The Student Distribution Calculator is used to compute values based on the Student’s t-distribution. This distribution is essential when working with small sample sizes and unknown population variances. It helps in hypothesis testing by comparing the sample mean to a hypothesized population mean and assessing the likelihood of observing a particular difference due to random chance.
Formula of Student Distribution Calculator
The Student Distribution Calculator uses two primary formulas: the t-statistic and degrees of freedom (df).
t-statistic
This formula calculates a value (t) that represents the difference between a sample mean (x) and a hypothesized population mean (μ) relative to the sample’s standard deviation (s) and size (n).
t = (x – μ) / [s / sqrt(n)]
Degrees of freedom (df)
This value reflects the number of independent pieces of information in your sample. It’s used alongside the t-statistic to find probabilities or critical values in a t-distribution table.
df = n – 1
Where to Use These Formulas
- The t-statistic is used for hypothesis testing. You compare a sample mean to a hypothesized population mean and assess the likelihood of observing such a difference due to random chance.
- After calculating the t-statistic, you look it up in a t-distribution table with the appropriate degrees of freedom to find p-values or critical values.
Pre-calculated Values Table
Below is a table of pre-calculated t-values for common degrees of freedom. This table helps you quickly find critical values without needing to calculate them each time.
| Degrees of Freedom (df) | t-value (95% confidence) | t-value (99% confidence) |
|---|---|---|
| 1 | 12.706 | 63.657 |
| 2 | 4.303 | 9.925 |
| 3 | 3.182 | 5.841 |
| 4 | 2.776 | 4.604 |
| 5 | 2.571 | 4.032 |
| 10 | 2.228 | 3.169 |
| 20 | 2.086 | 2.845 |
| 30 | 2.042 | 2.750 |
Example of Student Distribution Calculator
Let’s say you have a sample mean (x) of 50, a hypothesized population mean (μ) of 45, a sample standard deviation (s) of 10, and a sample size (n) of 16.
First, calculate the t-statistic:
t = (50 – 45) / [10 / sqrt(16)]
t = 5 / 2.5 = 2
Next, calculate the degrees of freedom:
df = n – 1
df = 16 – 1 = 15
Using a t-distribution table, you find that for df = 15 and a 95% confidence level, the critical t-value is approximately 2.131. Since your calculated t-value (2) is less than the critical t-value (2.131), you fail to reject the null hypothesis at the 95% confidence level.
Most Common FAQs
The Student’s t-distribution is a probability distribution used when the sample size is small and the population standard deviation is unknown. It is used in hypothesis testing to determine critical values.
Use the Student Distribution Calculator when you need to perform hypothesis testing with small sample sizes and unknown population variances. It helps in determining the probability of observing a given sample mean.
The t-statistic measures how far the sample mean is from the hypothesized population mean in terms of standard errors. A larger absolute t-value indicates a greater difference between the sample and population means.