Scalar Functions — Distributions / Transforms (83 Functions)

Database-independent scalar functions. Provides additional probability distributions (continuous and discrete), combinatorics, special functions, effect size conversions and interpretation, power analysis, and sample size calculations.

← Function Reference

Continuous Distribution Functions

Scalar functions. For each distribution, four variants are provided: PDF / CDF / Quantile / Rand.

Uniform Distribution

Function	Syntax	Description
`stat_uniform_pdf`	`stat_uniform_pdf(x [,a, b])`	Uniform distribution PDF (default [0,1])
`stat_uniform_cdf`	`stat_uniform_cdf(x [,a, b])`	Uniform distribution CDF
`stat_uniform_quantile`	`stat_uniform_quantile(p [,a, b])`	Uniform distribution quantile
`stat_uniform_rand`	`stat_uniform_rand([a, b])`	Uniform distribution random number

Exponential Distribution

Function	Syntax	Description
`stat_exponential_pdf`	`stat_exponential_pdf(x [,lambda])`	Exponential distribution PDF (default lambda=1)
`stat_exponential_cdf`	`stat_exponential_cdf(x [,lambda])`	Exponential distribution CDF
`stat_exponential_quantile`	`stat_exponential_quantile(p [,lambda])`	Exponential distribution quantile
`stat_exponential_rand`	`stat_exponential_rand([lambda])`	Exponential distribution random number

Gamma Distribution

Function	Syntax	Description
`stat_gamma_pdf`	`stat_gamma_pdf(x, shape, scale)`	Gamma distribution PDF
`stat_gamma_cdf`	`stat_gamma_cdf(x, shape, scale)`	Gamma distribution CDF
`stat_gamma_quantile`	`stat_gamma_quantile(p, shape, scale)`	Gamma distribution quantile
`stat_gamma_rand`	`stat_gamma_rand(shape, scale)`	Gamma distribution random number

Beta Distribution

Function	Syntax	Description
`stat_beta_pdf`	`stat_beta_pdf(x, alpha, beta)`	Beta distribution PDF
`stat_beta_cdf`	`stat_beta_cdf(x, alpha, beta)`	Beta distribution CDF
`stat_beta_quantile`	`stat_beta_quantile(p, alpha, beta)`	Beta distribution quantile
`stat_beta_rand`	`stat_beta_rand(alpha, beta)`	Beta distribution random number

Log-Normal Distribution

Function	Syntax	Description
`stat_lognormal_pdf`	`stat_lognormal_pdf(x [,mu, sigma])`	Log-normal distribution PDF (default mu=0, sigma=1)
`stat_lognormal_cdf`	`stat_lognormal_cdf(x [,mu, sigma])`	Log-normal distribution CDF
`stat_lognormal_quantile`	`stat_lognormal_quantile(p [,mu, sigma])`	Log-normal distribution quantile
`stat_lognormal_rand`	`stat_lognormal_rand([mu, sigma])`	Log-normal distribution random number

Weibull Distribution

Function	Syntax	Description
`stat_weibull_pdf`	`stat_weibull_pdf(x, shape, scale)`	Weibull distribution PDF
`stat_weibull_cdf`	`stat_weibull_cdf(x, shape, scale)`	Weibull distribution CDF
`stat_weibull_quantile`	`stat_weibull_quantile(p, shape, scale)`	Weibull distribution quantile
`stat_weibull_rand`	`stat_weibull_rand(shape, scale)`	Weibull distribution random number

-- Usage examples
SELECT stat_exponential_cdf(2.0, 0.5);       -- Exponential distribution CDF
SELECT stat_gamma_quantile(0.95, 2.0, 1.0);  -- Gamma distribution 95th percentile
SELECT stat_beta_pdf(0.3, 2, 5);             -- Beta distribution PDF
SELECT stat_weibull_cdf(3.0, 2.0, 1.5);     -- Weibull distribution CDF

Discrete Distribution Functions

For each distribution, four variants are provided: PMF / CDF / Quantile / Rand.

