Left Tailed Test Calculator
Run a left-tailed hypothesis test (H₀: μ ≥ μ₀ against the alternative Hₐ: μ < μ₀) without splitting the job across a table, a p-value lookup and a separate decision rule. Enter your test statistic directly, or use the optional sample-data mode to compute it from your sample mean, hypothesized mean, standard deviation and sample size: the tool returns the critical value, the p-value, the reject or fail-to-reject decision, a plain-English conclusion, and Cohen’s d when sample data is given.
Assumes a random sample, independent observations, and approximate normality (or n ≥ 30). A p-value is not the probability that H₀ is true.
Quick Answer
For a sample mean of 98 against a hypothesized mean of 100, with standard deviation 5 and n = 36 at α = 0.05, the t statistic is −2.4000 on 35 degrees of freedom, the critical value is −1.6896 and the p-value is 0.0109, so you reject the null hypothesis. Cohen’s d is −0.40, a small effect. The calculator is pre-filled with this exact case: open the sample-data section, click “Fill test statistic from sample”, then Calculate.
Key takeaways
- Left-tailed means the whole rejection region sits in the left tail: reject when the statistic falls below the critical value, equivalently when the p-value falls below α.
- The decision and the effect size are different statements. This example rejects H₀ and still only shows a small effect (d = −0.40), so report both numbers.
- Fill-from-sample always builds a t statistic and sets df = n − 1, even if you had picked the Z-test. Check the test type after filling.
- Very small p-values print as 0.0000 because the result rounds to four decimals. That is a display limit, not a claim that the probability is literally zero.
- Quick Answer
- How to run a left-tailed test with this calculator
- What “left-tailed” means, and when the tail is wrong
- Z-test or t-test?
- Worked example, checked against this exact tool
- Significant is not the same as important
- Assumptions, and four behaviours of this tool to know about
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
How to run a left-tailed test with this calculator
There are two routes in: type a statistic you already computed elsewhere, or let the sample-data mode compute it for you. Both routes end at the same Calculate button and the same four-line result.
H₀: μ ≥ μ₀ versus Hₐ: μ < μ₀. If your claim is not “below”, this is the wrong tail.
Type it in, or fill it from x̄, μ₀, s and n. The fill writes t to four decimals and sets df = n − 1.
Statistic against critical value, or p-value against α. The two comparisons always agree, boundary rounding aside.
Read the decision, then read Cohen’s d before calling the result important.
- Choose the test type. T-test for a sample standard deviation, Z-test only for a genuinely known population σ. Switching to Z hides the degrees of freedom field; switching back restores it.
- Set α. 0.05 is pre-filled. Leave the field blank and the tool quietly uses 0.05 anyway and writes it back into the field, so what you see after calculating is the α that was actually used.
- Cross-check a critical value on its own with the critical t score calculator when a coursework question asks for the boundary rather than the decision, and use the statistical significance calculator when the claim is about proportions or a two-group comparison instead of one mean.
- Report the interval as well in any write-up: the confidence interval calculator turns the same sample into the range statement that a one-sided test leaves out.
What “left-tailed” means, and when the tail is wrong
A left-tailed test is a one-sided test whose alternative hypothesis points down: the population mean is below the hypothesized value. All of α is spent in the left tail, which is what makes a one-sided test more sensitive in its chosen direction than a two-tailed test at the same α, and completely blind in the other direction.
That blindness is the discipline of the method. If the sample had come out above μ₀, this test could never reject, however extreme the value: a statistic of +2.4 on 35 degrees of freedom returns a p-value of 0.9891 here. Choose the tail from the research question before you see the data, the standard practice set out in the NIST/SEMATECH e-Handbook of Statistical Methods discussion of one-sided tests, and in the hypothesis testing chapter of OpenStax Introductory Statistics. Picking the tail after peeking at the result doubles the real false-alarm rate while the printed α stays politely at 0.05.
Z-test or t-test?
Use the t-test unless the population standard deviation is known from outside the sample, which in practice means a textbook problem or a tightly controlled measurement process. The t distribution has heavier tails at low degrees of freedom, so its critical values sit further out, and the difference shrinks as df grows.
| Test at α = 0.05, left-tailed | Critical value | What it tells you |
|---|---|---|
| Z-test (known σ) | −1.6449 | The normal boundary every one-sided 5% test is compared against |
| T-test, df = 35 (the worked example) | −1.6896 | Close to the z boundary; n = 36 is already a comfortable sample |
| T-test, df = 10 | −1.8125 | A small sample has to clear a visibly higher bar |
| T-test, df = 5 | −2.0150 | At df = 5 the same −1.8 statistic that rejects as a z fails as a t |
Every critical value in that table is this tool’s own output at α = 0.05. The pattern is the lesson: with small samples the t penalty is real, and quoting a z critical value for a t problem overstates the evidence. For tests built on variances and model comparisons rather than a single mean, the workflow changes distribution entirely, to the F statistic calculator.
