T-Statistic Calculator
Calculate the t-score for a one-sample mean test.
What is a T-Statistic?
The t-statistic (or t-score) is a ratio used in inferential statistics to determine if there is a significant difference between the means of two groups or between a sample mean and a hypothesized population mean. It is the core component of the T-test, which is used when the population standard deviation is unknown and the sample size is small.
The Formula for a One-Sample T-Statistic
The calculation for a one-sample t-test follows this mathematical formula:
- x̄ (Sample Mean): The average value calculated from your data set.
- μ (Hypothesized Mean): The value you are testing against (null hypothesis).
- s (Standard Deviation): The measure of variation or dispersion in your sample.
- n (Sample Size): The total number of observations in the sample.
- s / √n (Standard Error): The estimated standard deviation of the sample distribution of the mean.
How to Interpret the Result
A t-statistic measures how many standard errors the sample mean is away from the hypothesized population mean. Generally:
- Higher Absolute Value: Indicates a greater likelihood that the difference is statistically significant.
- Zero: Indicates the sample mean is exactly equal to the hypothesized mean.
- Degrees of Freedom (df): For a one-sample test, this is simply n – 1. This value is used alongside the t-statistic to find the p-value in a t-distribution table.
Example Calculation
Imagine a factory claims their lightbulbs last 1000 hours (μ = 1000). You test 25 bulbs (n = 25) and find a sample mean of 980 hours (x̄ = 980) with a standard deviation of 50 hours (s = 50).
- Standard Error = 50 / √25 = 10
- T-Statistic = (980 – 1000) / 10 = -2.0
- Degrees of Freedom = 25 – 1 = 24
You would then look up -2.0 in a T-table with 24 degrees of freedom to determine the p-value and check for significance.