Binomial Distribution

Function	Syntax	Description
`stat_binomial_pmf`	`stat_binomial_pmf(k, n, p)`	Binomial distribution PMF
`stat_binomial_cdf`	`stat_binomial_cdf(k, n, p)`	Binomial distribution CDF
`stat_binomial_quantile`	`stat_binomial_quantile(q, n, p)`	Binomial distribution quantile
`stat_binomial_rand`	`stat_binomial_rand(n, p)`	Binomial distribution random number

Poisson Distribution

Function	Syntax	Description
`stat_poisson_pmf`	`stat_poisson_pmf(k, lambda)`	Poisson distribution PMF
`stat_poisson_cdf`	`stat_poisson_cdf(k, lambda)`	Poisson distribution CDF
`stat_poisson_quantile`	`stat_poisson_quantile(q, lambda)`	Poisson distribution quantile
`stat_poisson_rand`	`stat_poisson_rand(lambda)`	Poisson distribution random number

Geometric Distribution

Function	Syntax	Description
`stat_geometric_pmf`	`stat_geometric_pmf(k, p)`	Geometric distribution PMF
`stat_geometric_cdf`	`stat_geometric_cdf(k, p)`	Geometric distribution CDF
`stat_geometric_quantile`	`stat_geometric_quantile(q, p)`	Geometric distribution quantile
`stat_geometric_rand`	`stat_geometric_rand(p)`	Geometric distribution random number

Negative Binomial Distribution

Function	Syntax	Description
`stat_nbinom_pmf`	`stat_nbinom_pmf(k, r, p)`	Negative binomial distribution PMF
`stat_nbinom_cdf`	`stat_nbinom_cdf(k, r, p)`	Negative binomial distribution CDF
`stat_nbinom_quantile`	`stat_nbinom_quantile(q, r, p)`	Negative binomial distribution quantile
`stat_nbinom_rand`	`stat_nbinom_rand(r, p)`	Negative binomial distribution random number

Hypergeometric Distribution

Function	Syntax	Description
`stat_hypergeom_pmf`	`stat_hypergeom_pmf(k, N, K, n)`	Hypergeometric distribution PMF
`stat_hypergeom_cdf`	`stat_hypergeom_cdf(k, N, K, n)`	Hypergeometric distribution CDF
`stat_hypergeom_quantile`	`stat_hypergeom_quantile(q, N, K, n)`	Hypergeometric distribution quantile
`stat_hypergeom_rand`	`stat_hypergeom_rand(N, K, n)`	Hypergeometric distribution random number

Bernoulli Distribution

Function	Syntax	Description
`stat_bernoulli_pmf`	`stat_bernoulli_pmf(k, p)`	Bernoulli distribution PMF
`stat_bernoulli_cdf`	`stat_bernoulli_cdf(k, p)`	Bernoulli distribution CDF
`stat_bernoulli_quantile`	`stat_bernoulli_quantile(q, p)`	Bernoulli distribution quantile
`stat_bernoulli_rand`	`stat_bernoulli_rand(p)`	Bernoulli distribution random number

Discrete Uniform Distribution

Function	Syntax	Description
`stat_duniform_pmf`	`stat_duniform_pmf(k, a, b)`	Discrete uniform distribution PMF
`stat_duniform_cdf`	`stat_duniform_cdf(k, a, b)`	Discrete uniform distribution CDF
`stat_duniform_quantile`	`stat_duniform_quantile(q, a, b)`	Discrete uniform distribution quantile
`stat_duniform_rand`	`stat_duniform_rand(a, b)`	Discrete uniform distribution random number

-- Usage examples
SELECT stat_binomial_pmf(3, 10, 0.5);       -- P(X=3) for Bin(10, 0.5)
SELECT stat_poisson_cdf(5, 3.0);             -- P(X<=5) for Poisson(3)
SELECT stat_hypergeom_pmf(4, 50, 10, 8);     -- Hypergeometric distribution

Combinatorics

Function	Syntax	Description
`stat_binomial_coef`	`stat_binomial_coef(n, k)`	Binomial coefficient C(n,k) (returns integer)
`stat_log_binomial_coef`	`stat_log_binomial_coef(n, k)`	Log binomial coefficient log C(n,k)
`stat_log_factorial`	`stat_log_factorial(n)`	Log factorial log(n!)

SELECT stat_binomial_coef(10, 3);       -- -> 120
SELECT stat_log_factorial(100);          -- -> 363.739...

