Common Statistical Symbols Reference

A lookup table covering the symbols you will encounter most often in research statistics. Symbols are shown in their proper mathematical typeset. Italicized symbols in APA papers are marked with a note.

Descriptive Statistics

Symbol Name Meaning / When Used
$$M$$ Sample mean The arithmetic average of a sample. APA uses M (italic). Some textbooks write \(\bar{X}\).
$$\mu$$ Population mean The true mean of the entire population. Greek letters denote population parameters.
$$SD$$ Sample standard deviation Average distance of scores from the sample mean. APA uses SD (italic). Some texts write s.
$$\sigma$$ Population standard deviation The true standard deviation of the population. \(\sigma^2\) is the population variance.
$$SE$$ Standard error (of the mean) The standard deviation of the sampling distribution. \(SE = SD / \sqrt{n}\). Used to build confidence intervals.
$$n$$ Sample size (subgroup) Number of participants in a subgroup or condition. Lowercase n.
$$N$$ Total sample size Total number of participants across all groups. Uppercase N.
$$\Sigma$$ Summation "Sum of." \(\Sigma X\) means add up all values of X. Appears in nearly every statistics formula.

Inferential Test Statistics

Symbol Name Meaning / When Used
$$t$$ t statistic Test statistic for t-tests. Measures how many standard errors the sample mean is from the null. Reported as t(df) = value.
$$F$$ F statistic Test statistic for ANOVA and regression. Ratio of between-group variance to within-group variance. Reported as F(df1, df2) = value.
$$\chi^2$$ Chi-square Test statistic for chi-square tests (independence, goodness of fit). Compares observed to expected frequencies. Reported as \(\chi^2\)(df, N = value) = value.
$$df$$ Degrees of freedom Number of values free to vary. Depends on the test: t-test df = N − 2; ANOVA has dfbetween and dfwithin.
$$p$$ p-value Probability of obtaining results at least as extreme as observed, assuming the null hypothesis is true. p < .05 is the conventional significance threshold.
$$CI$$ Confidence interval Range of plausible values for a parameter. A 95% CI means: if we repeated the study many times, 95% of intervals would contain the true value.

Hypothesis Testing & Error

Symbol Name Meaning / When Used
$$H_0$$ Null hypothesis States there is no effect or no difference. The hypothesis you are trying to reject. E.g., \(H_0\!: \mu_1 = \mu_2\).
$$H_1$$ Alternative hypothesis States there is an effect or a difference. Also written \(H_a\). E.g., \(H_1\!: \mu_1 \neq \mu_2\).
$$\alpha$$ Alpha (significance level) The threshold for rejecting \(H_0\). Conventionally set at .05. Also the probability of a Type I error (false positive).
$$\beta$$ Beta (Type II error rate) Probability of failing to reject \(H_0\) when it is false (false negative). Statistical power = \(1 - \beta\). Not the same as regression \(\beta\).

Effect Sizes & Relationships

Symbol Name Meaning / When Used
$$d$$ Cohen's d Standardized mean difference. The most common effect size for t-tests. Small = 0.2, medium = 0.5, large = 0.8.
$$r$$ Pearson correlation coefficient Measures the strength and direction of a linear relationship between two continuous variables. Ranges from −1 to +1.
$$R^2$$ Coefficient of determination Proportion of variance in the DV explained by the model. Ranges from 0 to 1. Used in regression.
$$\eta^2$$ Eta-squared Proportion of total variance explained by the IV. Effect size for ANOVA. Small = .01, medium = .06, large = .14.

Quick Memory Aids

  • Roman letters = sample statistics (M, SD, s, r, n). Greek letters = population parameters (\(\mu\), \(\sigma\), \(\rho\), \(\beta\)).
  • Lowercase n = subgroup size. Uppercase N = total sample.
  • \(\alpha\) is about Type I error (false alarm). \(\beta\) is about Type II error (miss). Power = \(1 - \beta\).
  • p-value: how surprised you should be if the null is true. Small p = surprising = reject \(H_0\).
  • r vs. R²: r is direction + strength; R² is variance explained. \(R^2 = r^2\) in simple regression.