Data Analysis and Statistics (SAT)

Measures of Central Tendency

Mean (Average)

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

Example: Test scores: 80, 85, 90, 95 $\text{Mean} = \frac{80+85+90+95}{4} = \frac{350}{4} = 87.5$

Median (Middle Value)

Steps:

1. Put numbers in order
2. Find the middle value
3. If even number of values, average the two middle ones

Example: 3, 7, 9, 12, 15 Median = 9 (middle value)

Example: 2, 5, 8, 11 Median = $\frac{5+8}{2} = 6.5$ (average of middle two)

Mode (Most Frequent)

The value that appears most often

Example: 2, 3, 3, 5, 7, 3, 9 Mode = 3 (appears three times)

Note: Can have no mode or multiple modes

Range

Maximum value - Minimum value

Example: 10, 15, 22, 18, 30 Range = $30 - 10 = 20$

Standard Deviation

Measures spread/variability

• Low standard deviation: Data close together
• High standard deviation: Data spread out

You don't need the formula! SAT gives context:

"Which dataset has greater variability?" → higher standard deviation

Reading Data from Graphs

Bar Graphs

• Height shows value
• Compare categories
• Look for trends

Line Graphs

• Shows change over time
• Slope indicates rate of change
• Look for increases/decreases

Scatter Plots

• Each point = one observation
• Look for patterns/trends
• Positive/negative correlation

Box Plots (Box-and-Whisker)

Shows 5-number summary:

1. Minimum
2. Q1 (25th percentile)
3. Median (Q2, 50th percentile)
4. Q3 (75th percentile)
5. Maximum

Interquartile Range (IQR): $Q3 - Q1$

Interpreting Data

Correlation vs. Causation

Correlation: Two things are related Causation: One CAUSES the other

SAT Trap: Just because things correlate doesn't mean one causes the other!

Example:

• Ice cream sales and drowning both increase in summer
• Correlation: YES
• Causation: NO (heat causes both, not each other)

Positive vs. Negative Correlation

Positive: Both increase together

• Example: Study time and test scores

Negative (Inverse): One increases, other decreases

• Example: Speed and travel time

No correlation: No clear pattern

Probability and Data

Basic Probability

$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Two-Way Tables

| | Group A | Group B | Total | |---|---------|---------|-------| | Yes | 30 | 20 | 50 | | No | 10 | 40 | 50 | | Total | 40 | 60 | 100 |

Questions:

• "What percent of Group A said Yes?" $\frac{30}{40} = 75\%$

• "What percent of all respondents said Yes?" $\frac{50}{100} = 50\%$

SAT Question Types

Type 1: Calculate Mean/Median

Given data, find the measure

Use formulas/procedures above

Type 2: Effect of Adding/Removing Data

"If we add a value of 100, how does the mean change?"

Calculate new mean with additional value

Type 3: Reading Graphs

"According to the graph, in which year...?"

Direct lookup from visual

Type 4: Interpreting Studies

"The study shows that X is associated with Y. Can we conclude X causes Y?"

Usually NO - correlation ≠ causation

SAT Strategies

Mean Tricks

If all values increase by 5, mean increases by 5 If all values double, mean doubles

Median is Resistant

Outliers don't affect median much Outliers greatly affect mean

Use Calculator

Sum, mean functions save time

Read Carefully

"What percent of Group A" vs "What percent of all"

Common SAT Traps

Trap 1: Mean vs. Median Confusion

Mean = average Median = middle

Trap 2: Percent vs. Number

30% ≠ 30 people

Trap 3: Correlation = Causation

Associated ≠ causes

Trap 4: Wrong Denominator

Percent of subgroup vs. percent of total

SAT Tips

• Median: Always order the data first!
• Mean: Sum ÷ count
• Outliers affect mean more than median
• Read graph labels carefully (units, scale)
• Check denominators for percent questions
• Correlation ≠ Causation on SAT!