Statistics and Data Interpretation

Data Analysis & Statistics

Part 1 of 7 — Mean, Median, Mode, Range

Mean (Average)

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Median

The middle value when data is ordered. For even number of values: average the two middle values.

$\{3, 5, 7, 9, 11\}$ → median = $7$

$\{3, 5, 7, 9\}$ → median = $(5 + 7)/2 = 6$

Mode

Most frequent value. Can have none, one, or multiple modes.

Range

Range $=$ max $-$ min

SAT Favorites 🎯

"Adding/removing a value": If the mean of 5 numbers is 20, their sum is 100. Add a 6th number = 38: new mean $= 138/6 = 23$.

"Which measure changes?": Adding an outlier affects the mean much more than the median.

Central Tendency 🎯

Worked Example 1 — Finding a Missing Value

The mean of 6 numbers is 12. Five of the numbers are 8, 10, 14, 15, 9. Find the 6th number.

Worked Example 2 — Weighted Average

A class of 20 students averages 75 on a test. A class of 30 students averages 85. What is the combined average?

Note: The combined average is NOT simply $(75 + 85)/2 = 80$. The larger class pulls the average closer to its mean.

Median from a Frequency Table

14 values total → median = average of 7th and 8th values. Count: positions 1–3 are 70, positions 4–8 are 80. Both 7th and 8th are 80, so median = 80.

Advanced Mean & Median 🎯

Mean vs. Median 🔍

Which measure of center is more appropriate?

Key Takeaways — Part 1

• On the SAT, "average" always means mean unless otherwise specified
• The weighted average is always closer to the group with more data points

Data Analysis & Statistics

Part 2 of 7 — Standard Deviation & Data Spread

Standard Deviation (σ)

Measures how spread out values are from the mean. You won't calculate it on the SAT, but you must understand it.

• Low SD: values are close to the mean (consistent data)
• High SD: values are far from the mean (varied data)
• SD = 0: all values are the same

Comparing Standard Deviations

Data A: $\{48, 49, 50, 51, 52\}$ → low SD (clustered near 50)

Data B: $\{10, 30, 50, 70, 90\}$ → high SD (spread out)

Both have mean 50, but B has a much larger standard deviation.

Effect of Transformations

Adding a constant shifts all data equally — spread doesn't change. Multiplying stretches the data — spread increases.

Standard Deviation 🎯

Worked Example 1 — Comparing Spread Visually

Two dot plots both have mean 50:

• Plot A: values at 48, 49, 50, 51, 52
• Plot B: values at 30, 40, 50, 60, 70

Plot B has values much farther from the mean → larger SD.

Worked Example 2 — Transformation Chain

A dataset has mean 40 and SD 6. Every value is tripled, then 5 is subtracted. Find the new mean and SD.

Adding/subtracting does NOT change SD. Multiplying DOES.

Percentiles and Quartiles

A value is an outlier if it's more than $1.5 \times \text{IQR}$ beyond $Q_1$ or $Q_3$.

Spread & Transformations 🎯

What Happens to the SD? 🔍

Determine the effect of each transformation on standard deviation.

Key Takeaways — Part 2

• SD is ALWAYS ≥ 0 (it can never be negative)
• On the SAT, you compare SDs visually — more clustered = lower SD

Data Analysis & Statistics

Part 3 of 7 — Scatterplots & Line of Best Fit

Scatterplots

Each point represents two measurements for one individual/item.

Correlation

• Positive: as $x$ increases, $y$ increases
• Negative: as $x$ increases, $y$ decreases
• No correlation: no clear pattern
• Strength: how closely points follow a line (strong vs. weak)

Line of Best Fit (Regression Line)

$y = ax + b$ where:

• $a$ (slope) = predicted change in $y$ for each 1-unit increase in $x$
• $b$ (y-intercept) = predicted $y$ when $x = 0$

Residuals

$\text{Residual} = \text{Actual} - \text{Predicted}$

• Positive residual: point is above the line
• Negative residual: point is below the line

SAT Strategy

"According to the line of best fit..." → plug into the equation and calculate.

