# What is my t score?

Computes the t score t = (x̄ − μ₀) ÷ (s ÷ √n) from a list of numbers or from the mean, standard deviation and sample size, or takes a t score you have, and gives its p-values and critical value.

- Page: https://www.acalculator.org/statistics/t-score-calculator
- JSON spec: https://www.acalculator.org/statistics/t-score-calculator.json
- Version: 6bf39e31d35e

## Default answer

Example with the default inputs (I have My numbers, Your numbers [5.1, 4.9, 5.6, 5.8, 6, 5.4, 5.3, 5.7], Hypothesized mean (μ₀) 5, Significance level (α) 0.05): Your t score is 3.63736 with 7 degrees of freedom; the two-tailed p-value is 0.00831621.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | I have | Your raw numbers, the summary statistics (mean, standard deviation, size), or a t score already. |
| data | Your numbers | The sample: at least 2 numbers, separated by commas, spaces or new lines. |
| mean | Sample mean (x̄) | The mean of the sample. |
| sd | Sample standard deviation (s) | The sample standard deviation, divided by n − 1. |
| n | Sample size (n) | How many observations are in the sample, at least 2. |
| mu | Hypothesized mean (μ₀) | The population mean you compare with, under the null hypothesis. |
| t | t score | The t score (t statistic) you already have. |
| df | Degrees of freedom | The degrees of freedom of the t score; n − 1 for one sample. Fractions are allowed (Welch). |
| alpha | Significance level (α) | The two-tailed significance level for the critical value, for example 0.05. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| t | t score | The t statistic: (x̄ − μ₀) ÷ (s ÷ √n). |
| df | Degrees of freedom | n − 1, or the value you typed. |
| pTwo | p-value, two-tailed | The chance of a t at least this far from 0 in either direction: 2 × P(T > \|t\|). |
| pRight | p-value, right-tailed | The chance of a t at least this large: P(T ≥ t). |
| pLeft | p-value, left-tailed | The chance of a t at most this small: P(T ≤ t). |
| critical | Critical value (two-tailed) | The t with P(T > t) = α ÷ 2; reject the null hypothesis at α when \|t\| is at least this. |
| decision | Decision at α (two-tailed) | Whether the two-tailed p-value is at most α. |
| mean | Sample mean | The mean of your numbers. |
| sd | Sample standard deviation | The standard deviation of your numbers, dividing by n − 1. |
| se | Standard error (s ÷ √n) | The standard deviation divided by the square root of the sample size. |
| count | Sample size | How many numbers are in the sample. |

## Method

t = (x̄ − μ₀) ÷ (s ÷ √n), df = n − 1; p-values from Student’s t distribution with df degrees of freedom.

## Assumptions

- The observations are independent and roughly normal, or the sample is large.
- s is the sample standard deviation, dividing by n − 1.
- The critical value is two-tailed: P(T > t) = α ÷ 2.

## Worked examples

1. mode = data, data = 5.1 or 4.9, mu = 5, alpha = 0.05 gives mean = 5.475, sd = 0.369362, t = 3.637357, df = 7, pTwo = 0.008316, pRight = 0.004158, critical = 2.364624. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2 Are the data consistent with the assumed process mean?: t = (Ȳ − μ₀) ÷ (s ÷ √N), N − 1 degrees of freedom (https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm, retrieved 2026-10-02).
2. mode = summary, mean = 105, sd = 12, n = 25, mu = 100, alpha = 0.05 gives t = 2.083333, df = 24, se = 2.4, pTwo = 0.048044, critical = 2.063899, decision = Significant: reject the null hypothesis at α = 0.05. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2 Are the data consistent with the assumed process mean?: t = (Ȳ − μ₀) ÷ (s ÷ √N), N − 1 degrees of freedom (https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm, retrieved 2026-10-02).
3. mode = t, t = 2, df = 10, alpha = 0.05 gives pTwo = 0.073388, pRight = 0.036694, pLeft = 0.963306. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2 Are the data consistent with the assumed process mean?: t = (Ȳ − μ₀) ÷ (s ÷ √N), N − 1 degrees of freedom (https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm, retrieved 2026-10-02).

## FAQ

### How do I calculate a t score?

Subtract the hypothesized mean from the sample mean, then divide by the standard error s ÷ √n: t = (x̄ − μ₀) ÷ (s ÷ √n). With x̄ = 105, μ₀ = 100, s = 12 and n = 25, the standard error is 12 ÷ 5 = 2.4, so t = 5 ÷ 2.4 = 2.083.

### What are the degrees of freedom?

For one sample, n − 1. Eight measurements have 7 degrees of freedom. Fewer degrees of freedom give a t distribution with fatter tails, so a larger t is needed to be significant.

### How do I get a p-value from a t score?

Choose "A t score", then type t and the degrees of freedom. The two-tailed p-value is the chance of a t at least as far from 0 in either direction: for t = 2 with 10 degrees of freedom it is 0.0734.

### What is the difference between a t score and a z score?

A z score divides by the population standard deviation, which you rarely know. A t score uses the sample standard deviation instead, so it follows Student’s t distribution, which approaches the normal distribution as n grows.

### What does the critical value mean?

It is the t beyond which a two-tailed result is significant at α. At α = 0.05 with 24 degrees of freedom it is 2.064, so t = 2.083 is just significant.

### Is this the bone density T-score?

No. A bone density T-score compares your bone density with a healthy young adult’s, in standard deviations. This page computes the t statistic of Student’s t-test.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2 Are the data consistent with the assumed process mean? (t = (Ȳ − μ₀) ÷ (s ÷ √N), N − 1 degrees of freedom; reject when |t| ≥ t(1 − α/2, N − 1)), retrieved 2026-10-02. https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t(1 − α/2, N − 1) × s ÷ √N), retrieved 2026-10-02. https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm