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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 🎯

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.

Step	Work
Total from mean	$6 \times 12 = 72$
Sum of 5 known	$8 + 10 + 14 + 15 + 9 = 56$

Advanced Mean & Median 🎯

Mean vs. Median 🔍

Which measure of center is more appropriate?

Key Takeaways — Part 1

Concept	Formula / Rule
Mean	Sum ÷ Count
Median	Middle value (avg of two middles if even count)
Mode	Most frequent value
Range	Max − Min
Find missing value	Sum = Mean × Count, then subtract known values
Weighted average	$\frac{n_1 \cdot \bar{x}_1 + n_2 \cdot \bar{x}_2}{n_1 + n_2}$

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

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?

Step	Work
Sum for class 1	$20 \times 75 = 1500$
Sum for class 2	$30 \times 85 = 2550$
Combined mean	$\frac{1500 + 2550}{20 + 30} = \frac{4050}{50} = 81$

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

Score	Frequency
70	3
80	5
90	4
100	2

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.

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

{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 🎯

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

Measure	Plot A	Plot B
Range	4	40
SD	Low	High

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.

Transformation	Mean	SD
Original	40	6
Multiply by 3	$40 \times 3 = 120$	$6 \times 3 = 18$
Subtract 5	$120 - 5 = 115$	$18$ (unchanged)

Adding/subtracting does NOT change SD. Multiplying DOES.

Percentiles and Quartiles

Measure	Meaning
$Q_1$ (25th percentile)	25% of data below
$Q_2$ (median, 50th)	50% of data below
$_{}$ (75th percentile)

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

Concept	Key Rule
SD measures	Spread / distance from mean
Add constant $k$	Mean + $k$ ; SD unchanged
Multiply by $k$	Mean × $k$ ; **SD × $
SD = 0	All values identical
IQR	$Q_3 - Q_1$ ; spread of middle 50%
Outlier threshold	Beyond $Q_1 - 1.5(\text{IQR})$ or $Q_3 + 1.5(\text{IQR})$

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

$a$ (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 🎯

Worked Example 1 — Interpreting Slope in Context

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

Component	Value	Interpretation
Slope	0.85	For each additional hour of practice, free-throw % is predicted to increase by 0.85 points
y-intercept	12	A player who practices 0 hours is predicted to have a 12% free-throw rate

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 $\hat{y} = 3(8) + 65 = 89$ .

Step	Work
Residual	$92 - 89 = 3$
Meaning	Student scored 3 points above prediction
On graph	Point is 3 units above the line

Correlation vs. Causation

Statement	Valid?
"Hours of study is correlated with higher grades"	✅ (describes relationship)
"Studying more hours causes higher grades"	⚠️ Only valid if from a controlled experiment
"The data proves studying improves grades"	❌ "Proves" is too strong for any study

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

Concept	Key Rule
Slope	Predicted change in $y$ per 1-unit increase in $x$
y-intercept	Predicted $y$ when $x = 0$
Residual	Actual − Predicted
Positive residual	Point above the line
Negative residual	Point below the line
$r$ close to ±1	Strong linear correlation
$r$ close to 0	Weak or no linear correlation

SAT Wording	Correct Response
"What does the slope represent?"	"For each additional [x-unit], [y] is predicted to change by [slope]"
"Does this prove causation?"	Only if randomized experiment
"Is the linear model appropriate?"	Check the residual plot for patterns

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?" → 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.

Step	Work
Sports AND clubs	30
Sports only	$55 - 30 = 25$
Clubs only	$80 - 30 = 50$
Neither	$120 - 25 - 30 - 50 = 15$

	In Club	Not in Club	Total
Sports	30	25	55
No Sports	50	15	65
Total	80	40	120

Worked Example 2 — Conditional vs. Joint

From the table above:

Question	Type	Calculation
P(sports AND club)	Joint	$30/120 = 25\%$
P(club	sports)	Conditional
P(sports	club)	Conditional

Two-Way Tables 🎯

	Cat	Dog
Male	30	50
Female	40	30

Independence in Two-Way Tables

Two events $A$ and $B$ are independent if $P(A|B) = P(A)$ .

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):

	Cat	Dog	Total
Male	$30/150 = 20\%$	$50/150 = 33.3\%$	$53.3\%$
Female

Common SAT Denominator Guide

Question asks	Denominator
"Of all survey respondents..."	Grand total
"Of those who prefer cats..."	Column total for cats
"Of the males surveyed..."	Row total for males
"What proportion of the total..."	Grand total

Advanced Two-Way Tables 🎯

Pick the Right Denominator 🔍

What denominator do you use for each question?

Key Takeaways — Part 4

Question Type	Denominator	Example
"Of all..."	Grand total	$P(\text{male and dog}) = 50/150$
"Of [group]..."	Group total	$P(\text{cat}
Joint probability	Grand total	$P(A \cap B)$
Conditional probability	Condition total	$P(A

Skill	How-to
Fill in table	Use row/column totals to find missing cells
Check independence	Compare $P(A
Convert to relative freq.	Divide each cell by grand total

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

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 🎯

Worked Example 1 — "At Least One" with Complement

A coin is flipped 3 times. What is the probability of getting at least one head?

Approach	Work
Direct	Count all cases with ≥ 1 head... tedious
Complement	$P(\text{at least 1 head}) = 1 - P(\text{no heads})$
Calculate	$P(\text{all tails}) = (1/2)^3 = 1/8$
Answer	$1 - 1/8 = 7/8$

Rule: "At least one" = $1$ − P(none). Always use the complement!

Worked Example 2 — "OR" with Overlap

In a class of 30: 18 play soccer, 12 play basketball, 5 play both. What is P(soccer OR basketball)?

Step	Work
Formula	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Worked Example 3 — Conditional from a Table

200 students surveyed: 80 passed math, 60 passed science, 40 passed both.

$P(\text{science} | \text{math}) = \frac{P(\text{both})}{P(\text{math})} = \frac{40/200}{80/200} = \frac{40}{80} = \frac{1}{2}$

Of students who passed math, half also passed science.

Advanced Probability 🎯

Probability Setup 🔍

Choose the correct formula or approach for each problem.

Key Takeaways — Part 5

Rule	Formula	When to Use
Basic probability	$P(A) = \text{favorable} / \text{total}$	Always
Complement	$P(\text{not } A) = 1 - P(A)$	"At least one," "none"
AND (independent)	$P(A) \times P(B)$	Events don't affect each other
OR	$P(A) + P(B) - P(A \cap B)$	"Either... or..."
Conditional	$P(A	B) = P(A \cap B) / P(B)$
Without replacement	Adjust denominator after each draw	Drawing from a bag/deck

"At least one" → use complement (much faster)
With replacement: probabilities stay the same. Without: they change
On the SAT, conditional probability often comes from two-way tables

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."

Check	Answer
Study type	Experiment (random assignment)
Random assignment?	Yes
Can conclude causation?	Yes — this is valid

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."

Check	Answer
Study type	Observational (no intervention)
Confound?	Maybe: health-conscious people both drink coffee and exercise
Valid conclusion?	"Coffee consumption is associated with more exercise" — NOT "causes"

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

Feature	Allows
Random sampling from population	Generalize results to the population
Random assignment to treatments	Conclude cause and effect
Both	Generalize + causation
Neither	Only describes the sample

SAT Conclusion Wording Guide

Study Design	Valid Conclusion Wording
Random sample, observational	"People who [X] tend to [Y]"
Random sample + random assignment	"[X] causes [Y] in the population"
Convenience sample, observational	"Among these participants, [X] is associated with [Y]"

Worked Example 3 — Margin of Error

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

Question	Answer
Confidence interval	$54\%$ to $62\%$
Can we say majority support?	Yes — even the low end ( $54\%$ ) exceeds $50\%$
If interval were $48\%$ to ?

Evaluating Conclusions 🎯

Identify the Study Type 🔍

Classify each scenario.

Key Takeaways — Part 6

Concept	Key Rule
Causation	Only from randomized experiments
Generalization	Only from random sampling
Association	Observational studies can show this
Confounding variable	Third variable explaining a correlation
Margin of error	Confidence interval = estimate ± margin
Larger sample	Smaller margin of error

Bias Type	Example
Selection	Surveying only library users
Response	Leading questions
Voluntary response	Online opt-in polls

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

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

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.