Special Functions

Function	Syntax	Description
`stat_lgamma`	`stat_lgamma(x)`	Log-gamma function ln Gamma(x)
`stat_tgamma`	`stat_tgamma(x)`	Gamma function Gamma(x)
`stat_beta_func`	`stat_beta_func(a, b)`	Beta function B(a,b)
`stat_lbeta`	`stat_lbeta(a, b)`	Log-beta function ln B(a,b)
`stat_erf`	`stat_erf(x)`	Error function erf(x)
`stat_erfc`	`stat_erfc(x)`	Complementary error function erfc(x)
`stat_logarithmic_mean`	`stat_logarithmic_mean(a, b)`	Logarithmic mean

SELECT stat_tgamma(5);          -- -> 24.0 (= 4!)
SELECT stat_lgamma(10);         -- -> 12.8018...
SELECT stat_erf(1.0);           -- -> 0.8427...
SELECT stat_logarithmic_mean(2, 8);  -- Logarithmic mean

Effect Size Conversions

Function	Syntax	Description
`stat_hedges_j`	`stat_hedges_j(n)`	Hedges correction factor J(n)
`stat_t_to_r`	`stat_t_to_r(t, df)`	t-value to correlation coefficient r conversion
`stat_d_to_r`	`stat_d_to_r(d)`	Cohen's d to correlation coefficient r conversion
`stat_r_to_d`	`stat_r_to_d(r)`	Correlation coefficient r to Cohen's d conversion
`stat_eta_squared_ef`	`stat_eta_squared_ef(ss_effect, ss_total)`	Eta-squared (effect size)
`stat_partial_eta_sq`	`stat_partial_eta_sq(F, df1, df2)`	Partial eta-squared
`stat_omega_squared_ef`	`stat_omega_squared_ef(ss_effect, ss_total, ms_error, df_effect)`	Omega-squared (effect size)
`stat_cohens_h`	`stat_cohens_h(p1, p2)`	Cohen's h (proportion difference effect size)

Effect Size Interpretation

Function	Syntax	Return	Description
`stat_interpret_d`	`stat_interpret_d(d)`	TEXT	Cohen's d interpretation (small/medium/large)
`stat_interpret_r`	`stat_interpret_r(r)`	TEXT	Correlation coefficient interpretation
`stat_interpret_eta2`	`stat_interpret_eta2(eta2)`	TEXT	Eta-squared interpretation

SELECT stat_d_to_r(0.5);           -- Cohen's d to r conversion
SELECT stat_interpret_d(0.8);       -- -> 'large'
SELECT stat_interpret_r(0.3);       -- -> 'medium'
SELECT stat_cohens_h(0.6, 0.4);    -- Proportion difference effect size

Power Analysis

Function	Syntax	Description
`stat_power_t1`	`stat_power_t1(d, n, alpha)`	One-sample t-test power
`stat_n_t1`	`stat_n_t1(d, power, alpha)`	One-sample t-test required sample size
`stat_power_t2`	`stat_power_t2(d, n1, n2, alpha)`	Two-sample t-test power
`stat_n_t2`	`stat_n_t2(d, power, alpha)`	Two-sample t-test required sample size
`stat_power_prop`	`stat_power_prop(p1, p2, n, alpha)`	Proportion test power
`stat_n_prop`	`stat_n_prop(p1, p2, power, alpha)`	Proportion test required sample size

-- Power for effect size d=0.5, n=30, alpha=0.05
SELECT stat_power_t1(0.5, 30, 0.05);

-- Required n for effect size d=0.5, power 80%, alpha=0.05
SELECT stat_n_t1(0.5, 0.8, 0.05);

-- Required sample size for a two-group proportion test
SELECT stat_n_prop(0.5, 0.3, 0.8, 0.05);

Sample Size / Margin of Error

Function	Syntax	Description
`stat_moe_prop`	`stat_moe_prop(x, n [,confidence])`	Margin of error for proportion
`stat_moe_prop_worst`	`stat_moe_prop_worst(n [,confidence])`	Worst-case margin of error (p=0.5)
`stat_n_moe_prop`	`stat_n_moe_prop(moe [,confidence [,p]])`	Required sample size for proportion estimation
`stat_n_moe_mean`	`stat_n_moe_mean(moe, sigma [,confidence])`	Required sample size for mean estimation

-- Margin of error for 200 successes out of n=400 (95% confidence interval)
SELECT stat_moe_prop(200, 400);

-- Sample size needed to achieve a margin of error within 3%
SELECT stat_n_moe_prop(0.03);

-- Mean estimation: margin of error within 2.0, sigma=15
SELECT stat_n_moe_mean(2.0, 15.0);