Two-Tailed Hypothesis Testing: Formula & Example

Q: What does hypothesis testing two-tailed mean?

It means the alternative hypothesis is “not equal,” so the rejection region is split between the two tails, with α/2 in each. A result far above or far below the null value counts as evidence against H 0 .

Q: What are the two-tailed critical values at α = 0.05?

For a standard normal reference, ±1.96. For a t distribution they depend on the degrees of freedom: ±2.0639 with 24 degrees of freedom and ±1.9673 with 326 (the NIST example).

Q: How does the two-tailed p-value relate to the one-tailed p-value?

For a symmetric reference distribution such as t, the two-tailed p-value is twice the one-tailed value for the direction observed, as in Example 2 (0.0285 and 0.0569).

A two-tailed hypothesis test asks whether a parameter differs from a stated value in either direction. You split the significance level α across both tails of the test statistic’s distribution and reject the null hypothesis when the statistic lands far enough out on either side. This page gives the rule, a worked example with every step of the arithmetic, and the code to reproduce it.

Definition in one paragraph

In a two-tailed test (also called a two-sided or non-directional test), the alternative hypothesis states only that the parameter is not equal to the null value. For two means, NIST/SEMATECH writes the hypotheses as H₀: μ₁ = μ₂ against H_a: μ₁ ≠ μ₂. Because a difference in either direction counts as evidence against H₀, the rejection region sits in both tails. If you only want to detect an increase or only a decrease, that is a one-tailed test; the one-tailed vs two-tailed test comparison covers when each is legitimate. For the null hypothesis itself, see null hypothesis explained.

Hypothesis testing two-tailed: the rejection rule and formula

The NIST/SEMATECH e-Handbook gives the rejection regions for the two-sample t-test. Its test statistic is

T = (Ȳ₁ − Ȳ₂) / √(s₁²/N₁ + s₂²/N₂)

Ȳ₁, Ȳ₂: the two sample means.
s₁², s₂²: the two sample variances.
N₁, N₂: the two sample sizes.
α: the significance level you fixed before looking at the data (for example 0.05).
ν: the degrees of freedom of the t distribution.
t_1−α/2,ν: the critical value that leaves α/2 in the upper tail.

The handbook’s rejection regions for the three possible alternatives are:

Alternative hypothesis	Reject H₀ if	Tails
μ₁ ≠ μ₂	\|T\| > t_1−α/2,ν	Two
μ₁ > μ₂	T > t_1−α,ν	One (upper)
μ₁ < μ₂	T < t_α,ν	One (lower)

The vertical bars mean absolute value, so |T| > t_1−α/2,ν is the same as “T is above the upper critical value or below its negative.” With α = 0.05, each tail holds 0.025. When the reference distribution is the standard normal, that cutoff is z = ±1.96; for a t distribution it depends on the degrees of freedom and is always larger in size than 1.96. To look up or compute these cutoffs, see how to find a critical value and what the t distribution is.

Worked example with real numbers

Example 1: two-sample test on published data (NIST)

The NIST/SEMATECH handbook tests whether the mean fuel economy (miles per gallon) of U.S. and Japanese cars differs, using the AUTO83B data set:

U.S. cars: N₁ = 249, mean = 20.14458, standard deviation = 6.41470
Japanese cars: N₂ = 79, mean = 30.48101, standard deviation = 6.10771
Test statistic T = −12.62059, degrees of freedom ν = 326, α = 0.05

The two-tailed critical value is t_1−α/2,ν = 1.9673. Because |T| = 12.62059 is greater than 1.9673, the handbook rejects H₀ and concludes the means differ. For contrast, the upper one-tailed critical value at the same α and ν is 1.6495: the two-tailed cutoff is further out because only 0.025 is placed in each tail instead of 0.05 in one.

Example 2: one-sample test, arithmetic shown

Suppose a sample of n = 25 has mean x̄ = 52.4 and standard deviation s = 6, and you test H₀: μ = 50 against H_a: μ ≠ 50 at α = 0.05.

Standard error: s/√n = 6/√25 = 6/5 = 1.2
Test statistic: t = (52.4 − 50)/1.2 = 2.4/1.2 = 2.000
Degrees of freedom: n − 1 = 24
Two-tailed critical value: t_0.975,24 = 2.0639, so the rejection region is t > 2.0639 or t < −2.0639
Decision: |2.000| < 2.0639, so you do not reject H₀
Two-tailed p-value: 2 × P(T₂₄ > 2.000) = 2 × 0.02847 = 0.0569, which is above 0.05, matching the decision
95% confidence interval: 52.4 ± 2.0639 × 1.2 = 52.4 ± 2.48, or (49.92, 54.88); it contains 50

The same data tested one-tailed (H_a: μ > 50) give p = 0.0285 and would be rejected at 0.05. That is exactly why the direction must be fixed before seeing the data (see mistakes below). All figures in Example 2 were computed with SciPy and double-checked by hand.