Question	Work
Predicted score at $x = 10$	$1.5(10) + 50 = 65$
Predicted score at $x = 16$	$1.5(16) + 50 = 74$
If scores multiplied by 2, new SD of study hours?	SD of $x$ is unchanged (transformation was on $y$ , not $x$ )

Worked Example 2 — Two-Way Table + Probability

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

	Pass	Fail	Total
Male	15	5	20
Female	20	10	30
Total	35	15	50

Question	Answer
P(pass)	$35/50 = 70\%$
P(pass	male)
P(male	pass)
Are gender & passing independent?	$P(\text{pass}) = 70% \neq P(\text{pass}

Mixed Review 🎯

SAT Data & Statistics Cheat Sheet

Topic	Key Formula / Concept
Mean	$\bar{x} = \frac{\sum x}{n}$ ; Sum = Mean × Count
Median	Middle value; sort first
SD	Spread from mean; add → same; multiply → changes
Slope	Predicted $\Delta y$ per unit $\Delta x$
Residual	Actual − Predicted
Complement	$P(\text{not A}) = 1 - P(A)$
OR (with overlap)	$P(A) + P(B) - P(A \cap B)$
AND (independent)	$P(A) \times P(B)$
Conditional	$P(A
Causation	Only from randomized experiments
Generalization	Only from random sampling

Common Mistakes to Avoid

Mistake	Fix
Using wrong denominator in two-way table	Re-read "of [group]" to find denominator
Saying correlation = causation	Use "associated with" unless random assignment
Forgetting to subtract overlap in P(A or B)	Always check: can A and B happen together?
Confusing "all values" with "the mean"	Adding 10 to every value ≠ adding 10 to just the mean
Ignoring "without replacement"	After removing one item, total decreases by 1

SAT Challenge Round 🎯

Quick Concept Check 🔍

Match each scenario to the correct statistical concept.

Key Takeaways — Part 7

Part	Core Skill
1	Mean, median, mode — and how outliers affect them
2	Standard deviation — comparing spread, effect of transformations
3	Scatterplots — slope interpretation, residuals, correlation
4	Two-way tables — joint, marginal, conditional frequencies
5	Probability — complement, AND/OR, conditional
6	Study design — causation vs. association, bias types
7	Review — combining all skills for SAT questions

Top 5 SAT Data & Statistics Rules

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

Statistics and Data Interpretation

Statistics and Data Interpretation - Complete Interactive Lesson

Part 1: Mean, Median, Mode

Data Analysis & Statistics

Mean (Average)

Median

Mode

Range

SAT Favorites 🎯

Worked Example 1 — Finding a Missing Value

Key Takeaways — Part 1

Part 2: Data Displays

Data Analysis & Statistics

Standard Deviation (σ)

Comparing Standard Deviations

Part 3: Interpreting Tables

Data Analysis & Statistics

Scatterplots

Correlation

Line of Best Fit (Regression Line)

Part 4: Standard Deviation

Data Analysis & Statistics

Reading Two-Way Tables

Types of Questions

Part 5: Statistical Inference

Data Analysis & Statistics

Basic Probability

Complement

Part 6: Problem-Solving Workshop

Data Analysis & Statistics

Types of Studies

Random Sampling

Bias

SAT Wording to Watch For

Part 7: Review & Applications

Data Analysis & Statistics

Data & Statistics Quick Reference

Worked Example 2 — Weighted Average

Median from a Frequency Table

Effect of Transformations

Worked Example 1 — Comparing Spread Visually

Worked Example 2 — Transformation Chain

Percentiles and Quartiles

Key Takeaways — Part 2

Residuals

SAT Strategy

Worked Example 1 — Interpreting Slope in Context

Worked Example 2 — Residual Analysis

Correlation vs. Causation

Choosing a Model

Key Takeaways — Part 3

SAT Trap ⚠️

Worked Example 1 — Filling in a Two-Way Table

Worked Example 2 — Conditional vs. Joint

Independence in Two-Way Tables

Worked Example 3 — Relative Frequency Table

Common SAT Denominator Guide

Key Takeaways — Part 4

"AND" (Intersection)

"OR" (Union)

Conditional Probability

Worked Example 1 — "At Least One" with Complement

Worked Example 2 — "OR" with Overlap

Worked Example 3 — Conditional from a Table

Key Takeaways — Part 5

Worked Example 1 — Is the Conclusion Valid?

Worked Example 2 — Why Only Association?

Margin of Error

Generalizability vs. Causation — Two Separate Questions

SAT Conclusion Wording Guide

Worked Example 3 — Margin of Error

Key Takeaways — Part 6

Common SAT Question Types

Worked Example 1 — Multi-Concept Problem

Worked Example 2 — Two-Way Table + Probability

SAT Data & Statistics Cheat Sheet

Common Mistakes to Avoid

Key Takeaways — Part 7

Top 5 SAT Data & Statistics Rules