# Z Table - Standard Normal Table, t, Chi-Square & F Tables

> Interactive z table with left, right, mean-to-z and central areas, shaded charts and inverse lookup, plus t, chi-square and F critical value tables.

- URL: https://qilingtools.com/en/math/normal-table
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Statistics](https://qilingtools.com/en/tags/statistics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

This is a statistics table you can click. The standard normal table is laid out as in textbooks — rows give z to one decimal place, columns the second decimal, from −3.49 to 3.49 — and its entries can show the left tail Φ(z) = P(Z ≤ z), the right tail P(Z ≥ z), the area from the mean P(0 ≤ Z ≤ z) or the central area P(−z ≤ Z ≤ z). Click any cell to highlight its row and column and see the shaded area on the bell curve. The inverse lookup turns an area into z (0.975 → 1.96) and points you to the nearest cell. The t table lists critical values for one- and two-tailed α with 1–30, 40–120 and ∞ degrees of freedom; the chi-square table covers right-tail probabilities from 0.995 to 0.005; the F table offers α = 0.10, 0.05, 0.025, 0.01 and 0.005, with infinite degrees of freedom too. Values not in the tables can be computed for any df and α. Everything is calculated from the incomplete gamma and beta functions, with quantiles found by root finding, and matches printed tables to four decimal places. The probability calculator handles normal (any μ and σ), t, chi-square and F distributions for left-tail, right-tail, between and two-tailed probabilities, with a shaded chart.

## How to use

1. Choose the standard normal, t, chi-square or F table; for the z table pick what the entries mean
2. Click a cell to see the critical value and its shaded area, or type a df and α to compute one that isn’t listed
3. Use the probability calculator to find the probability of a range, or the critical value for a probability

## Features

- Standard normal table for z from −3.49 to 3.49, with four area conventions
- Click a cell to highlight its row and column and draw the shaded area
- Inverse lookup from an area to z, with the nearest table cell
- t table with one- and two-tailed α side by side, including ∞ df
- Chi-square table with ten common right-tail probabilities
- F tables for five α levels, including infinite df
- Critical values for any df and α
- Normal, t, chi-square and F probability calculator: left, right, between, two-tailed

## FAQ

### How do I read a z table?

Split z into two parts: the units and first decimal pick the row, the second decimal picks the column. For z = 1.96 go to row 1.9 and column 0.06 to read 0.9750, so P(Z ≤ 1.96) = 0.9750. For negative z use the negative half of the table or the symmetry Φ(−z) = 1 − Φ(z). For a general normal N(μ, σ²), standardise first with z = (x − μ)/σ.

### What’s the difference between one-tailed and two-tailed t critical values?

A one-tailed test puts the whole rejection region on one side, so the critical value satisfies P(T > t) = α. A two-tailed test splits α between both tails, so P(|T| > t) = α, which is the one-tailed value for α/2. That’s why two-tailed α = 0.05 and one-tailed α = 0.025 share a column — 2.228 for 10 degrees of freedom. The two header rows show both.

### Is α in the chi-square table a left or right tail?

It’s the right tail throughout: each entry c satisfies P(χ² > c) = α. Goodness-of-fit and independence tests use the right-tail column for your α, such as 0.05. A confidence interval for a variance needs both α/2 and 1 − α/2 — with 10 degrees of freedom and 95% confidence, 3.247 from the 0.975 column and 20.483 from the 0.025 column.

### What are df₁ and df₂ in the F table?

df₁ is the numerator degrees of freedom (columns) and df₂ the denominator degrees of freedom (rows). In one-way ANOVA df₁ is the number of groups minus 1 and df₂ the total sample size minus the number of groups; when comparing two variances they are each sample’s n − 1. The table lists right-tail critical values; for the left tail use F₁₋α(d₁, d₂) = 1 / Fα(d₂, d₁).

## Related tools

- [Graphing Calculator](https://qilingtools.com/en/math/graph): Free online function graphing calculator: type sin(x), x^2 or 1/x and see the curve at once, with several functions, parametric and polar curves, PNG export.
- [Scientific Calculator](https://qilingtools.com/en/math/calculator): Free online scientific calculator: type 2*(3+4)^2, sin(pi/6) or log(8,2) and see the answer as you type, with degrees or radians and a history list.
- [Equation Solver](https://qilingtools.com/en/math/equation): Free online equation solver: enter coefficients for linear, quadratic (with discriminant, vertex and complex roots) and cubic equations, plus 2×2 systems.
- [Fraction Calculator](https://qilingtools.com/en/math/fraction): Free online fraction calculator: add, subtract, multiply and divide fractions with automatic reduction, plus fraction ↔ decimal with repeating digits.
- [Percentage Calculator](https://qilingtools.com/en/math/percentage): Free online percentage calculator: X% of Y, X as a percent of Y, increase and decrease, a value after a percentage change and the original behind it.
- [Prime Factorization](https://qilingtools.com/en/math/prime): Free online prime factorization: factor an integer with exponents, list every divisor and their sum, test primality with neighbouring primes, or list a range.
- [GCD and LCM](https://qilingtools.com/en/math/gcd-lcm): Free online GCD and LCM calculator: works with two or more integers, says whether they are coprime, shows each prime factorization and every Euclidean step.
- [Statistics Calculator](https://qilingtools.com/en/math/statistics): Free online statistics calculator: paste a data set for the mean, median, mode, range, quartiles, variance, standard deviation, standard error and outliers.