Data Literacy Exam 1

X

the information you use to make a prediciton

Y

the outcome of what you want to predict

fitted line

summarizes the relationship between X and Y

Prediciton

the estimated value of Y for a particular X

Linear regression equation

Ŷ = α̂ + β̂X

Ŷ

predicted outcome is the value of Y the model predicts

α̂ (alpha hat)

intercept, the predicted Y when X=0

β̂ (beta hat)

the predicted change in Y when X increases by one unit

least squares

The model chooses the line that minimizes the sum of squared prediction errors by finding the line of best fit.

prediction error

vertical distance between an actual data point and the fitted line.

purpose of squaring errors

prevents positive and negative errors from canceling and penalizes larger errors more heavily.

lm()

fits a linear regression model

abline ()

adds the fitted line to a scatterplot

Y ~ X

predict Y using X

correlation

tells us the direction and strength of a linear relationship

positive correlation

as X increases, Y tends to increase

Negative correlation

as X increases, Y tends to decrease

Correlation near zero

little linear association, although a curved relationship may still exist

Correlation of +1 or −1

a perfectly linear relationship

linear probability model

A linear regression with a binary outcome

logistic regression

designed for binary outcomes

R²

coefficient of determination, measures the proportion of variation in Y explained by the model

coefficient of determination (R²)

R² = cor(X,Y)²

Higher R²

the model explains more of the variation in Y

R² = 1

the model explains 100% of the variation

R² = 0

the model explains none of the variation through a linear relationship

Straight line

learns two coefficients and captures the overall trend

Connect-the-dots

joins neighboring training observations, following individual data points much more closely

overfitting

when a model fits the peculiarities of a sample rather than the true underlying pattern, causing poor performance on new data

k-fold cross-validation:

cut the data into 𝑘 groups; each takes a turn as the test set; average the 𝑘 test errors, the lowest cross-validated error wins

neural network

many bent lines, glued together and stacked in layers

LASSO and ridge regression

many X’s, plus a penalty that shrinks the ̂𝛽s toward 0 to prevent overfitting

Decision trees and random forests

split the data into groups (e.g., under/over 30) and predict each group’s mean; a forest averages hundreds of trees

Large language models (ChatGPT, Claude)

neural networks with billions of ̂𝛽s, trained to predict the next word

features

predictors (𝑋)

target, label

outcome (𝑌)

weights and bias

slope(s) ̂𝛽 and intercept ̂𝛼

training a model

fitting it, e.g., with lm()

loss function

what we minimize: the sum of squared prediction errors

regression vs. classification

predicting a number vs. predicting a category

supervised vs. unsupervised learning

we observe 𝑌 vs. we only observe 𝑋 (e.g., finding groups of similar voters)

MEASURE:

To infer population characteristics via survey research (ex. what proportion of constituents support a particular policy)

PREDICT:

to make predictions (ex. who is the most likely candidate to win an upcoming election?)

EXPLAIN:

To estimate the causal effect of a treatment on an outcome (ex. what is the effect of small classrooms on student performance?)

Dataset

A collection of observations and variables

Observation

The unit or case represented by a row

Variable

A characteristic measured in the data

Unit of observation

The entity each observation represents

Unit of measurement

The scale or units in which a variable is measured

Binary variable

A variable with two possible values, often 0 and 1

Categorical variable

Places observations into categories

Quantitative variable

Measures a numerical quantity

The distribution of a variable

• Which values occur

• How often they occur

The distribution of a variable: center

Measured with:

• Mean

• Median

The distribution of a variable: spread

Variability in the data.

Measured with:

• Range

• Variance

• Standard deviation

The distribution of a variable: skew

Direction of a long tail.

• Right-skewed - tail right

• Left-skewed - tail left

The distribution of a variable: peaks

• Unimodal = one peak

• Bimodal = two peaks

Mean

Mean = (Sum of all values) / (Number of values)​

Mean of a Binary Variable

For a binary variable coded 1 and 0

Median

Middle observation after sorting

Mode

Most common value

Use the median when:

• Data are skewed

• There are outliers

Treatment Variable

The intervention or condition.

