Statistical Formulas

Measures of Central Tendency

Arithmetic Mean

\(\bar{x} = \frac{\sum(x)}{n}\)

Measures of Dispersion

Sample Variance

\(s^2 = \frac{\sum(x-\bar{x})^2}{n-1}\)

Sample Standard Deviation

\(s = \sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}\)

Z-Score

\(z = \frac{x - \bar{x}}{s}\)

Correlation and Covariance

\(r = \frac{\sum(X-\bar{X})(Y-\bar{Y})}{(s_x)(s_y)(N-1)}\)

Inferential Statistics

Single Sample T-Test

\(t = \frac{\bar{x}-\mu}{\sigma_{\bar{x}}}\)

  • \(t\) = The t-statistic, representing how far the sample mean is from the population mean (in standard error units).

  • \(\bar{X}\) = The sample mean of your data.

  • \(\mu_0\) = The hypothesized population mean (the value you’re comparing your sample to).

  • \(\sigma_{\bar{X}}\) = The standard error of the mean, calculated as:

    • \(\sigma_{\bar{X}} = \frac{s}{\sqrt{n}}\)

Paired-Samples T-Test

\(t = \frac{\bar{D}}{\sigma_{\bar{D}}}\)

  • \(\bar{D}\) = Mean difference between T1 and T2

  • \(\sigma_{\bar{D}}\) = Standard Error of the mean difference

    • \(\frac{\sigma_{\bar{D}}}{n}\)

Independent-Samples T-Test

\(t = \frac{\bar{X}_1-\bar{X}_2}{\sqrt{\frac{s^2_1}{N_1}-\frac{s^2_2}{N_2}}}\)

  • \(t\) = The t-statistic, representing how far apart the two sample means are, relative to the variability in the data.

  • \(\bar{X}_1\) and \(\bar{X}_2\) = The means of the two independent samples (Group 1 and Group 2).

  • \(s_1^2\) and \(s_2^2\) = The sample variances of the two groups.

  • \(n_1\) and \(n_2\) = The sample sizes of the two groups.

One-Way ANOVA (Between)

\(F = \frac{MS_{\text{between}}}{MS_{\text{within}}} = \frac{\frac{SS_{\text{between}}}{df_{\text{between}}}}{\frac{SS_{\text{within}}}{df_{\text{within}}}}\)

  • \(F\) = Test statistic for the ANOVA

  • \(MS_{between}\) = Mean Square Between Groups

  • \(MS_{within}\) = Mean Square Within Groups (also called Error)

  • \(SS_{between}\) = Sum of Squares Between Groups

  • \(SS_{within}\) = Sum of Squares Within Groups

  • \(df_{between}\) = Degrees of Freedom Between Groups

  • \(df_{within}\) = Degrees of Freedom Within Groups