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Continuous Distribution Calculator

Probability & Inference Quantile Workbench

Solve for tail probabilities and critical values across the foundational sampling distributions without looking at printed textbook tables.

👨‍🏫 Professor Mode: Guided Learning Experience

New to continuous probability? Enable Professor Mode for step-by-step guidance on density curves, critical values, and Student's t heavy tails!

OVERVIEW & WHAT THIS TOOL SOLVES

Every hypothesis test, confidence interval, and risk threshold in marketing analytics relies on computing tail areas or critical cutoffs under a continuous probability distribution. Looking up static textbook tables hides how degrees of freedom, scale, and tail direction reshape tail risk. The Continuous Distribution Calculator computes exact probabilities and inverse quantiles across the Normal, Student's t, Chi-Square (χ²), and Fisher's F distributions with live shaded density curves.

🎯 Questions It Answers

  • What proportion of campaigns, customers, or test statistics fall below, above, between, or outside a given cutoff?
  • What critical value (quantile) marks the top 5%, 1%, or two-tailed 95% boundary of a distribution?
  • How much heavier are the tails of Student's t at small sample sizes compared to the standard Normal bell curve?

📂 What Data You Need

  • Distribution Family: Choose Standard Normal (Z), Custom Normal in your own units (μ, σ, such as ROAS or dollars), Student's t (df), Chi-Square χ² (df), or Fisher's F (df₁, df₂)
  • Calculation Direction: Forward mode (Cutoff x → Probability p) or Inverse Quantile mode (Probability p → Cutoff x)
  • Tail Region: Left tail P(X ≤ x), Right tail P(X ≥ x), Central/Between P(a ≤ X ≤ b), or Two-Tailed

💡 What You'll Discover

  • Interactive Shaded Density Plot: Exact visual area under the PDF curve matching your chosen tail region
  • Standardized & Raw Cutoffs: The Custom Normal tab reads a cutoff in your metric's units beside its z-score (P(ROAS ≥ 4.00x) is the same area as P(Z ≥ 2.00)); the other tabs read Z/t/χ²/F scores and tail probabilities
  • Small-Sample Tail Comparison: Direct demonstration of how degrees of freedom widen critical thresholds
🔍 Key Nuances & Common Traps to Watch For
  • Raw Units and z Describe the Same Area: Under a normal model with mean 2.80x and standard deviation 0.60x, a 4.00x ROAS sits at z = (4.00 − 2.80) / 0.60 = 2.00, so P(ROAS ≥ 4.00x) and P(Z ≥ 2.00) are one shaded area read on two scales. Going back, a z-score becomes a cutoff with x = μ + z × σ.
  • Point Probability vs. Interval Area: For any continuous distribution, the probability of hitting a single exact decimal to infinite precision is zero (P(X = x) = 0); probability always equals the area under the density curve over an interval.
  • One-Tailed vs. Two-Tailed Critical Values: At α = 0.05, a one-tailed Normal cutoff puts all 5% in one tail (z = 1.645), whereas a two-tailed cutoff splits 2.5% into each tail (z = ±1.960).
  • When Student's t Converges to Normal: Below df = 30, estimating the standard deviation from sample data adds tail uncertainty and pushes critical cutoffs outward; as df → ∞, the t distribution converges to the standard Normal.
📐 Under the Hood: Model Equations & Variable Definitions

1. Normal Density, Z-Standardization & Cumulative Probability

\[ z = \frac{x - \mu}{\sigma}, \qquad f(x \mid \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right), \qquad F(x) = P(X \le x) = \int_{-\infty}^{x} f(u)\,du \]

💬 Read it in plain English: The Z-score expresses how many standard deviations σ a value x sits above or below the mean μ, while the cumulative distribution function F(x) integrates the density curve to give the left-tail probability.

x, a, b
Cutoff Threshold(s)YOUR DATA ⓘ
Boundary value(s) on the horizontal axis defining the probability region.
μ, σ
Population Mean & Standard DeviationYOUR DATA ⓘ
Center μ and spread σ of the Normal distribution.
z
Standardized Normal ScoreOUTPUT ⓘ
Distance from the mean measured in standard-deviation units.
f(x), F(x)
Density (PDF) & Cumulative Area (CDF)OUTPUT ⓘ
Curve height f(x) and integrated left-tail probability P(X ≤ x).

2. Sampling Distributions for Inference (t, χ², F) & Inverse Quantile Function

\[ t_{\nu} = \frac{Z}{\sqrt{V / \nu}}, \qquad \chi^2_{\nu} = \sum_{i=1}^{\nu} Z_i^2, \qquad F_{\nu_1,\nu_2} = \frac{\chi^2_{\nu_1}/\nu_1}{\chi^2_{\nu_2}/\nu_2}, \qquad x_p = F^{-1}(p) \]

💬 Read it in plain English: Chi-square sums ν squared standard normals, Student's t divides a standard normal by a scaled chi-square root, Fisher's F compares two scaled chi-squares, and the inverse CDF F⁻¹(p) finds the exact cutoff x_p holding probability p.

ν (df), ν1, ν2
Degrees of FreedomYOUR DATA ⓘ
Shape parameters governing tail heaviness and skewness in t, χ², and F distributions.
p, α
Target Probability / Tail AreaYOUR DATA ⓘ
Cumulative probability p or significance tail area α supplied in Inverse Quantile mode.
xp = F−1(p)
Inverse Quantile / Critical ValueINFERENCE ⓘ
Threshold value x where the cumulative probability equals p.

⚡ Real-World Business Presets

Mode:

Calculated Probability Area

P(Z ≥ 1.645)
0.0500
5.00% of distribution area

Probability Density Curve & Shaded Region

Region of Interest & Parameters