APSM 4375: Sport Analytics - Comprehensive Statistics Review

Comprehensive vocabulary flashcards covering the core concepts of descriptive statistics, distributions, data visualization, Pearson's correlation, simple and multiple linear regression, t-tests, and one-way ANOVA across sports analytics applications.

Analysis of Variance (ANOVA)

A statistical hypothesis test used to determine whether there are statistically significant differences among the means of three or more independent groups simultaneously in a single test, thereby preventing inflation of the false-positive error rate.

Multiple Comparisons Problem

The cumulative inflation of the overall false-positive rate ($\alpha$) that occurs when running multiple individual pairwise tests; for example, conducting 6 pairwise t-tests each at $\alpha = .05$ yields an overall probability of approximately $26\%$ of finding at least one false positive.

Between-Group Variance

In ANOVA, a measure of how much the individual group means differ from one another, addressing the question: 'Are the groups far apart?'

Within-Group Variance

In ANOVA, a measure of the variability or dispersion of individual observations within their respective groups, addressing the question: 'How noisy is the data within each group?'

F-Ratio

The test statistic for ANOVA, defined mathematically as $F = \frac{\text{Between-Group Variance}}{\text{Within-Group Variance}}$, named after Ronald Fisher; values substantially greater than 1 ($F > 1$) indicate group differences exceed random within-group noise.

Tukey's Honestly Significant Difference (HSD)

A post-hoc test conducted after a statistically significant ANOVA to compare every possible pair of group means while strictly controlling the family-wise error rate at $5\%$.

Levene's Test

An inferential test used to assess the assumption of homogeneity of variance across groups ($H_0$: variances are equal); if statistically significant ($p < .05$), the assumption is violated, requiring Welch's ANOVA and Games-Howell post-hoc tests.

Pearson's Correlation Coefficient ($r$)

A standardized measure of the strength and direction of a linear relationship between two continuous variables, ranging strictly from $-1.0$ (perfect negative linear relationship) to $+1.0$ (perfect positive linear relationship), with $0.0$ indicating no linear relationship.

Coefficient of Determination ($r^2$)

The square of Pearson's correlation coefficient, representing the proportion or percentage of total variance in the dependent variable ($Y$) that is explained or accounted for by the independent variable ($X$).

Reverse Causality

A causal fallacy where two variables $X$ and $Y$ are correlated, but changes in $Y$ actually drive changes in $X$ rather than vice versa (e.g., winning games causing increased social media engagement rather than social media spending causing winning).

Third Variable Problem

A situation where an unobserved confounding variable ($Z$) causes changes in both observed variables ($X$ and $Y$), creating a non-causal or spurious statistical correlation between them (e.g., summer heat driving both ice cream consumption and drowning rates).

Non-Linear Relationship (Aging Curve)

A curved relationship where the rate or direction of change alters across levels of the predictor variable—such as sprint speed increasing to an athletic prime before declining with age—resulting in a Pearson's $r$ near zero despite a strong, predictable association.

Normal Distribution

A symmetrical, bell-shaped continuous probability distribution where the mean, median, and mode are identical and centered, with observations tapering symmetrically into two tails.

Right-Skewed Distribution

An asymmetrical distribution with a long tail extending toward higher positive values, causing the arithmetic mean to be pulled higher than the median ($\text{Mean} > \text{Median}$), typical of athlete salaries and transfer fees.

Left-Skewed Distribution

An asymmetrical distribution with a long tail extending toward lower values, causing the arithmetic mean to be pulled lower than the median ($\text{Mean} < \text{Median}$), such as the age of Olympic gymnasts.

Empirical Rule (68-95-99.7 Rule)

A statistical principle for normally distributed data stating that approximately $68\%$ of observations fall within $\pm 1\text{ SD}$ of the mean, $95\%$ fall within $\pm 2\text{ SD}$, and $99.7\%$ fall within $\pm 3\text{ SD}$.

Box Plot

A standardized visual display of a dataset showing the median (central line), the interquartile range (IQR, middle $50\%$ of data), whiskers extending up to $1.5 \times \text{IQR}$, and individual points plotted beyond whiskers representing outliers.

