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Probability Distribution Visualizer — PDF / CDF, P(a ≤ X ≤ b) Area, Inverse Quantile (z / t / χ² / F)

Probability distribution visualizer — normal/t/chi-sq/F/binomial/poisson/exp/uniform PDF+CDF, P(a≤X≤b) area, z/p table replacement.

  • Runs locally
  • Category Calculator
  • Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Pick a distribution, set its parameters, drag [a, b] to compute P(a ≤ X ≤ b), or read a critical value off the Inverse CDF panel.

P(a ≤ X ≤ b)
0.9545
95.45% of total mass
F(a) = P(X ≤ a)
0.0227501
F(b) = P(X ≤ b)
0.97725
-4-202400.09970.1990.2990.399ab
Slice [a, b]: [a = -2, b = 2]
Moments (analytic)
Mean E[X]
0
Variance Var(X)
1
Skewness
0
Excess kurtosis
0
Std dev = 1
Inverse CDF — type a probability, read a critical value
For a two-sided 95% interval enter 0.975 (upper 2.5% bound). For a one-sided 5% test enter 0.95.
x such that F(x) = p: 1.95996
PDF at x: 0.398942
CDF at x: 0.5
Presets
All parameters and probabilities are computed in your browser. Nothing is uploaded or logged.

What this tool does

A live PDF / CDF visualizer for the eight probability distributions every intro statistics, econometrics, and data-analysis course uses: the continuous normal N(μ, σ²), Student's t(df), chi-squared χ²(df), Fisher F(df₁, df₂), exponential Exp(λ), continuous uniform U(a, b), plus the discrete binomial B(n, p) and Poisson Poi(λ). Every density and mass function is implemented in plain JavaScript (no SciPy, no WASM, no API call) using the underlying special functions — log-Γ via Lanczos, regularized incomplete gamma for χ² / Poisson CDF, regularized incomplete beta for t / F / binomial CDF, Abramowitz–Stegun erfc for the normal — so the numbers match R, NumPy, and Excel to 6+ decimals across the entire valid parameter range. The SVG canvas plots either the PDF / PMF or the CDF, lets you drag two handles to define a slice [a, b], and shows the shaded probability P(a ≤ X ≤ b) updating in real time along with the proportion of total mass that slice represents. An inverse-CDF (quantile) panel goes the other direction: type a probability and read off the matching critical value — that is the operation that replaces the z-table at the back of your textbook (z₀.₀₂₅ = 1.95996, t₀.₀₂₅(15) = 2.13145, χ²₀.₉₅(10) = 18.307, and so on). The summary card always shows the four moments (mean, variance, skewness, excess kurtosis) computed from the analytic formula for each family, so you can sanity-check homework answers without cracking open a table. Six one-click presets cover the canonical examples (standard normal, IQ N(100, 15²), 100 coin flips, customer arrivals, 95% confidence band, χ² goodness-of-fit at df=4). Every computation runs locally in your browser — your parameters, your probabilities, nothing is uploaded.

Tool details

Input
Files + Numbers
The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output
Live result + Copy + Preview
The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy
Browser-side processing
The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share
Shareable URL state
Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget
Initial JS <= 38 KB
No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit
Calculator · Student
Category and role tags drive related tools, internal links, and quick fit checks.

How to use

  1. 1. Input

    Paste or drop your content into the tool panel.

  2. 2. Process

    Click the button. All processing is local in your browser.

  3. 3. Copy / Download

    Copy the result or download to disk in one click.

How Probability Distribution Visualizer fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

  • Getting a realistic range before a purchase, plan, workout, or schedule decision.
  • Comparing scenarios by changing one input at a time.
  • Turning rough assumptions into a number you can discuss.

Calculation checks

  • Double-check units, dates, rates, and rounding assumptions.
  • Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
  • Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

  1. 1 Percentage Calculator 5 common percentage calculations — "x% of y", "x is what% of y", percentage change, increase/decrease — instant, browser-only Open
  2. 2 Permutation & Combination Calculator nPr, nCr, n!, n^r and combinations-with-repetition — exact BigInt results with the formula and worked steps — browser-only Open
  3. 3 Statistics Basic Calculator Basic statistics calculator — mean/median/mode/variance/std-dev/quartiles/range/IQR/skewness/kurtosis + histogram + box plot, paste any numbers. Open

Real-world use cases

  • Replace the z-table at the back of your textbook

    Your stats homework asks for z₀.₀₂₅ (the upper-2.5% z-value for a two-sided 95% confidence interval). Pick "Normal", keep μ = 0 and σ = 1, open the Inverse CDF panel, type 0.975, and read 1.95996. Type 0.995 instead and you get 2.57583 — the z₀.₀₀₅ that bounds a 99% interval. Same workflow gets you z₀.₀₅ = 1.64485 for a one-sided 5% test. No more flipping to the appendix and squinting at four-decimal table cells.