Worked example, checked against this exact tool
A filling line is specified to average 100 units. A sample of 36 runs averages 98 with a standard deviation of 5, and the question is whether the true mean has dropped below spec, tested at α = 0.05.
- Open “Compute statistic from sample data”, enter x̄ = 98, μ₀ = 100, s = 5, n = 36, and click “Fill test statistic from sample”.
- The tool writes t = (98 − 100) ÷ (5 ÷ √36) = −2.4000 and sets df = 35 and the test type to T-test.
- Click Calculate: critical value −1.6896, p-value 0.0109, decision reject the null hypothesis.
- Cohen’s d prints as −0.40 (small effect), and the conclusion line reads that there is statistically significant evidence at the 5% level that the population mean is below μ₀.
Two variations show how the pieces move. Keep the same statistic and tighten α to 0.01 and the critical value moves to −2.4377, so the same data now fails to reject: the p-value of 0.0109 sits just above the stricter line, a result worth expecting rather than treating as a contradiction. Run the statistic as a Z-test instead and the critical value relaxes to −1.6449 with a p-value of 0.0082 for a statistic of −2.4, the normal distribution’s thinner tails flattering the evidence slightly. Neither variation changes the data; both change the standard the data is held to.
Significant is not the same as important
A p-value measures compatibility with H₀, not size. Cohen’s d, (x̄ − μ₀) ÷ s, measures the size of the difference in standard deviation units, and this tool bands it exactly as follows when it prints the label next to your result.
| |d| range the tool uses | Label it prints |
|---|---|
| Below 0.2 | negligible |
| 0.2 to below 0.5 | small |
| 0.5 to below 0.8 | medium |
| 0.8 and above | large |
Those bands descend from Jacob Cohen’s conventional benchmarks, offered by Cohen himself as rough rules for the behavioural sciences, not as laws of nature. The worked example lands at |d| = 0.40, small, on a result that is statistically significant: with n = 36, a two-unit shortfall on a five-unit spread is detectable without being dramatic. The American Statistical Association’s statement on p-values makes the companion point formally: a p-value does not measure the size of an effect or the importance of a result, which is why this page prints the two side by side.
Assumptions, and four behaviours of this tool to know about
The test assumes a random sample, independent observations, and a sampling distribution that is roughly normal, which the central limit theorem supplies around n = 30 and up, or which roughly normal data supplies at smaller n. A skewed small sample breaks the p-value before the arithmetic starts, and no calculator can repair that after the fact.
- Cohen’s d sticks to the last sample you filled. Fill from sample data, calculate, then type a different statistic by hand and calculate again: the d line still prints, computed from the earlier sample. It describes that sample, not the statistic you just typed. Reset clears it.
- Fill always builds a t statistic. Even with the Z-test selected, filling writes a t, switches the type to T-test and sets df = n − 1. If your σ is genuinely known, re-select Z after filling and enter the statistic the z formula gives.
- Degrees of freedom are taken as typed. A fractional df such as 2.5 is accepted and used as-is, which suits Welch-style adjusted degrees of freedom, and there is no check that df matches n − 1 when you type the statistic directly.
- Boundary results can look contradictory at four decimals. The p-value and critical value are computed by numerical approximation and printed rounded. At an exact boundary, a statistic of 0 with α = 0.5 as a Z-test prints statistic 0.0000, critical value 0.0000 and p 0.5000 alongside a reject decision, because the unrounded values sit a hair on the reject side. Trust the decision line over the rounded digits at an exact tie.
What competing calculators miss
The left-tailed corner of statistics tooling is mostly split workflows and partial answers:
- SageCalculator Left Tailed Test Calculator: the closest namesake, with the z and t formulas printed and an example that mirrors this page’s sample (48 against 50). Gap: its example is worked as a z problem from the start, and the workflow offers no effect size alongside the decision.
- Cal67 Hypothesis Testing for One Population Mean: thorough variable tables and a clean z example (bulb life, z = −1.768, reject). Gap: the tail is one mode inside a general tool, so the left-tailed reading, statistic below critical value, has to be assembled from general instructions.
