Statistics Summary Calculator

Discrete Distrubutions

Distribution

Formula

Mean μ

Variance σ²

Moment Generating Function

Bernoulli Distribution

p if k = 1, 1 - p if k = 0, else 0

p(1 - p)

1 - p + pe^t

Binomial Distribution

f(k;n,p) =	n! * p^k(1 - p)^{n - k}
	k!(n - k)!

np(1 - p)

(1 - p + pe^t)ⁿ

Negative Binomial Distribution

f(n;k,p) =	(n - 1)! * p^k(1 - p)^{n - k}
	(k - 1)!(n - k)!

k(1 - p)
p²

(1 - p)^r
(1 - pe^t)^r

Geometric Distribution

P(x = n) = p * (1 - p)^{(n - 1)}

pe^t
1 - (1 - p)e^t

Poisson Distribution

P(k; λ) =	λ^k
	e^λk!

e^{λ(e^t - 1)}

Hypergeometric Distribution

P(x;n,N,k) =	(_kC_x) * (_{N - k}C_{n - x})
	_NC_n

nk(N - k)(N - n)
N²(N - 1)

N/A

Multinomial Distribution

ƒ(x₀!·x₁!·x₂...&middot_n;x;n,θ₀,θ₁,θ₂,...,θ_n) =	n!(θ₀ · θ₁ · θ₂ · ... · θ_n)
	x₀ · x₁ · x₂ · ... · x_n

np_i

np_{i(1 - p_i)}

(Σp_ie^t_i)ⁿ

Uniform Distribution

P(k; λ) =	λ^k
	e^λk!

½a + b

(b - a)²
12

e^tb - e^ta
t(b - a)

Test Statistic Decision Tree:

Type

Keywords

Sample Size

Use Test

Z-score

Mean, Average

Greater than 30 or population σ is known

z =	X - μ
	σ/√n

t-score

Mean, Average

30 or less: population σ is not known

t =	x - μ
	s/√n

proportion-score

Proportion (Test p), Fraction, Percentage, Rate, Probability

more than 30

z =	p^ - p₀
	√p₀q₀/n

Variance σ²

Variance, Variability, Spread

N/A

χ² =	(n - 1)s²
	σ²

Equal Variances σ₁² = σ₂²

Equal Variances, Ratio or Difference in Variances

N/A

F =	σ₁²
	σ₂²

Author

David Miller

David Miller brings 15 years of experience in global economics, personal finance strategy, and market dynamics. He specializes in turning complex economic trends into actionable insights for everyday readers.