  • Get the p-value of a t-test without looking up a table

    You ran a paired t-test on n=13 measurements (df = 12) and got t = 2.40. Pick "Student t", set df = 12, type 2.40 into the x box: F(2.40) = 0.98316. The right-tail p is 0.01684, the two-tail p is 2 × 0.01684 = 0.03368. Both are under 5%, so you reject H₀ at the 5% level and you can quote 0.034 directly in your write-up — the calculator already rounded to four decimals for you.

  • Visualize the 68–95–99.7 rule on the IQ distribution

    Pick "Normal", μ = 100, σ = 15 (the standard IQ parametrization), drag the [a, b] handles to 85 and 115: the shaded area reads 0.6827 — that is the "68% within one σ" rule, made visual. Drag to 70 and 130 to see 0.9545 (the "95% within 2σ" rule), and to 55 and 145 to see 0.9973 (the "99.7% within 3σ" rule). Students stop memorizing the rule and start trusting it because they just dragged it out themselves.

  • Decide if 7 heads in 10 coin flips is "weird"

    Pick "Binomial", n = 10, p = 0.5. The bars peak at k = 5 as expected. Drag [a, b] to [7, 10] and the shaded mass is P(X ≥ 7) = 0.1719 — about a 1-in-6 chance under a fair coin. That is not weird at all (you would need P &lt; 0.05 for "statistically significant evidence the coin is biased"), and your students can see exactly why instead of taking "n=10 is too small" on faith.

  • Run a chi-squared goodness-of-fit at alpha 0.05 and find the critical value

    You computed χ² = 11.2 on 4 degrees of freedom. Pick "Chi- squared", df = 4. The Inverse CDF panel at 0.95 gives critical value 9.488. Because your test statistic 11.2 &gt; 9.488, you reject the null at the 5% level. Read F(11.2) = 0.97565 — the right-tail p-value is 1 − 0.97565 = 0.02435. A full goodness-of-fit conclusion in under 30 seconds, no appendix table needed.

  • Find the time until 90% of customers have arrived using Exp(lambda)

    A coffee shop sees 30 customers per hour on average, so the inter-arrival time is Exp(λ = 30/hour) ≈ Exp(0.5/minute). Pick "Exponential", λ = 0.5, open Inverse CDF, type 0.9, get x ≈ 4.605 minutes. So 90% of the next customer's wait times are under ~4.6 minutes. Drag [a, b] to [0, 1] to see that P(wait &lt; 1 minute) ≈ 0.393 — almost 40% of arrivals come back-to-back within a minute, which is the queueing fact that justifies a second register at peak.

Common pitfalls

  • Using the one-tail p-value when the textbook question asks for a two-tail test (or vice versa). Two-sided p = 2 × min(F(x), 1 − F(x)) for symmetric distributions (z, t); one-sided right-tail p = 1 − F(x). For χ² and F goodness-of-fit / ANOVA the test is intrinsically one-sided right-tail — never multiply by two there.

  • Confusing 95% confidence with the 95% quantile. A two-sided 95% confidence interval uses the upper 2.5% quantile (z₀.₀₂₅ = 1.96, not z₀.₀₅ = 1.645). If you typed 0.95 into the Inverse CDF box and got 1.645, you asked for the one-sided 5% bound — type 0.975 for the two-sided answer.

  • Reading a discrete distribution's "P(X = k)" off the CDF instead of the PMF. For binomial / Poisson, P(X = k) is the height of the bar at integer k (the PMF), not F(k). The CDF gives you P(X ≤ k), which includes everything to the left of and at k. Switch the PDF / CDF toggle if you're not sure which view you're looking at.

  • Using the normal approximation to the binomial outside its safe range. The rule of thumb is np ≥ 5 AND n(1 − p) ≥ 5; below that the approximation is visibly off, especially in the tails (which is where you usually need accuracy for a p-value). The page lets you overlay both curves so you can see the discrepancy instead of guessing.

Privacy

The eight distribution implementations (PDF / PMF, CDF, inverse CDF, moments, slice area) all run inside your browser tab in JavaScript. Your parameter inputs, the [a, b] handles you drag, and the probabilities you type into the inverse-CDF panel are never sent to a server, never logged, and never written to localStorage. The selected distribution and its parameters are synced into the URL query string only so a teacher can share a single link that reproduces a worked example for the class — if that is sensitive (it usually is not), open the page in an incognito tab and your view will reset to the default standard normal. The page works offline once loaded.

FAQ

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Made by Toolora · 100% client-side · Updated 2026-05-29