🔔 Normal Distribution Calculator — Z Table

The normal distribution is the single most important distribution in statistics, describing countless measurements and — via the Central Limit Theorem — sampling distributions in general.

Z-score

\[z = \frac{x - \mu}{\sigma}, \qquad \Phi(z) = P(Z \le z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-t^2/2}\, dt\]

Critical Z Values

Confidence / p	One-tailed z	Two-tailed z
90% (α = .1)	1.282	1.645
95% (α = .05)	1.645	1.960
97.5% (α = .025)	1.960	2.241
99% (α = .01)	2.326	2.576
99.5% (α = .005)	2.576	2.807
99.9% (α = .001)	3.090	3.291

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📐 t-Distribution Calculator — Critical t Table

Student's t-distribution is the correct reference distribution whenever a sample's standard deviation is estimated rather than known — the case for almost every real t-test.

t-statistic

\[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \qquad df = n - 1\]

Critical t Values

df	t.90 (α=.10)	t.95 (α=.05)	t.975 (α=.025)	t.99 (α=.01)
1	3.078	6.314	12.706	31.821
2	1.886	2.920	4.303	6.965
3	1.638	2.353	3.182	4.541
4	1.533	2.132	2.776	3.747
5	1.476	2.015	2.571	3.365
6	1.440	1.943	2.447	3.143
7	1.415	1.895	2.365	2.998
8	1.397	1.860	2.306	2.896
9	1.383	1.833	2.262	2.821
10	1.372	1.812	2.228	2.764
12	1.356	1.782	2.179	2.681
15	1.341	1.753	2.131	2.602
20	1.325	1.725	2.086	2.528
24	1.318	1.711	2.064	2.492
30	1.310	1.697	2.042	2.457
40	1.303	1.684	2.021	2.423
60	1.296	1.671	2.000	2.390
120	1.289	1.658	1.980	2.358

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📊 Chi-Square Calculator — Critical Value Table

The chi-square distribution arises as the sum of squared standard-normal variables — the natural reference distribution for the test of independence, goodness-of-fit, and variance confidence intervals.

Chi-square goodness-of-fit

\[\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}\]

Critical χ² Values

df	χ².90 (α=.10)	χ².95 (α=.05)	χ².975 (α=.025)	χ².99 (α=.01)
1	2.706	3.841	5.024	6.635
2	4.605	5.991	7.378	9.210
3	6.251	7.815	9.348	11.345
4	7.779	9.488	11.143	13.277
5	9.236	11.070	12.833	15.086
6	10.645	12.592	14.449	16.812
7	12.017	14.067	16.013	18.475
8	13.362	15.507	17.535	20.090
9	14.684	16.919	19.023	21.666
10	15.987	18.307	20.483	23.209
15	22.307	24.996	27.488	30.578
20	28.412	31.410	34.170	37.566
25	34.382	37.652	40.646	44.314
30	40.256	43.773	46.979	50.892
40	51.805	55.758	59.342	63.691
50	63.167	67.505	71.420	76.154
60	74.397	79.082	83.298	88.379
80	96.578	101.879	106.629	112.329
100	118.498	124.342	129.561	135.807

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📈 F-Distribution Calculator — Critical F Table

The F-distribution is the ratio of two independent chi-square variables — the reference distribution for the omnibus test in ANOVA and multiple regression.

One-way ANOVA F-ratio

\[F = \frac{MS_{between}}{MS_{within}} = \frac{SS_{between}/df_1}{SS_{within}/df_2}\]

Critical F Values (α = .05)

df1 \ df2	5	10	20	30	60
1	6.608	4.965	4.351	4.171	4.001
2	5.786	4.103	3.493	3.316	3.150
3	5.409	3.708	3.098	2.922	2.758
4	5.192	3.478	2.866	2.690	2.525
5	5.050	3.326	2.711	2.534	2.368
6	4.950	3.217	2.599	2.421	2.254
8	4.818	3.072	2.447	2.266	2.097
10	4.735	2.978	2.348	2.165	1.993
12	4.678	2.913	2.278	2.092	1.917
15	4.619	2.845	2.203	2.015	1.836
20	4.558	2.774	2.124	1.932	1.748

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🎲 Binomial Distribution Calculator

The binomial distribution models the number of successes in n independent yes/no trials, each with the same probability p — fully specified by n and p, and discrete rather than continuous.

Binomial PMF

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad k = 0, 1, \dots, n\]

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🔢 Poisson Distribution Calculator

The Poisson distribution models the count of independent events in a fixed interval given a known average rate λ, with the distinctive property that its variance always equals its mean.

Poisson PMF

\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad k = 0, 1, 2, \dots\]

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🧮 Statistics Calculators

🔔 Normal Distribution Calculator — Z Table

Critical Z Values

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📐 t-Distribution Calculator — Critical t Table

Critical t Values

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📊 Chi-Square Calculator — Critical Value Table

Critical χ² Values

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📈 F-Distribution Calculator — Critical F Table

Critical F Values (α = .05)

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🎲 Binomial Distribution Calculator

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🔢 Poisson Distribution Calculator

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