"The slope of the line means..." → interpret as "for each additional [x-unit], the [y-quantity] increases/decreases by [slope]."

Scatterplots 🎯

Worked Example 1 — Interpreting Slope in Context

Regression: $\hat{y} = 0.85x + 12$, where $x$ = hours practiced per week and $y$ = free-throw percentage.

SAT phrasing: "The slope means that for each additional hour of weekly practice, the predicted free-throw percentage increases by 0.85 percentage points."

Worked Example 2 — Residual Analysis

A student studies 8 hours and scores 92. The regression line predicts $\hat{y} = 3(8) + 65 = 89$.

Correlation vs. Causation

Choosing a Model

If a residual plot shows a clear curve, a linear model is NOT the best fit — try quadratic or exponential.

Scatterplot Interpretation 🎯

Interpret the Regression 🔍

For the equation $\hat{y} = 2.5x + 40$ where $x$ = study hours and $y$ = test score:

Key Takeaways — Part 3

Data Analysis & Statistics

Part 4 of 7 — Two-Way Tables

Reading Two-Way Tables

Types of Questions

Marginal frequency: What percent prefer dogs? $80/150 ≈ 53.3\%$

Joint frequency: What percent are female AND prefer cats? $40/150 ≈ 26.7\%$

Conditional frequency: Of those who prefer cats, what percent are female? $40/70 ≈ 57.1\%$

SAT Trap ⚠️

"What fraction of cat owners are female?" → denominator = cat total = 70 → $40/70 ≈ 57.1\%$

"What fraction of females own cats?" → denominator = female total = 70 → $40/70 ≈ 57.1\%$

In this example they happen to give the same answer because both totals are 70, but in general they're different! The trick is identifying the correct denominator (row total, column total, or grand total).

Worked Example 1 — Filling in a Two-Way Table

120 students: 55 play sports, 80 are in clubs. 30 do both. Complete the table.

Worked Example 2 — Conditional vs. Joint

From the table above:

Two-Way Tables 🎯

Independence in Two-Way Tables

Two variables show no association when each group has the SAME rate — e.g., juniors and seniors choosing pizza at equal percentages. The SAT asks this as "do the data support an association?", always comparing group rates in words.

From the pet table: $P(\text{cat}) = 70/150 ≈ 46.7\%$. But $P(\text{cat | male}) = 30/80 = 37.5\%$. Since $37.5\% \neq 46.7\%$, gender and pet preference are not independent.

Worked Example 3 — Relative Frequency Table

Convert a two-way table to relative frequencies (divide everything by grand total):

Common SAT Denominator Guide

Advanced Two-Way Tables 🎯

Pick the Right Denominator 🔍

What denominator do you use for each question?

Key Takeaways — Part 4

• The SAT's #1 trap in two-way tables is using the wrong denominator
• Always re-read the question to identify the "of" phrase — that gives you the denominator

Data Analysis & Statistics

Part 5 of 7 — Probability, the SAT Way

The digital SAT tests probability as relative frequency — a count divided by a total, almost always read from a table or described in words. You will NOT see formal notation like P(A|B) or unions/intersections on the test.

Probability as a fraction

$\text{Probability} = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Always between 0 (impossible) and 1 (certain).

"NOT" questions (complements)

The probability an event does NOT happen is 1 minus the probability it does.

Example: If the probability a customer orders coffee is $\frac{7}{20}$, the probability a customer does NOT order coffee is $1 - \frac{7}{20} = \frac{13}{20}$.

The SAT's favorite setup: "selected at random from…"

Most SAT probability questions give you a two-way table and ask about a person "selected at random" — sometimes from everyone, sometimes from just one row or column. The entire skill is choosing the right denominator.

Probability 🎯

Worked Example 1 — Selected From Everyone

The table shows 200 students by grade and lunch choice.

If a student is selected at random, what is the probability the student is a junior who chose pizza?

One cell over the grand total: $\frac{60}{200} = \frac{3}{10}$.

