Advanced Stats symbols and definitions

X

A score or measured value on variable X

Y

A second variable or outcome score, often the variable being predicted

Xᵢ

The score of the ith observation or person on X

Σ

Summation sign: add all values that follow it

N

Number of scores or participants in the population or total data set

n

Number of scores or participants in a sample; sample size

X̄

Sample mean (sample average): X̄ = ΣX / n

μ

Population mean: the true average for the whole population

s

Sample standard deviation: typical spread of sample scores around X̄

s²

Sample variance: the square of the sample standard deviation

σ

Population standard deviation: true population spread

σ²

Population variance: the square of the population standard deviation

z

Standard score: distance of a score from the mean in standard-deviation units

df

Degrees of freedom: independent information available for estimating a quantity

SE

Standard error: expected sample-to-sample variability of a statistic

SEₓ̄

Standard error of the mean; commonly estimated as s / √n

p

Probability or observed significance value, depending on context

α

Preset Type I error rate or significance criterion, commonly .05

β

Type II error probability: failing to reject a false null hypothesis

1 − β

Statistical power: probability of detecting a real effect

H₀

Null hypothesis: usually no population difference, effect, or relationship

Hₐ or H₁

Alternative hypothesis: a population difference, effect, or relationship exists

t

t-test statistic: observed difference relative to chance variability

tcrit

Critical t cutoff for deciding whether to reject H₀

F

Ratio of two variance estimates; often F = MSeffect / MSerror

χ²

Chi-square statistic; compares observed frequencies with expected frequencies

O

Observed frequency: count actually found in a category or cell

E

Expected frequency: count predicted under the null hypothesis

D

Difference score, commonly D = X₁ − X₂

D̄

Mean of the difference scores

sD

Standard deviation of difference scores

CI

Confidence interval: an interval estimate of a population parameter

LL

Lower limit of a confidence interval

UL

Upper limit of a confidence interval

≠

Not equal to; commonly used for a two-tailed alternative hypothesis

>

Greater than; used in a directional alternative hypothesis

<

Less than; used in a directional alternative hypothesis

P(A)

Probability that event A occurs

P(A|B)

Probability that A occurs given that B has occurred

A ∩ B

Intersection: both events A and B occur

A ∪ B

Union: A occurs, B occurs, or both occur

q

Complement probability; q = 1 − p

SS

Sum of squares: measure of squared variability

SStotal

Total variability of all scores around the grand mean

SSbetween

Variability attributable to differences among group or treatment means

SSwithin

Variability among people within groups; unexplained or error variation

SSerror

Error or unexplained sum of squares

SSsubjects

Variability attributable to individual differences; especially used in repeated-measures ANOVA

MS

Mean square: a sum of squares divided by its degrees of freedom

MSbetween

Between-groups mean square: SSbetween / dfbetween

MSwithin

Within-groups mean square: SSwithin / dfwithin

MSerror

Error mean square used as the denominator of an F ratio

dfbetween

Between-groups degrees of freedom; usually k − 1 in one-way ANOVA

dfwithin

Within-groups degrees of freedom; usually N − k in one-way ANOVA

k

Number of groups or treatment levels in a one-way ANOVA

A, B

Factors A and B in a factorial design

A × B

Interaction: the effect of A depends on the level of B

a, b

Number of levels of Factors A and B

η²

Eta squared: proportion of total variance accounted for by an effect

ηp²

Partial eta squared: proportion of effect-plus-error variance linked to an effect

ω²

Omega squared: less biased estimate of population variance explained

d

Cohen’s d: standardized mean difference in standard-deviation units

g

Hedges’ g: small-sample-corrected standardized mean difference

r

Pearson sample correlation: direction and strength of a linear relationship

ρ

Population correlation: population counterpart of sample r

rXY

Correlation between variables X and Y

r²

Coefficient of determination: proportion of variance associated with a linear relationship

R

Multiple correlation between observed Y and Y predicted from several predictors

R²

Multiple coefficient of determination: proportion of variance in Y explained by predictors together

Adjusted R²

R² corrected for sample size and number of predictors

rXY.Z

Partial correlation of X and Y while controlling Z

rs

Spearman rank-order correlation

τ

Kendall’s tau: rank-based correlation coefficient

φ

Phi coefficient: association between two dichotomous variables

b

Unstandardized regression slope: predicted change in Y for a one-unit increase in X

b₀

Regression intercept: predicted Y when all predictors equal zero

b₁, b₂, …

Regression slopes for Predictor 1, Predictor 2, and so forth

β

Standardized regression coefficient; do not confuse with Type II error probability

Ŷ

Predicted value of Y from a regression equation

Y′

Another notation for predicted Y

e

Regression residual or prediction error

Y − Ŷ

Residual: observed Y minus predicted Y

SEb

Standard error of a regression coefficient

SEest

Standard error of estimate: typical prediction-error size

U

Mann–Whitney U statistic for comparing two independent groups with ranks

W

Wilcoxon signed-rank statistic for paired/ranked difference data

H

Kruskal–Wallis statistic for comparing three or more independent groups with ranks

T

Test statistic or rank sum; exact meaning depends on the nonparametric test

λ

Noncentrality parameter, used in power calculations and noncentral distributions

Λ

Wilks’s lambda: a multivariate test statistic, often used in MANOVA

MSE

Mean squared error: average unexplained squared variation

SSE

Sum of squared errors: total squared residual or unexplained variation

SSR

Regression sum of squares: variation explained by a regression model

SST

Total sum of squares: total variation in the outcome

p̂

Sample proportion

π

Population proportion