Mean (Arithmetic Average)
x̄ = (Σxᵢ) / n
The sum of all values divided by the number of values.
Variables
x̄= Sample mean
Σxᵢ= Sum of all values
n= Number of values
Reference Guide
Statistics Formula Reference
A comprehensive collection of essential statistics formulas with explanations, variable definitions, and links to related calculators and guides.
Central Tendency
Measures of the center of a dataset
Weighted Mean
x̄ᵥ = (Σwᵢxᵢ) / (Σwᵢ)
Average where each value has an associated weight.
Variables
wᵢ= Weight of each value
xᵢ= Data values
Median Position
Position = (n + 1) / 2
The middle value when data is sorted. For even n, average the two middle values.
Variables
n= Number of values
Measures of Spread
Formulas describing data variability
Population Variance
σ² = Σ(xᵢ - μ)² / N
Average of squared deviations from the population mean.
Variables
σ²= Population variance
μ= Population mean
N= Population size
Sample Variance
s² = Σ(xᵢ - x̄)² / (n - 1)
Unbiased estimator of population variance using Bessel's correction.
Variables
s²= Sample variance
x̄= Sample mean
n-1= Degrees of freedom
Standard Deviation
σ = √σ² or s = √s²
Square root of variance, measuring spread in original units.
Variables
σ= Population standard deviation
s= Sample standard deviation
Coefficient of Variation
CV = (s / x̄) × 100%
Relative measure of variability, useful for comparing datasets.
Variables
CV= Coefficient of variation
s= Standard deviation
x̄= Mean
Probability
Fundamental probability formulas
Basic Probability
P(A) = n(A) / n(S)
Probability equals favorable outcomes divided by total outcomes.
Variables
P(A)= Probability of event A
n(A)= Number of favorable outcomes
n(S)= Total number of outcomes
Permutations
P(n,r) = n! / (n-r)!
Number of arrangements where order matters.
Variables
n= Total items
r= Items chosen
!= Factorial
Combinations
C(n,r) = n! / (r!(n-r)!)
Number of selections where order doesn't matter.
Variables
n= Total items
r= Items chosen
Binomial Probability
P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
Probability of exactly k successes in n independent trials.
Variables
n= Number of trials
k= Number of successes
p= Probability of success
Z-Scores & Normal Distribution
Standardization and normal distribution formulas
Z-Score
z = (x - μ) / σ
Number of standard deviations a value is from the mean.
Variables
z= Z-score
x= Individual value
μ= Mean
σ= Standard deviation
Standard Error of Mean
SE = σ / √n
Standard deviation of the sampling distribution of the mean.
Variables
SE= Standard error
σ= Population standard deviation
n= Sample size
Correlation & Regression
Relationship and prediction formulas
Pearson Correlation Coefficient
r = Σ[(xᵢ-x̄)(yᵢ-ȳ)] / √[Σ(xᵢ-x̄)²Σ(yᵢ-ȳ)²]
Measures linear relationship strength between two variables (-1 to 1).
Variables
r= Correlation coefficient
x̄, ȳ= Means of x and y
Coefficient of Determination
R² = r²
Proportion of variance in y explained by x.
Variables
R²= Coefficient of determination (0 to 1)
Linear Regression Slope
b = Σ[(xᵢ-x̄)(yᵢ-ȳ)] / Σ(xᵢ-x̄)²
The slope of the best-fit line.
Variables
b= Slope
x̄, ȳ= Means
Linear Regression Intercept
a = ȳ - b × x̄
The y-intercept of the regression line.
Variables
a= Y-intercept
b= Slope
Confidence Intervals
Interval estimation formulas
Confidence Interval for Mean (σ known)
CI = x̄ ± z*(σ/√n)
Interval estimate when population standard deviation is known.
Variables
x̄= Sample mean
z*= Critical z-value
σ/√n= Standard error
Confidence Interval for Mean (σ unknown)
CI = x̄ ± t*(s/√n)
Interval estimate using t-distribution when σ is unknown.
Variables
t*= Critical t-value
s= Sample standard deviation
n-1= Degrees of freedom
Margin of Error
E = z* × (σ/√n)
Half-width of the confidence interval.
Variables
E= Margin of error
z*= Critical value
Sample Size Determination
Formulas for planning studies
Sample Size for Mean
n = (z*σ/E)²
Required sample size for estimating a population mean.
Variables
n= Required sample size
z*= Critical value
σ= Population standard deviation
E= Desired margin of error
Sample Size for Proportion
n = p(1-p)(z*/E)²
Required sample size for estimating a population proportion.
Variables
p= Estimated proportion
E= Desired margin of error
Hypothesis Testing
Test statistics formulas
Chi-Square Test Statistic
χ² = Σ[(O-E)²/E]
Measures difference between observed and expected frequencies.
Variables
O= Observed frequency
E= Expected frequency
ANOVA F-Statistic
F = MSB / MSW
Ratio of between-group variance to within-group variance.
Variables
MSB= Mean square between groups
MSW= Mean square within groups
SS= Sum of squares
df= Degrees of freedom
T-Test Statistic
t = (x̄ - μ₀) / (s/√n)
Tests whether sample mean differs from hypothesized value.
Variables
x̄= Sample mean
μ₀= Hypothesized mean
s/√n= Standard error
Common Statistical Symbols
μ (mu)Population mean
σ (sigma)Population standard deviation
x̄ (x-bar)Sample mean
sSample standard deviation
nSample size
NPopulation size
Σ (sigma)Sum of values
α (alpha)Significance level
β (beta)Type II error probability
ρ (rho)Population correlation
rSample correlation
H₀Null hypothesis
H₁ or HₐAlternative hypothesis
dfDegrees of freedom
p-valueProbability value
CIConfidence interval