🎯⭐ INTERACTIVE LESSON

Statistics and Data Interpretation

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Statistics and Data Interpretation - Complete Interactive Lesson

Part 1: Mean, Median, Mode

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 🎯

Key Takeaways — Part 1

Mean: sum ÷ count. Use "sum = mean × count" to find missing totals
Median: middle value (or average of two middles)
Outliers affect the mean much more than the median
For "add a value" problems: recalculate sum, then divide by new count

Part 2: Data Displays

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)

Part 3: Interpreting Tables

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)

where:

Part 4: Standard Deviation

Data Analysis & Statistics

Part 4 of 7 — Two-Way Tables

Reading Two-Way Tables

	Cat	Dog	Total
Male	30	50	80
Female	40	30	70
Total	70	80	150

Types of Questions

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

Joint frequency: What percent are female AND prefer cats?

Part 5: Statistical Inference

Data Analysis & Statistics

Part 5 of 7 — Probability

Basic Probability

$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

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

Complement

Part 6: Problem-Solving Workshop

Data Analysis & Statistics

Part 6 of 7 — Sampling and Study Design

Types of Studies

Type	Description	Can show causation?
Observational	Observe without intervention	No (only association)
Experiment	Randomly assign treatments	Yes!
Survey	Ask questions	No (only opinion)

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.

Part 7: Review & Applications

Data Analysis & Statistics

Part 7 of 7 — Review & SAT Mixed Practice

Data & Statistics Quick Reference

Concept	Key Point
Mean	Sum ÷ count; affected by outliers
Median	Middle value; resistant to outliers
SD	Measures spread; add → same, multiply → changes
Scatterplot slope	Predicted change in $y$ per unit $x$
Residual	Actual − predicted
Two-way tables	Watch the denominator!
Probability AND	Multiply (if independent)
Probability OR	Add, then subtract overlap

{10, 30, 50, 70, 90}

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

Effect of Transformations

Transformation	Mean	SD
Add $k$ to all values	Mean $+ k$	Same
Multiply all by $k$	Mean $\times k$	SD $\times

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

Standard Deviation 🎯

Key Takeaways — Part 2

SD measures spread — you compare, not calculate on the SAT
More spread out data = higher SD; identical values = SD of 0
Adding a constant: mean changes, SD does NOT change
Multiplying by a constant: both mean and SD are multiplied

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

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

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

"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 🎯

Key Takeaways — Part 3

Slope = predicted $y$ -change per unit $x$ -change
Residual = actual − predicted (positive = above line, negative = below)
Correlation ≠ causation (the SAT tests this!)
Interpret slope in context: "for each additional ___, the ___ is predicted to ___"

40/150 \approx 26.7%

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

"What fraction of cat owners are female?" → $40/70$ (denominator = cat total)

"What fraction of females own cats?" → $40/70$ also... wait, here denominator = female total = $40/70$

WRONG! "Of females, what fraction own cats?" → $40/70$ is NOT right. Female total = 70, cat females = 40. So $40/70 ≈ 57.1\%$ .

The trick is identifying the correct denominator (row total, column total, or grand total).

Two-Way Tables 🎯

Key Takeaways — Part 4

"Of all...": denominator = grand total
"Of [group]...": denominator = that group's total (row or column)
Conditional: "Of those who ___" tells you the denominator
Always identify the correct denominator before calculating

P (not A) = 1 - P (A)

"AND" (Intersection)

If events are independent: $P(A \text{ and } B) = P(A) \times P(B)$

Example: Coin flip AND die roll: $P(\text{heads and 6}) = (1/2)(1/6) = 1/12$

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

If events are mutually exclusive (can't happen together): $P(A \text{ or } B) = P(A) + P(B)$

Conditional Probability

$P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$

"Probability of A given B" — restrict your attention to only the B outcomes.

Probability 🎯

Key Takeaways — Part 5

$P(A) = \text{favorable}/\text{total}$ , always between 0 and 1
Complement: $P(\text{not } A) = 1 - P(A)$ — useful when "not" is easier
"AND" (independent): multiply probabilities
"OR": add probabilities, subtract overlap
Conditional: restrict the total to the given condition

Study Design 🎯

Key Takeaways — Part 6

Only randomized experiments can show causation
Observational studies show association, not causation
Random sampling → results generalize to the population
Watch for bias: selection, response, and voluntary response bias
Confounding variables can create misleading correlations

Common SAT Question Types

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

Mixed Review 🎯

Key Takeaways — Part 7

Statistics questions are about 25-30% of SAT math — they're heavily tested
Always identify whether a question asks about the whole population or a subgroup
Interpret slopes, intercepts, and residuals in the context of the problem
Correlation ≠ causation — this appears on almost every SAT

Statistics and Data Interpretation

Effect of Transformations

Key Takeaways — Part 2

Residuals

SAT Strategy

Key Takeaways — Part 3

SAT Trap ⚠️

Key Takeaways — Part 4

"AND" (Intersection)

"OR" (Union)

Conditional Probability

Key Takeaways — Part 5

Key Takeaways — Part 6

Common SAT Question Types

Key Takeaways — Part 7