Mean

The arithmetic average of a dataset, calculated by dividing the sum of all observed values by the sample size ($n$); it is highly sensitive to extreme outliers and skewed distributions.

Median

The middle value of an ordered dataset when sorted from lowest to highest; because it is resistant to extreme values, it is preferred over the mean when describing skewed distributions.

Mode

The most frequently occurring value or categorical observation within a dataset.

Standard Deviation

A measure of dispersion expressing the average distance or spread of data points from the mean in the original measurement units; used in sports analytics to quantify athlete consistency versus streakiness.

Z-Score

A standardized positional score indicating how many standard deviations an observation falls above or below the mean, calculated as $z = \frac{X - \mu}{\sigma}$.

Sabermetrics

The empirical, statistical analysis of baseball records, popularized by the 2002 Oakland Athletics who exploited market inefficiencies by prioritizing on-base percentage (OBP) over traditional batting average on a limited budget.

Simple Linear Regression

A statistical technique used to model and predict the value of a continuous dependent variable ($Y$) based on a single continuous independent predictor ($X$) using the linear equation $\hat{Y} = a + bX$.

Regression Slope ($b$)

The coefficient in a regression model indicating the predicted change in the dependent variable ($Y$) for every one-unit increase in the independent variable ($X$).

Regression Intercept ($a$)

The constant value in a regression equation representing the predicted value of the dependent variable ($Y$) when the independent variable ($X$) equals zero.

Standard Error of the Estimate

A metric assessing regression model accuracy by quantifying the standard deviation of residuals, representing the typical prediction error margin expressed in the original units of the dependent variable.

Regression to the Mean

The statistical phenomenon first identified by Francis Galton whereby extreme measurements or performances on an initial trial naturally tend to be followed by measurements closer to the long-term average.

Multiple Linear Regression

A regression extension modeling a continuous dependent outcome ($Y$) using two or more independent predictors ($\hat{Y} = a + b_1X_1 + b_2X_2 + \dots + b_kX_k$), isolating the unique contribution of each predictor while holding all other predictors constant.

Standardized Beta ($\text{Beta}$)

A regression slope converted into standard deviation units, allowing direct comparison of the relative predictive importance of independent variables measured on entirely different scales.

Adjusted $R^2$

A modified version of the coefficient of determination that incorporates a mathematical penalty for adding extraneous predictors that do not significantly improve model performance.

Homoscedasticity

A core regression assumption requiring that the variance of the residuals (errors) remains constant across all predicted values of the dependent variable, appearing as an even, random scatter on a diagnostic residual plot.

Heteroscedasticity

A violation of the homoscedasticity assumption in regression where residual spread systematically fans out or varies across predicted levels, rendering standard errors and $p$-values unreliable.

Multicollinearity

A regression modeling problem occurring when two or more independent predictors are highly correlated with each other ($r > .80$), making it difficult or impossible for the model to isolate their unique individual effects on $Y$.

Extrapolation (Regression)

The hazardous practice of using an estimated regression equation to make predictions for values of the independent variable ($X$) that lie outside the range of values used to fit the model.

Null Hypothesis ($H_0$)

The default statistical hypothesis positing that no true difference, effect, or relationship exists in the population (e.g., $\mu_1 = \mu_2$), presumed true until sufficient empirical evidence demonstrates otherwise.

Alternative Hypothesis ($H_1$)

The competing statistical hypothesis stating that a real difference, effect, or non-zero relationship exists in the underlying population.

$p$-Value

The conditional probability of observing test results at least as extreme as the sample data, assuming the null hypothesis ($H_0$) is true; values below $.05$ ($p < .05$) are designated as statistically significant.

Independent Samples t-Test

A parametric hypothesis test used to compare the means of exactly two separate, mutually exclusive groups on a single continuous outcome variable.

Paired Samples t-Test

A hypothesis test comparing two repeated measurements taken from the same subjects (e.g., pre-test vs. post-test) or matched pairs, which removes subject-to-subject variability and provides greater statistical power than independent testing.

t-Statistic

The calculated test statistic in a t-test, representing the ratio of the observed difference between group means to the standard error of that difference ($t = \frac{\bar{X}_1 - \bar{X}_2}{\text{SE}_{\text{diff}}}$).