χ² Statistic
P-Value
Critical Value
Degrees of Freedom
Data Analysis Table
| Category | Observed (O) | Expected (E) | (O-E)² | (O-E)²/E |
|---|
ℹ️ How to read this chart?
- The Curve (PDF): Represents the probability of getting a specific Chi-Square value just by random chance. Imagine running the experiment 10,000 times with a perfectly fair system: this curve shows how the results would be distributed. Most results would be near 0 (perfect fit), but occasionally you'd get higher values just by luck.
- X-Axis: The Chi-Square Statistic value (0 means perfect fit, higher means worse fit).
- Blue Line: Your calculated statistic. Where does your data fall?
- Red Line (Critical Value): The "Limit". If your blue line is to the right of this red line (in the shaded area), your result is too extreme to be just luck (probability < 5%).
📉 Intuitive: How is this curve drawn?
Think of this curve as a "Pile of Possibilities".
1. Simulation: Imagine we simulate a "null" world (e.g., a perfectly
fair
die) millions of times.
2. Calculate: For each simulation, we calculate the Chi-Square score
(measure of "weirdness").
3. Stack: We stack all those scores up. The curve is highest where the
scores are most common.
4. The Tail: The long tail to the right represents those rare,
super-weird
outcomes that shouldn't happen often if everything is fair.
🧮 How is P-Value calculated?
The P-Value represents the area under the curve to the right of your
statistic.
Mathematically, it is the integral of the probability density function (PDF) from your
χ² value to infinity:
P = ∫ f(x; k) dx
In this app, we use a numerical approximation (series expansion of the
Incomplete Gamma Function) to calculate this area.
📊 What Does This Mean?
Select a scenario to begin exploring hypothesis testing!