Library / Statistics / STA-05
Statistics
One-sample t statistic
Compares a sample mean with a claimed value μ₀, measured in standard errors.
- Ref
- STA-05
- Rev
- v1
- Field
- Statistics
- Result
- t
x̄ is the sample mean, μ₀ the claimed (null) mean, s the sample standard deviation and n the sample size. The t value counts how many standard errors the sample mean lies from the claim. Compare |t| with the critical value of the t distribution for n − 1 degrees of freedom (about 2.06 for 95 % with 25 readings).
Compares a sample mean with a claimed value μ₀, measured in standard errors. Large |t| is evidence that the claim is wrong; compare it with the t-table for n − 1 degrees of freedom.
Calculator
- Standard error s/√n1.6
- Degrees of freedom n − 124
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The default t = 1.25 is well below the 95 % critical value of about 2.06 for 24 degrees of freedom, so the sample does not contradict the claimed mean.
- A result that is not significant does not prove the claim; with few readings a real difference can go undetected.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x̄ Sample mean | 0 to 200 | −31.25 | 93.75 | 125 | +10% | +260 % |
| μ₀ Claimed mean | 0 to 200 | −92.5 | 32.5 | 125 | +10% | −250 % |
| s Sample standard deviation | 0.1 to 50 | 0.2 | 100 | 99.8 | +10% | −9.09 % |
| n Sample size | 2 to 1,000 | 0.3536 | 7.906 | 7.552 | +1 | +1.98 % |
Sweep of Sample mean
| Sample mean | t statistic |
|---|---|
| 0 | −31.25 |
| 25 | −15.63 |
| 50 | 0 |
| 75 | 15.63 |
| 100 | 31.25 |
| 125 | 46.88 |
| 150 | 62.5 |
| 175 | 78.13 |
| 200 | 93.75 |
Sweep one variable
Variables and how each one contributes
t statistic
What the equation gives you. Everything else feeds it.
Sample mean
The mean of the sample. Enters only through the gap x̄ − μ₀. At the default the gap is just 2, so a 10 % rise in x̄ (to 57.2) more than triples t, to about 4.5.
Claimed mean
The mean being claimed or specified. Moving μ₀ towards x̄ shrinks |t|. A 10 % rise in μ₀ (to 55) turns the gap negative and t becomes about −1.9.
Sample standard deviation
The standard deviation of the readings. Inverse: noisier data give a smaller |t| for the same gap.
Sample size
The number of readings. Inverse square root in the standard error: four times as many readings double |t| for the same gap and scatter.
Worked example with the values above
- Gap from the claim: x̄ − μ₀ = 52 − 50 = 2.
- Standard error: s/√n = 8 / 5 = 1.6.
- t = 2 / 1.6 = 1.25.
Assumptions and limits
- Readings are independent and approximately normally distributed.
- The sample is random, with no systematic bias.
Common mistakes
- Using σ of the population when only the sample s is available.
- Treating a significant t as proof the difference is large enough to matter.
Statistics and uncertainty
Compare |t| with the critical value of the t distribution with n − 1 degrees of freedom. A large |t| means the sample mean is unlikely to come from a population with mean μ₀; it does not prove the difference matters in practice.
Where it is used
- Testing whether a new batch meets a specified mean strength or accuracy.
- Checking whether a system has changed after a modification.
Maths you need
- Standard error: s/√n
- Hypothesis testing: null claim and critical value