Exam Formulae and Statistical Tables Reference

Vocabulary and reference flashcards covering statistical formulas, concepts, and lookup tables for normal, t, and chi-square distributions.

Mean ($M$)

The arithmetic average of a group of scores, defined by the formula: $M = \frac{\sum X}{N}$.

Population Variance ($SD^2$)

The average of the squared deviations from the mean in a population, given by: $SD^2 = \frac{\sum (X - M)^2}{N}$.

Population Standard Deviation ($SD$)

The square root of the population variance, representing the spread of scores: $SD = \sqrt{\frac{\sum (X - M)^2}{N}}$.

$Z$ Score (Formula)

The number of standard deviations a raw score lies above or below the mean: $Z = \frac{X - M}{SD}$.

Raw Score ($X$) from $Z$ Score

The formula used to convert a standardized $Z$ score back into an original raw score: $X = (Z)(SD) + M$.

Pearson Correlation Coefficient ($r$)

A measure of the linear relationship between two variables, computed as the average product of paired $Z$ scores: $r = \frac{\sum Z_X Z_Y}{N}$.

$Z$ Score for a Sample Mean ($Z_M$)

The standardized distance between a sample mean and the population mean of a distribution of means: $Z_M = \frac{M - \mu_M}{\sigma_M}$.

Standard Error of the Mean ($\sigma_M$)

The standard deviation of the distribution of means when population standard deviation is known: $\sigma_M = \frac{\sigma}{\sqrt{N}}$.

Variance of the Distribution of Means ($\sigma_M^2$)

The variance of a distribution of sample means, calculated as: $\sigma_M^2 = \frac{\sigma^2}{N}$.

Confidence Interval ($CI$)

A range of estimates calculated around a sample mean that is likely to include the population mean: $CI = M \pm (Z)(\sigma_M)$.

Estimated Population Variance ($S^2$)

An unbiased estimate of the population variance computed from sample deviations: $S^2 = \frac{\sum (X - M)^2}{N - 1}$.

Estimated Population Standard Deviation ($S$)

The square root of the estimated population variance: $S = \sqrt{\frac{\sum (X - M)^2}{N - 1}}$.

Estimated Standard Error of the Mean ($S_M$)

The standard error estimated from sample variance: $S_M = \frac{S}{\sqrt{N}} = \sqrt{\frac{S^2}{N}}$.

Single-Sample $t$ Statistic

The test statistic used to compare an observed sample mean to a population mean with unknown variance: $t = \frac{M - \mu}{S_M}$.

Degrees of Freedom ($df$) for a Single-Sample $t$ Test

The number of independent values that are free to vary in a sample of size $N$: $df = N - 1$.

Pooled Variance ($S_{\text{pooled}}^2$)

A weighted average of the sample variances from two groups based on their respective degrees of freedom: $S_{\text{pooled}}^2 = \left(\frac{df_1}{df_{\text{total}}}\right)(S_1^2) + \left(\frac{df_2}{df_{\text{total}}}\right)(S_2^2)$.

Standard Error of the Difference ($S_{\text{difference}}$)

The estimated standard deviation of the distribution of differences between two independent sample means: $S_{\text{difference}} = \sqrt{S_{M_1}^2 + S_{M_2}^2} = \sqrt{\frac{S_1^2}{N_1} + \frac{S_2^2}{N_2}}$.

Independent-Samples $t$ Statistic

The test statistic comparing the difference between two sample means against the standard error of the difference: $t = \frac{M_1 - M_2}{S_{\text{difference}}}$.

Degrees of Freedom ($df$) for an Independent-Samples $t$ Test

The combined degrees of freedom from two independent samples: $df = (N_1 - 1) + (N_2 - 1)$.

Eta Squared ($\eta^2$)

An effect size measure representing the proportion of total variance accounted for by the group differences: $\eta^2 = \frac{t^2}{t^2 + df}$.

Expected Frequency ($E$)

The frequency expected in a cell of a contingency table assuming independence between variables: $E = \frac{R \times C}{N}$, where $R$ is the row total and $C$ is the column total.

Chi-Square Statistic ($\chi^2$)

A test statistic that sums the normalized squared discrepancies between observed and expected frequencies: $\chi^2 = \sum \frac{(O - E)^2}{E}$.

Degrees of Freedom ($df$) for Goodness-of-Fit Test

The degrees of freedom for a single-variable chi-square test: $df = \text{Categories} - 1$.

Degrees of Freedom ($df$) for Chi-Square Test of Independence

The degrees of freedom for a two-way contingency table: $df = (\text{Columns} - 1) \times (\text{Rows} - 1)$.

Phi Coefficient ($\phi$)

A measure of effect size for a $2 \times 2$ chi-square contingency table: $\phi = \sqrt{\frac{\chi^2}{N}}$.

Table 1 (Normal Curve Areas)

A table providing the percentage of the normal curve between the mean and given $Z$ scores ($\% \text{ Mean to } Z$) along with the percentage of scores in the tail ($\% \text{ in Tail}$).

Table 2 (Cutoff Scores for the $t$ Distribution)

A reference table specifying the critical cutoff values for one-tailed and two-tailed hypothesis tests at significance levels of $0.10$, $0.05$, and $0.01$ across varying degrees of freedom ($df$).

Table 4 (Cutoff Scores for the Chi-Square Distribution)

A reference table providing critical cutoff scores for the chi-square ($\chi^2$) distribution across degrees of freedom ($df = 1$ through $10$) at significance levels of $0.10$, $0.05$, and $0.01$.