Worked Example 2 — Selected From One Group (the key SAT move)

If a student is selected at random from those who chose salad, what is the probability the student is a senior?

The words "from those who chose salad" shrink your world to the Salad column: $\frac{40}{70} = \frac{4}{7}$.

Rule: the group named after "from" becomes your denominator. This is how the SAT asks conditional probability — in words, never with notation.

Worked Example 3 — Comparing Groups

Which group was more likely to choose pizza?

Juniors: $\frac{60}{90} \approx 67\%$. Seniors: $\frac{70}{110} \approx 64\%$. Juniors were slightly more likely. Compare each group's RATE, never the raw counts.

Table Probability 🎯

Choose the Right Denominator 🔍

For each question about a school survey table, pick the correct denominator.

Key Takeaways — Part 5

• The SAT asks probability in words from tables — no P(A|B), no union/intersection symbols
• The word "from" is the whole game: it names your denominator
• Compare groups by their rates (fractions), never raw counts

Data Analysis & Statistics

Part 6 of 7 — Sampling and Study Design

Types of Studies

Random Sampling

A sample is representative if every member of the population has an equal chance of being selected.

• Random sample: conclusions can be generalized to the population
• Convenience sample (e.g., only surveying friends): results may be biased

Bias

• Selection bias: sample doesn't represent the population
• Response bias: wording of questions influences answers
• Voluntary response bias: only people with strong opinions respond

SAT Wording to Watch For

❌ "The study proves that X causes Y" — only experiments with random assignment can suggest causation.

✓ "The study suggests an association between X and Y" — appropriate for observational studies.

Worked Example 1 — Is the Conclusion Valid?

A researcher gives Vitamin C to 50 volunteers and a placebo to 50 others (randomly assigned). The Vitamin C group had fewer colds. Conclusion: "Vitamin C reduces colds."

Worked Example 2 — Why Only Association?

A study surveys 1,000 adults and finds that coffee drinkers exercise more. Conclusion: "Coffee causes people to exercise."

Margin of Error

When the SAT says "95% confidence interval is $52\% \pm 3\%$":

• Plausible range: $49\%$ to $55\%$
• Larger sample → smaller margin of error
• The margin does NOT mean 3% of people changed their mind

Study Design 🎯

Generalizability vs. Causation — Two Separate Questions

SAT Conclusion Wording Guide

Worked Example 3 — Margin of Error

A poll of 400 voters: $58\% \pm 4\%$ support a candidate.

Evaluating Conclusions 🎯

Identify the Study Type 🔍

Classify each scenario.

Key Takeaways — Part 6

• On the SAT, the wrong answer often claims causation from an observational study — always check!

Data Analysis & Statistics

Part 7 of 7 — Review & SAT Mixed Practice

Data & Statistics Quick Reference

Common SAT Question Types

1. Calculate the mean/median from given data
2. Interpret a slope or y-intercept in context
3. Read a two-way table for conditional probability
4. Evaluate whether a study conclusion is valid
5. Compare standard deviations visually

Worked Example 1 — Multi-Concept Problem

A survey of 200 students found a regression equation $\hat{y} = 1.5x + 50$ relating weekly study hours ($x$) to test scores ($y$). The mean study time was 10 hours with SD 3.

Worked Example 2 — Two-Way Table + Probability

In a class: 15 males passed, 5 males failed, 20 females passed, 10 females failed.

Mixed Review 🎯

SAT Data & Statistics Cheat Sheet

Common Mistakes to Avoid

SAT Challenge Round 🎯

Quick Concept Check 🔍

Match each scenario to the correct statistical concept.

Key Takeaways — Part 7

Top 5 SAT Data & Statistics Rules

1. Mean = Sum ÷ Count — use Sum = Mean × Count to find missing values
2. Correlation ≠ Causation — only experiments prove cause
3. Watch the denominator — "of males" vs. "of all" changes the answer
4. "At least one" = 1 − P(none) — always use the complement
5. Slope in context — "For each additional [x], [y] is predicted to [increase/decrease] by [slope]"