Unit 2: Exploring Two-Variable Data

Notes from every Collegeboard daily video turned into flashcards

Segmented bar chart characteristics

Bars with spaces in between measured by relative frequency on the x axis. Each bar has sections indicated by a key.

Mosaic Plot

Same as segmented bar chart but bars touch and the width of the bars is proportional to the range size.

Scatterplot

Represents 2 quantitative variables. Scale shown with tickmarks and there must be a title.

Side-by-side bar graph

Values are measured in relative frequency (percents are taken from a 2-way table) and each of the relative frequencies of one x-variable are graphed side-by-side touching but separate from the next x-variable range.

Variable Association

If the distribution of conditional relative frequencies differ for each group.

Joint relative frequency

Cell frequency divided by total for entire table

Marginal relative frequency

Row + Column totals in a 2-way table divided by the table total

Conditional relative frequency

Relative frequency for a specific part of the 2-way table

x-variable name

Explanatory (x can explain y)

y-variable name

Response

Positive/Negative association

Positive: as x increases, y tends to increase

Negative: as x increases, y tends to decrease

Forms

Linear or non-linearU

Unusual Features

Clusters and apparent outliers

Strong/Weak strength

Strong: Data closely follows pattern

Weak: doesn’t closely follow but there is still a pattern

Determines how good of a predictor x is of y

Correlation Coefficient - r

Gives direction and strength of a linear relationship with the closeness to 1/-1 indicating a strong correlation

Alone - Doesn’t provide information for claims on form or features (need scatterplot for that)

Causation Fallacies

Some variables aren’t considered or correlations are completely coincidental

Linear Regression Model

Y-hat = a + bx

Y-hat = predicted value

Extrapolation

Unreliable. Prediction made outside the current data’s interval, meaning the current trend is not guaranteed to continue

Residuals

r = observed y - predicted y

r = y - y-hat

[positive = underestimate]

Residual plot (scatterplot)

y-axis plots residual values

Good fit = apparent randomness centered at zero with no clear patterns

Bad fit = curved pattern (not random)

Least Squares Regression Line (LSRL)

Linear model that minimizes the sum of r²

LSRL Properties

(x-hat, y-hat) will be represented on the linear model

Slope and Correlation —> b = r(S_y/S_x) [b = slope S = Standard deviation of either y or x]

Coefficient of Determination (r²)

Proportion of variation in the y-variable that is explained by the x-variable (how good the model is)

r and r² ranges

-1≤ r ≤1

0≤ r² ≤1

Effect of removing low/high-leverage points

Low: LSRL slope/y-intercept doesn’t change much

High: Substantial shift from original LSRL

High-leverage points

Unusually large or small x-values from x-bar (mean). Not high-leverage if close to x-bar

Low-leverage points

Close to (x-bar, y-bar), and especially close to x-bar

Influential points

Points that if removed would result in a big change in slope, y-intercept, and/or correlation (outliers, high-leverage points, both)

Log transformations

Taking the log of a variable makes high values less extreme and reduces skew (for x-variable logs) and is used to turn non-linear data into linear data