Examples:

• Receiving campaign mail

• Taking a drug

Outcome Variable

What is measured afterward.

Examples:

• Voting

• Recovery

Individual Causal Effect

Y(1)−Y(0)

the difference between a person's outcome with treatment, Y(1), and without treatment, Y(0)

Average Causal Effect (ACE)

Average of individual causal effects across all units.

E[Y(1)−Y(0)]

Fundamental Problem of Causal Inference

For any person we can observe:

• Treatment outcome OR

• Control outcome

But never both simultaneously.

Therefore: We never directly observe individual causal effects.

Selection Effects

People who choose treatment often differ from those who do not.

Example: People who exercise may already be healthier.

The difference you're seeing might be due to who chose the treatment, not the treatment itself.

Randomized Experiment

Researchers assign treatment randomly.

Advantages:

• Eliminates selection effects

• Strong causal evidence

Observational Study

Researcher only observes existing choices.

Disadvantages:

• Confounding

• Selection bias

Difference-in-Means Estimator

overline(Y)_T−overline(Y)_C​

Treatment-group average minus control-group average.

Measures estimated treatment effect.

When is Difference-in-Means Valid?

when:

E[Y(0)|T=1]=E[Y(0)∣T=0]

Meaning groups are comparable before treatment.

Best way to guarantee this:

Random assignment

Checking Comparability

Compare pre-treatment characteristics:

• Age

• Income

• Education

• Gender

• Prior behavior

Groups should look similar before treatment.

Percentage Points vs Percent

Example:

Treatment group voting: 45%

Control group voting: 40%

Difference:

45%−40%=5

Pros of Randomized Controlled Experiments

• Strong causal evidence

• Minimizes selection bias

• Easy interpretation

Cons of Randomized Controlled Experiments

• Expensive

• Ethical concerns

• Sometimes impractical

• Limited external validity

Random Assignment

Creates comparable treatment/control groups.

Helps causal inference.

Random Sampling

Creates representative samples.

Helps population inference.

These are different concepts.

Valid Population Inferences need:

• Representative sample

• Probability sampling

• Low nonresponse

Without representativeness, results may not generalize.

Nonresponse Bias

People who respond differ from those who don't.

Social Desirability Bias

People give socially acceptable answers rather than truthful answers.

Selection Bias

Sample differs systematically from population.

List Experiments Purpose:

Reduce social desirability bias when asking sensitive questions.

Respondents reveal only how many statements are true, not which ones.

Tables

Counts or percentages

Histogram

Shows distribution using bins

Density Plot

Smoothed histogram

Box Plot

Displays:

• Median

• Quartiles

• Outliers

Variance

Average squared distance from the mean.

s²

Standard Deviation

Square root of variance.

s

Interpretation: Typical distance from the mean.

Scatter Plot

Plots:

• X variable on horizontal axis

• Y variable on vertical axis

Used to visualize relationships.

Curved Relationships

Strong nonlinear relationships can have correlation near 0.

Outliers

One outlier can dramatically change correlation.

Always plot your data.

Ecological Fallacy

Occurs when conclusions about individuals are drawn from aggregate data.

Example: State-level patterns may not apply to individuals.

The unit of observation matters.

Correlation Is Not Causation

Correlation alone does not establish causality because:

• Reverse causation possible

• Confounding variables possible

Least Squares Method

∑(Y_i​−Y^_i​)²

β̂ (Slope)

Expected change in Y from a one-unit increase in X.

Units:

(Units of Y)/(Units of X)​

Binary Outcome Variables

• Predictions are probabilities

• Coefficients often interpreted in percentage points

Example:

β^=.03

A one-unit increase in X increases probability by 3 percentage points.

Extrapolation

Predicting beyond observed data.

Dangerous because the relationship may not continue.

Example: Estimating income for age 150.

R² (Coefficient of Determination)

Measures proportion of variation in Y explained by X.

Range:

0≤R²≤1

Interpreting R² = 0.4

Means:

40% of the variation in Y is explained by the model.

The remaining 60% is unexplained.