T-Value Calculator (One-Sample)
Calculation Results
Understanding the T-Value Formula
The T-value (also known as the T-score) is a ratio used in statistics to determine the difference between a sample mean and a hypothesized population mean, relative to the variation in the sample data. It is the core component of the Student's T-test.
The Formula
t = (x̄ – μ) / (s / √n)
Where:
- x̄ (Sample Mean): The average value calculated from your collected data sample.
- μ (Population Mean): The theoretical or hypothesized value you are testing against.
- s (Sample Standard Deviation): The measure of spread or dispersion in your sample.
- n (Sample Size): The total number of observations or data points in your sample.
- s / √n: This is known as the Standard Error of the Mean (SE).
How to Interpret T-Values
A T-value tells you how many standard errors the sample mean is away from the population mean.
- High T-Value: Indicates that the sample mean is significantly different from the population mean, suggesting the results are likely not due to chance.
- Low T-Value (Close to 0): Indicates that the sample mean is very similar to the population mean, suggesting any difference is likely due to random variation.
Real-World Example
Imagine a lightbulb manufacturer claims their bulbs last 1,000 hours (μ = 1000). You test 25 bulbs (n = 25) and find they have an average life of 980 hours (x̄ = 980) with a standard deviation of 50 hours (s = 50).
Step 1: Calculate Standard Error
SE = 50 / √25 = 50 / 5 = 10
Step 2: Calculate T-Value
t = (980 – 1000) / 10 = -20 / 10 = -2.00
In this case, the T-value is -2.00, and the degrees of freedom would be 24 (n – 1). You would then look up this value in a T-distribution table to determine the p-value and statistical significance.
Why Use T-Value instead of Z-Score?
We use the T-distribution when the population standard deviation is unknown and the sample size is small (typically n < 30). As the sample size increases, the T-distribution becomes almost identical to the normal Z-distribution.