 1.645 (t* = 1.812 at df = 10).

Should I use t or z?

Use t when the population standard deviation is unknown and estimated from the sample (the usual case for means). Use z when σ is known or for proportions. For z-scores and normal areas, see z-table.com and its printable z chart.

Related t-table pages