Statistics Review: Central Tendency, Variability, Z-Scores, Probability, and Hypothesis Testing

A set of vocabulary flashcards covering fundamental statistical concepts, formulas, and terminology from the review lecture.

Balance Point

The mean of a distribution, which is the measure of central tendency where all scores balance.

Mode

The measure of central tendency defined as the score or category that occurs with the greatest frequency in a data set.

Positively Skewed Distribution

A distribution shape in which the measures of central tendency from left to right fall in the order of Mode, Median, and Mean.

Negatively Skewed Distribution

A distribution shape in which scores pile up at the high end and trail off toward the lower values.

Variability

A measure of the degree to which the scores in a distribution are clustered together or spread apart.

Standard Deviation

A measure of variability indicating how far, on average, a score in the distribution deviates from the mean.

Sum of Squares ($SS$)

The sum of the squared deviations from the mean, calculated as $SS = \text{Σ}(X - M)^2$ or using the computational formula $SS = \text{Σ}X^2 - \frac{(\text{Σ}X)^2}{n}$.

Population Variance ($\sigma^2$)

The mean of the squared deviations in a population, calculated as $\sigma^2 = \frac{SS}{N}$.

Sample Variance ($s^2$)

The sample sum of squares divided by the degrees of freedom, calculated as $s^2 = \frac{SS}{n - 1}$.

$z$-Score

A standardized value where the sign ($+$ or $-$) identifies whether the score is above or below the mean, and the numerical value corresponds to the number of standard deviations between $X$ and the mean.

Purposes of $z$-Scores

1. To specify an exact location of an $X$-value within a distribution. 2. To standardize a distribution so that different distributions can be made comparable.

Independent Random Sample

A sample obtained using a process that gives every individual an equal chance of being selected and keeps the probability of being selected constant over a series of selections.

Alpha Level ($\alpha$)

The level of significance chosen for a hypothesis test, which represents the probability of making a Type I error.

Type I Error

The error of rejecting the null hypothesis ($H_0$) when the null hypothesis is actually true.

Type II Error

The error of retaining the null hypothesis ($H_0$) when the null hypothesis is actually false.

Statistical Power

The probability of correctly rejecting a false null hypothesis, which increases with a larger sample size, stronger treatments, a less stringent $\alpha$, or a one-tailed test.

One-Sample $z$-Test

The statistical test used when comparing a sample mean to a population mean when both the population mean ($\mu$) and the population variance ($\sigma^2$) are known.

Standard Error of the Mean ($\sigma_M$)

The standard deviation of the sampling distribution of sample means, which serves as the denominator in the one-sample $z$-test formula: $z = \frac{M - \mu}{\sigma_M}$.

Pooled Variance ($s_p^2$)

A combined estimate of variance for two samples, calculated as $s_p^2 = \frac{SS_1 + SS_2}{df_1 + df_2} = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{(n_1 - 1) + (n_2 - 1)}$.

Estimated Standard Error of the Difference ($s_{\text{diff}}$)

The estimated standard error used in an independent-samples $t$-test when sample sizes differ, calculated as $s_{\text{diff}} = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}$.

Mean Difference ($M_D$)

The average of the difference scores ($D = X_2 - X_1$) across individuals in a repeated-measures design, computed as $M_D = \frac{\text{Σ}D}{n}$.