T Critical Value Calculator

User Guide: T Critical Value Calculator

How to Use the Calculator

This calculator helps you find critical values for four common statistical distributions: t-Distribution, z-Distribution (Normal), Chi-Square (χ²), and F-Distribution. Follow these steps:

1. Select the Distribution Type

• t-Distribution: Used for small sample sizes or when population variance is unknown.
• z-Distribution (Normal): Used when population variance is known or for large sample sizes.
• Chi-Square (χ²): Used for goodness-of-fit tests and tests of independence.
• F-Distribution: Used in ANOVA and comparing variances.

2. Enter Degrees of Freedom (df)

• For t-Distribution & Chi-Square: Enter a single df value (≥1).
• For F-Distribution: Enter two df values (df1 = numerator, df2 = denominator).

3. Choose Tail Type

• Two-tailed: Used for non-directional hypotheses (e.g., "not equal to").
• Left-tailed: Used when testing if a value is significantly lower.
• Right-tailed: Used when testing if a value is significantly higher.

4. Set Significance Level (α)

• Common values: 0.05 (5%), 0.01 (1%), or 0.10 (10%).
• For two-tailed tests, the calculator automatically splits α in half (e.g., α=0.05 becomes 0.025 per tail).

5. Click "Calculate Critical Value"

• The calculator will display:
• The critical value (threshold for rejecting the null hypothesis).
• A visual graph showing the distribution and critical region.
• Details about the distribution, df, α, and tail type.

---

Explanation of Terms used in T Critical Value Calculator

1. Critical Value

• The cutoff point that defines the rejection region in hypothesis testing.
• If your test statistic exceeds this value, you reject the null hypothesis.

2. Degrees of Freedom (df)

• t-Distribution: df = sample size - 1.
• Chi-Square: df = (rows - 1) * (columns - 1) for contingency tables.
• F-Distribution: df1 = between-group variability, df2 = within-group variability.

3. Significance Level (α)

• The probability of rejecting the null hypothesis when it is true (Type I error).
• Common choices: 0.05 (5%), 0.01 (1%), 0.10 (10%).

4. Tail Type

• Two-tailed: Tests for differences in either direction (≠).
• Left-tailed: Tests if a value is significantly lower (<).
• Right-tailed: Tests if a value is significantly higher (>).

5. Distribution Types

Distribution	Use Case	Key Feature
t-Distribution	Small samples, unknown σ	Wider tails than normal
z-Distribution	Known σ or large samples	Symmetric, bell-shaped
Chi-Square (χ²)	Categorical data tests	Right-skewed
F-Distribution	ANOVA, variance tests	Two df parameters

---

Example Use Cases

1. t-Test (Two-Tailed, α = 0.05, df = 20)

• Inputs:
• Distribution = t-Distribution
• df = 20
• Tail = Two-tailed
• α = 0.05
• Output: Critical Value ≈ ±2.086

2. Chi-Square Test (Right-Tailed, α = 0.01, df = 5)

• Inputs:
• Distribution = Chi-Square
• df = 5
• Tail = Right-tailed
• α = 0.01
• Output: Critical Value ≈ 15.086

3. F-Test (Right-Tailed, α = 0.05, df1 = 3, df2 = 20)

• Inputs:
• Distribution = F-Distribution
• df1 = 3, df2 = 20
• Tail = Right-tailed
• α = 0.05
• Output: Critical Value ≈ 3.098

---

Troubleshooting

• "NaN" or incorrect values?
• Check if df or α is outside valid ranges (e.g., df ≥ 1, 0 < α < 0.5).
• Graph not updating?
• Ensure all inputs are valid numbers.
• Need more precision?
• For exact values, use statistical software like R or Python’s scipy.stats.

---