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