- Pearson Critical Value Calculator: polished, covers z, t, chi-square and F with a shaded rejection region. Gap: it stops at the critical value. There is no p-value, no decision on your statistic and no conclusion sentence to paste into a report.
- VRC Academy t Test p Value Calculator: honest p-value interpretation table and a good misinterpretations list. Gap: p-value only, all three tails in one form, no critical value, and its worked example is two-tailed, so left-tailed users translate everything themselves.
- BraveCalculator P Value Approach Calculator: a friendly four-step z walkthrough. Gap: z-only in the worked material, with strength-of-evidence labels that encourage reading p as importance, the exact habit the ASA statement warns against.
- The gap this page fills: one dedicated left-tailed workflow, critical value and p-value and decision and a written conclusion in a single result, a fill-from-sample mode that derives the statistic students are actually given, and Cohen’s d printed next to the decision with its bands stated openly.
Frequently asked questions
When the test statistic falls below the critical value, which is the same event as the p-value falling below your significance level. At α = 0.05 with 35 degrees of freedom the critical value is −1.6896, so a t statistic of −2.4000 rejects and a t statistic of −1.0000 does not.
It is the probability, computed as if the null hypothesis were true, of getting a test statistic at least as far to the left as the one you observed. For t = −2.4000 on 35 degrees of freedom it is 0.0109. It is not the probability that the null hypothesis is true, and a small p-value is evidence against the null, not proof it is false.
Use the Z-test only when the population standard deviation is genuinely known, which is rare outside textbooks and controlled processes. With a standard deviation estimated from your sample, use the t-test with n minus 1 degrees of freedom. The two nearly agree once the sample passes about 100, because the t distribution converges to the normal.
Cohen’s d is the standardized size of the difference, (sample mean minus hypothesized mean) divided by the standard deviation. This tool labels its absolute value negligible below 0.2, small from 0.2 to below 0.5, medium from 0.5 to below 0.8, and large at 0.8 and above. It prints only when you fill the statistic from sample data, because a bare test statistic carries no effect size. Significance says an effect is detectable; d says whether it is big.
0.05 is the working convention in most fields. Use 0.01 when a false alarm is expensive, such as releasing a product batch or publishing a claim others will build on, and 0.10 only for exploratory screening where missing a real effect costs more than chasing a false one. Decide before you run the test: the same statistic of −2.4000 on 35 degrees of freedom rejects at 0.05 and fails at 0.01, because its p-value of 0.0109 sits between the two.
Yes, that is the primary input. If you already have a z or t value from software, a textbook or another calculator, type it in, set alpha and the degrees of freedom for a t-test, and calculate. The sample-data section is optional and exists for the common case where you have the raw summary statistics instead.
For a one-sample t-test, the sample size minus 1: a sample of 36 has 35 degrees of freedom. Filling the statistic from sample data sets this field for you. The tool accepts the value as typed, including fractional degrees of freedom from Welch-style adjustments, and the Z-test ignores the field entirely.
No. A small p-value says the data would be unusual if the null hypothesis and the test’s assumptions held; it does not give the probability that the null is true, and it says nothing about whether the effect is large enough to matter. That is why this calculator pairs every decision with Cohen’s d when sample data is available, and why broken assumptions, such as a heavily skewed small sample, can produce a small p-value that means very little.
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References
- NIST/SEMATECH. e-Handbook of Statistical Methods: One-Sample t-Test. Accessed October 22, 2026.
- OpenStax, Rice University. Introductory Statistics 2e: Null and Alternative Hypotheses. Accessed October 22, 2026.
- Wasserstein, Ronald L., and Nicole A. Lazar. The ASA Statement on p-Values: Context, Process, and Purpose, The American Statistician. Accessed October 22, 2026.
- Student (William Sealy Gosset). The Probable Error of a Mean, Biometrika. Accessed October 22, 2026.
- Lakens, Daniel. Calculating and Reporting Effect Sizes to Facilitate Cumulative Science, Frontiers in Psychology. Accessed October 22, 2026.
- Penn State Eberly College of Science, STAT 415 Online Notes. The T-Test For One Mean. Accessed October 22, 2026.
- Lane, David M., Online Statistics Education. Logic of Hypothesis Testing. Accessed October 22, 2026.
- Khan Academy. Significance Tests (One Sample). Accessed October 22, 2026.