T Distribution Formula Properties and How to Calculate with Examples
What is T-Distribution?
T-distribution, also known as student’s t-distribution, is a statistical methodology of evaluating or calculating the mean of a data set that is normally distributed.
A normally distributed data is the one with a “bell-shaped” or “an inverted U shaped” curve. The shape of the distribution implies that slope is concentrated at the center, at the mean value, and slopes downwards towards either side of the curve where the extreme values are. T-distribution and t-test form a part of inferential statistics. The concept of t-distribution was developed by William Sealy Gosset.
The aim of t-distribution is used to test the hypothesis and whether it should be accepted or rejected. It is used to estimate the mean of a population that is normally distributed. It is commonly used when the sample size is small (not less than 20) and when the variance or standard deviation is unknown. It is used to compute the probabilities with the sample mean.
the Formula Used to Calculate the T-value is Given Below:
t - is the t-test score,
x̅ - is the mean of the sample,
μ - is the mean of the population,
s - is the calculated or given standard deviation of the sample,
N - is the sample size
When the values of the above-given variables are provided, then one can simply calculate the t-score.
Let’s work on some examples to understand this better.
Example:
Question
There is a class of 25 students and the mean score of their test is 60 out of 100, with standard deviation 4 marks from the mean. While other students of the school have a mean score of 50 on the same test. What will be the t-score for calculating the probability that school students scored not less than 60 in their tests?
Solution
Let us begin assembling the values given in the question. From the question we can infer that the sample here is the class students and the population consists of all the students in the school.
The samples size of the class (N) - 25
Mean score of the class (x̅) - 60
Mean score of the population ( μ) - 50The standard deviation of the sample (s) - 4
Since we have got all the values that are required to calculate the t-score, we can simply insert them in the formula below
t = ( x̅ - μ) ÷ (s / √N),
t= (60 - 50) ÷ (4 / √25)
t= 10 ÷ 0.8
t= 12.5
The t-value obtained here leads to the cumulative probability from the t-distribution table from where you can find the log value of this t-score with the degrees of freedom, the sample means, the population means and standard deviation for this sample.
FAQs on T Distribution in Statistics Explained Clearly
1. What is the t distribution in statistics?
The t distribution is a probability distribution used to estimate population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which account for extra variability in small samples.
- It is symmetric and bell-shaped.
- It depends on degrees of freedom (df).
- As sample size increases, it approaches the normal distribution.
2. What is the formula for the t statistic?
The formula for the t statistic is t = (x̄ − μ) / (s / √n). This formula is used in hypothesis testing when the population standard deviation is unknown.
- x̄ = sample mean
- μ = population mean
- s = sample standard deviation
- n = sample size
3. When should you use the t distribution instead of the normal distribution?
You should use the t distribution when the sample size is small (typically n < 30) and the population standard deviation is unknown. It adjusts for additional uncertainty in estimating the standard deviation.
- Population standard deviation is unknown.
- Data is approximately normally distributed.
- Sample size is small.
4. What are degrees of freedom in the t distribution?
In a t distribution, degrees of freedom (df) represent the number of independent values that can vary in the calculation, usually given by df = n − 1. Degrees of freedom determine the exact shape of the t curve.
- Smaller df → heavier tails.
- Larger df → closer to normal distribution.
- As df → ∞, t distribution ≈ normal distribution.
5. How do you calculate a confidence interval using the t distribution?
A confidence interval using the t distribution is calculated as x̄ ± t* (s / √n). Here, t* is the critical t value for the chosen confidence level and degrees of freedom.
- Step 1: Find sample mean (x̄).
- Step 2: Compute standard error (s / √n).
- Step 3: Find critical value t* from t table.
- Step 4: Calculate margin of error and form interval.
6. What is the difference between the t distribution and the normal distribution?
The main difference is that the t distribution has heavier tails than the normal distribution, accounting for extra uncertainty in small samples.
- T distribution depends on degrees of freedom.
- Normal distribution uses known population standard deviation.
- T distribution is wider and flatter for small df.
7. Can you give an example of calculating a t statistic?
Yes, for example, if x̄ = 20, μ = 18, s = 4, and n = 16, then the t statistic is t = (20 − 18) / (4 / √16) = 2 / 1 = 2.
- Standard error = 4 / 4 = 1
- t = 2 / 1 = 2
- Degrees of freedom = 16 − 1 = 15
8. What is a t-test and how is it related to the t distribution?
A t-test is a hypothesis test that uses the t distribution to determine whether a sample mean significantly differs from a population mean or another sample mean.
- One-sample t-test
- Independent two-sample t-test
- Paired sample t-test
9. Why does the t distribution have heavier tails?
The t distribution has heavier tails because it accounts for the extra variability introduced when estimating the population standard deviation from a small sample.
- Sample standard deviation varies from sample to sample.
- This increases uncertainty.
- Heavier tails allow for more extreme values.
10. How do you find the critical t value from a t table?
To find a critical t value, locate the row for the correct degrees of freedom and the column for the desired significance level (α).
- Step 1: Compute df = n − 1.
- Step 2: Choose confidence level (e.g., 95%).
- Step 3: Find α (for 95%, α = 0.05).
- Step 4: Read the corresponding t value from the table.
