1. What is the T-Distribution?

The t-distribution is a probability distribution that estimates the population parameters when the sample size is small and/or when the population variance is unknown. It is similar to the normal distribution but has heavier tails.

2. How Does the T-Statistic Work?

The calculator uses the t-statistic formula:

Where:

• \( \bar{x} \) — Sample mean
• \( \mu \) — Population mean
• \( s \) — Sample standard deviation
• \( n \) — Sample size

Explanation: The t-statistic measures how many standard errors the sample mean is from the population mean. Higher absolute values indicate greater deviation.

3. Importance of T-Distribution

Details: The t-distribution is crucial for hypothesis testing (t-tests) and constructing confidence intervals when dealing with small sample sizes (typically n < 30) or unknown population standard deviation.

4. Using the Calculator

Tips: Enter all values in consistent units. Sample size must be ≥1, and standard deviation must be >0.

5. Frequently Asked Questions (FAQ)

Q1: When should I use t-distribution instead of normal distribution?
A: Use t-distribution when sample size is small (<30) or population standard deviation is unknown.

Q2: What are degrees of freedom in t-distribution?
A: Degrees of freedom equal sample size minus one (n-1). It affects the shape of the t-distribution.

Q3: How does t-distribution differ from normal distribution?
A: T-distribution has heavier tails than normal distribution, especially with small sample sizes. As n increases, t-distribution approaches normal distribution.

Q4: What is a typical critical t-value?
A: For α=0.05 and df=10, two-tailed critical t-value is approximately ±2.228. Values vary by degrees of freedom and significance level.

Q5: Can I use this for paired t-tests?
A: Yes, the same formula applies where the "sample" represents differences between paired measurements.