Calculator
Example Data Table
| Scenario | Input | Value | Meaning |
|---|---|---|---|
| One sample | Sample mean | 47 | Observed average score |
| One sample | Null mean | 50 | Claimed population mean |
| One sample | Sample standard deviation | 7.5 | Sample spread |
| One sample | Sample size | 16 | Number of observations |
| Decision | Alpha | 0.05 | Left-tail rejection cutoff |
Formula Used
One sample or paired test:
t = (x̄ - μ0) / (s / √n)
df = n - 1
Two sample Welch test:
t = ((x̄1 - x̄2) - D0) / √((s1² / n1) + (s2² / n2))
df = ((s1² / n1 + s2² / n2)²) / (((s1² / n1)² / (n1 - 1)) + ((s2² / n2)² / (n2 - 1)))
Two sample pooled test:
sp² = (((n1 - 1)s1²) + ((n2 - 1)s2²)) / (n1 + n2 - 2)
t = ((x̄1 - x̄2) - D0) / (sp × √(1 / n1 + 1 / n2))
Left-tail p value: p = P(Tdf ≤ t)
Reject H0 when p ≤ alpha, or when t ≤ critical t.
How to Use This Calculator
- Select one sample, paired, or two sample testing.
- Choose summary statistics or raw pasted observations.
- Enter alpha, null value, sample sizes, means, and deviations.
- For two samples, choose Welch or pooled variance.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to save the same results.
Left Tailed T Test Guide
What This Calculator Does
A left tailed t test checks whether a population mean, paired difference, or difference between two means is smaller than a claimed value. It is useful when the research question has a clear lower direction. The calculator supports one sample, paired, and two sample designs. It also accepts summary statistics or raw observations.
Why the Left Tail Matters
In a left tailed test, only unusually low test statistics count as evidence against the null hypothesis. This is different from a two tailed test. A two tailed test looks for change in either direction. Here, the alternative claim is focused on a decrease.
Input Choices
Summary mode is best when you already know the sample size, mean, and standard deviation. Raw data mode is useful when your observations are still in a list. The calculator computes the needed mean and sample deviation automatically. For paired tests, it subtracts the second value from the first value in each pair.
Interpreting the Output
The t statistic measures how far the estimate sits below the null value after standard error is considered. The p value gives the probability of getting a t statistic this small or smaller, assuming the null hypothesis is true. A smaller p value means stronger evidence.
Welch and Pooled Options
Welch testing is safer when two samples may have different variances. Pooled testing assumes equal population variances. Use the pooled option only when that assumption is reasonable. Otherwise, Welch is usually the better default for independent samples.
Decision Rule
Compare the p value with alpha. If p is less than or equal to alpha, reject the null hypothesis. If p is greater than alpha, do not reject it. This does not prove the null is true. It only means the evidence was not strong enough for the selected significance level.
FAQs
What is a left tailed t test?
A left tailed t test checks whether a mean or mean difference is significantly less than a null value. It focuses only on the lower side of the t distribution.
When should I use a one sample test?
Use it when one sample mean is compared with one claimed population mean. The sample should be numeric and reasonably suited for t testing.
When should I use a paired test?
Use a paired test when measurements are linked. Common examples include before and after values, matched subjects, or repeated measurements on the same items.
When should I use Welch testing?
Use Welch testing for two independent samples when equal variance is uncertain. It adjusts degrees of freedom and is often safer than pooled testing.
What does the p value mean?
The p value is the probability of a t statistic this small or smaller if the null hypothesis is true. Smaller values support the left tailed claim.
What alpha value should I enter?
Many studies use 0.05. Some use 0.01 for stricter testing or 0.10 for exploratory work. Choose alpha before reviewing results.
Can I enter raw data?
Yes. Paste numbers separated by commas, spaces, semicolons, or line breaks. The calculator computes sample statistics from those values.
Why is my result not significant?
The sample may not be far enough below the null value. Small sample size, high variation, or a strict alpha can also raise the p value.