Interpretation: how to read the result

What you see	Reading
\|T\| greater than the two-tailed critical value	Reject H₀. The statistic is in the rejection region in whichever tail it falls.
\|T\| less than or equal to the critical value	Do not reject H₀. This is not proof the values are equal.
Two-tailed p ≤ α	Reject H₀ (equivalent to the critical-value rule).
Confidence interval (1 − α) excludes the null value	Reject H₀ at level α; if it contains the null value, do not reject.
Negative T with a large \|T\|	Still significant: the sign shows direction (sample 1 lower, as in Example 1), the rule uses \|T\|.

For a statistic with a symmetric distribution such as t or z, the two-tailed p-value is twice the one-tailed p-value for the observed direction, as Example 2 shows (0.0285 × 2 ≈ 0.0569). For more on reading the statistic itself, see how to interpret the t-statistic.

Assumptions and common mistakes

Choosing the direction after seeing the data. Switching to a one-tailed test because the sample moved one way halves the p-value and inflates the false-positive rate. Decide, and record, the alternative beforehand.
Using a signed T in the comparison. The two-tailed rule compares |T| to the critical value, or checks both tails.
Using the one-tailed critical value. At α = 0.05 and ν = 326 that is 1.6495, not 1.9673; using it rejects too easily.
Reading “fail to reject” as “no difference.” It means the data did not provide enough evidence at level α.
Ignoring the test’s own conditions. The t statistic assumes independent observations and sampling from an approximately normal population (or a large enough sample); check the conditions of the specific test you use. How to choose a statistical test helps you pick one.

How to report a two-tailed test (APA-style example)

A common reporting pattern gives the statistic, degrees of freedom, exact p-value, the test’s sidedness if it is not the default, and an interval:

A two-tailed one-sample t test showed that the mean (M = 52.40, SD = 6.00) did not differ significantly from 50, t(24) = 2.00, p = .057, 95% CI [49.92, 54.88].

Report the exact p-value rather than “p < .05” when you have it, and state the test as two-tailed (or note any one-tailed choice and its pre-specified justification). See how to report confidence intervals for the interval part.

How to calculate a two-tailed test in Excel, R or Python

Excel

Microsoft documents T.INV.2T(probability, deg_freedom) as the two-tailed inverse of the Student’s t distribution; it finds the value that T.DIST.2T(x, deg_freedom) maps to the probability you supply, so T.DIST.2T gives the two-tailed p-value. For Example 2:

=T.INV.2T(0.05, 24)        -> critical value, about 2.0639
=T.DIST.2T(ABS(2), 24)     -> two-tailed p-value, about 0.0569

R

The R documentation for t.test lists alternative = c("two.sided", "less", "greater"), with “two.sided” as the default and mu (default 0) as the null value.

x <- c(52, 48, 55, 51, 57, 49, 53, 54)
t.test(x, mu = 50, alternative = "two.sided")   # t = 2.2238, df = 7, p-value = 0.0615
qt(0.975, 24)                                   # 2.0639 (two-tailed critical value, df = 24)

Python (SciPy)

SciPy’s ttest_1samp(a, popmean, ..., alternative='two-sided') defaults to the two-sided alternative, which tests whether the mean differs from popmean, and returns the statistic, p-value, degrees of freedom and a confidence interval.

from scipy import stats
x = [52, 48, 55, 51, 57, 49, 53, 54]
res = stats.ttest_1samp(x, popmean=50, alternative="two-sided")
print(res.statistic, res.pvalue)      # 2.2238  0.0615

# Example 2 from summary statistics
n, m, mu, s = 25, 52.4, 50, 6
t = (m - mu) / (s / n**0.5)           # 2.0
p = 2 * stats.t.sf(abs(t), n - 1)     # 0.0569
crit = stats.t.ppf(0.975, n - 1)      # 2.0639

FAQ

What does hypothesis testing two-tailed mean?

It means the alternative hypothesis is “not equal,” so the rejection region is split between the two tails, with α/2 in each. A result far above or far below the null value counts as evidence against H₀.

What are the two-tailed critical values at α = 0.05?