How Many Decimal Places in Statistics?

The rule: round once, at the end

There are two separate questions here, and mixing them up is the whole problem.

• How much precision do you carry while calculating? All of it. Every digit the calculator holds, at every intermediate step.
• How much precision do you report? Enough for a reader to follow and check your work, and no more. Usually two to four decimals.

Rounding early is not a small approximation that washes out. The rounded value goes into a division, then that quotient goes into a tail probability, and each step can magnify the error rather than shrink it. A standard error rounded from 1.7527 to 1.8 is a change of under 3%; by the time it reaches the p-value it has become a 12% change, which is enough to move a result from one side of $\alpha$ to the other.

The practical version of the rule: never retype a number you just computed. Chain the calculation, or store the intermediate value and recall it. Retyping a rounded readout is the single most common way students introduce this error.

How many digits to report for each quantity

Use this as the default, and follow any instruction in the problem that overrides it.

Two entries deserve a comment. Expected counts genuinely are not whole numbers, and rounding them changes every term of the chi-square statistic, so leave them alone. And the decimal degrees of freedom from a two-sample t procedure are correct, not a display glitch. Why df is a decimal in a two-sample t-test explains where it comes from.

One formatting habit worth keeping: report a proportion as a proportion (0.308) or as a percent (30.8%), and say which. Do not report 0.308% for a proportion of 0.308.

Why rounding a z or t moves the p-value

A p-value is a tail area, and out in the tail a small shift in the test statistic is a large fraction of an already small area. That is why the digits you keep in $z$ or $t$ show up in the p-value you report.

Watch what a two-decimal round does to a two-sided z-test:

Rounding to two decimals moved the p-value in the third decimal place, from 0.0328 to 0.0332. Rounding to one decimal moved that same third decimal place more than seven times as far, from 0.033 to 0.036. Neither changed the decision here, but neither was harmless either, and reporting 0.036 when the answer is 0.0328 is simply wrong.

Rounding earlier than the test statistic is worse, because the error gets divided into a small number. Suppose 37 of 120 sampled households recycle and you test $H_0: p = 0.25$. The sample proportion is $\hat{p} = 37/120 = 0.308333$, and the standard error under the null is $\sqrt{0.25(0.75)/120} = 0.039528$. Full precision gives $z = 1.4757$ and $p = 0.1400$. Round $\hat{p}$ to 0.31 first and you get $z = 1.5179$ and $p = 0.1290$, an error of about 8% in the reported p-value, from a rounding that looked completely innocent.

The rounding that flips a decision

Here is the case that should end the habit for good.

A random sample of 30 measurements has $\bar{x} = 53.6$ and $s = 9.6$. Test $H_0: \mu = 50$ against $H_a: \mu \neq 50$ at $\alpha = 0.05$, with $df = 29$.

Full precision. $SE = \frac{9.6}{\sqrt{30}} = \frac{9.6}{5.477226} = 1.752712$, so $t = \frac{3.6}{1.752712} = 2.0540$ and $p = 0.0491$. Since $0.0491 \leq 0.05$, you reject $H_0$.

Standard error rounded to 1.8 first. Then $t = \frac{3.6}{1.8} = 2.0000$ and $p = 0.0549$. Since $0.0549 > 0.05$, you fail to reject $H_0$.

Same data, opposite conclusions, and the only difference is that someone wrote down 1.8 instead of 1.752712. The critical value here is $t^* = 2.045$ for $df = 29$, and the true statistic 2.0540 clears it by less than one hundredth. Rounding the standard error to one decimal is enough to lose that. Rounding it to four decimals is not, and neither is rounding the test statistic itself to two: $\frac{3.6}{1.7527} = 2.0540$ still clears $t^*$, and so does a reported $t = 2.05$.

The honest reading of this example is not that the rounding was the difference between right and wrong science. A p-value of 0.0491 and a p-value of 0.0549 are nearly the same evidence, which is the point of what is a good p-value. It is that your arithmetic should not be what decides the sentence you write. Compute it correctly, report 0.0491, and note that the result is marginal.

Keeping full precision on a calculator

Three habits on a TI-84 remove almost all of this risk.

• Use `Ans`. The last computed value is available as `Ans` (press `2nd` then `(-)`), and it carries every digit the calculator has, not the ones on screen. Compute the standard error, then divide directly into it.
• Store, do not retype. `STO` (the arrow key above `ON`) writes a value into a letter variable. Store the standard error in a variable and use that variable in the next line.
• Let the built-in test do the whole thing. `STAT` then `TESTS` runs the one-sample t-test, the one-proportion z-test, and the chi-square tests end to end at full precision, so no intermediate value is ever rounded. How to run a t-test on the TI-84 walks through the menus, and reading calculator output covers what the display is telling you.

After `1-Var Stats`, the summary values live under `VARS` then `5:Statistics`, so you can pull $\bar{x}$ and $s$ into a later calculation without copying digits by hand.

The same principle applies to any tool. On this site, the p-value calculator and the one-sample t-test calculator compute from the inputs you type at full precision, so type the raw numbers rather than a rounded statistic you worked out yourself.

Showing enough digits in written work

On free-response questions your solution is read as an argument, not just checked as a final number, so the digits you show are part of the communication. A reader should be able to take your written values and reproduce your next line.

That gives a simple standard: show the quantity that goes into the next step at enough precision that redoing the step lands on your answer. If you write $SE = 1.75$ and then $t = 2.054$, the two lines do not agree, and a reader has to guess whether you rounded or made an error. Writing $SE = 1.7527$ removes the doubt.

A few conventions worth adopting.

• Keep units on every reported quantity: 6.07 hours, 3.2 mm Hg, 30.8%.
• Round confidence interval endpoints to the same number of decimals as each other, and to a precision the context can support.
• Write $p < 0.0001$ when the p-value underflows the display. Writing $p = 0$ claims something impossible.
• Do not round a value and then use the rounded value later in the same problem. Report the rounded version; keep computing with the full one.
• If a problem says "round to three decimal places," that instruction applies to the final answer, not to the working.

The AP Statistics free-response guide covers how the rest of a written answer should be laid out.