SAT Math – Problem Solving & Data Analysis
Center, Spread, and Shape of Distributions
Understanding statistical measures and data distribution patterns
Understanding distributions is the foundation of data analysis and statistical reasoning. On the SAT, you'll interpret data sets by analyzing their center (typical value), spread (variability), and shape (pattern), then use these characteristics to draw conclusions, make comparisons, and evaluate claims.
Success requires knowing when to use mean versus median, understanding how outliers affect measures of center, interpreting standard deviation and interquartile range, and recognizing symmetric, skewed, and normal distributions. These aren't just test concepts—they're the tools scientists, economists, and data analysts use to make sense of real-world information.
Understanding Distributions
Measures of Center
Center describes the "typical" or "middle" value in a data set. Three main measures exist:
Median: The middle value when ordered
Mode: The most frequently occurring value
Measures of Spread
Spread (variability) describes how dispersed or clustered the data values are.
Interquartile Range (IQR): Q3 - Q1 (middle 50%)
Standard Deviation: Average distance from mean
Shape of Distribution
Shape describes the overall pattern of the data distribution.
Skewed Right: Long tail to right (mean > median)
Skewed Left: Long tail to left (mean < median)
Normal: Bell-shaped, symmetric distribution
Essential Formulas & Concepts
Mean (Average)
\(\text{Mean} = \bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}\)
Sum all values and divide by the count
Median
To find median:
1. Order data from smallest to largest
2. If odd count: middle value
3. If even count: average of two middle values
Interquartile Range (IQR)
\(\text{IQR} = Q_3 - Q_1\)
Q1: Median of lower half (25th percentile)
Q2: Overall median (50th percentile)
Q3: Median of upper half (75th percentile)
Standard Deviation
Conceptual understanding: Measures average distance from mean
• Larger SD = more spread out data
• Smaller SD = more clustered data
SAT rarely requires calculation, focuses on interpretation
Interpreting Distribution Shape
Symmetric Distribution
Data is evenly distributed around the center
Key feature: Mean ≈ Median
Example: Heights in a large population, test scores
Right-Skewed (Positively Skewed)
Long tail extends to the right (high values)
Key feature: Mean > Median (pulled by high outliers)
Example: Income, home prices, test scores with ceiling
Left-Skewed (Negatively Skewed)
Long tail extends to the left (low values)
Key feature: Mean < Median (pulled by low outliers)
Example: Age at retirement, exam scores with many high scorers
Common Pitfalls & Expert Tips
❌ Confusing mean and median
Mean is affected by outliers; median is not. When distribution is skewed or has outliers, median better represents "typical" value.
❌ Forgetting to order data for median
You MUST arrange data in order before finding median or quartiles. The middle position of unordered data is meaningless!
❌ Mixing up skew direction
Right-skewed means the tail points right (toward higher values), NOT that most data is on the right. The bulk is on the left with a tail to the right!
❌ Thinking range accounts for all variation
Range only uses two values (max and min). IQR and standard deviation better describe overall spread because they use all or most data points.
✓ Expert Tip: Check mean vs. median for shape
If mean > median, distribution is right-skewed. If mean < median, left-skewed. If mean ≈ median, likely symmetric. This is a quick diagnostic!
✓ Expert Tip: Use median for skewed data
When describing "typical" values in skewed distributions (like income or home prices), median is more representative than mean.
✓ Expert Tip: IQR is resistant to outliers
IQR only uses middle 50% of data, so extreme values don't affect it. This makes it more robust than range for describing spread.
Fully Worked SAT-Style Examples
A student receives the following test scores: 78, 85, 92, 88, 95. What are the mean and median scores?
Solution:
Finding Mean:
\(\text{Mean} = \frac{78 + 85 + 92 + 88 + 95}{5} = \frac{438}{5} = 87.6\)
Finding Median:
Step 1: Order the data: 78, 85, 88, 92, 95
Step 2: Find middle value (5 data points, so 3rd value)
Median = 88
Answer:
Mean = 87.6
Median = 88
Five houses on a street sold for: $200,000, $210,000, $205,000, $215,000, and $1,500,000. What are the mean and median prices? Which better represents a "typical" price?
Solution:
Calculate Mean:
\(\text{Mean} = \frac{200{,}000 + 210{,}000 + 205{,}000 + 215{,}000 + 1{,}500{,}000}{5}\)
\(= \frac{2{,}330{,}000}{5} = \$466{,}000\)
Calculate Median:
Ordered: $200,000, $205,000, $210,000, $215,000, $1,500,000
Median = $210,000 (middle value)
Analysis:
The $1,500,000 house is an outlier that pulls the mean up to $466,000
The median of $210,000 better represents a "typical" house price
Four of five houses sold for around $210,000, not $466,000!
Answer:
Mean = $466,000; Median = $210,000
Median better represents typical price due to outlier
Find the interquartile range (IQR) for the data set: 12, 15, 18, 20, 22, 25, 28, 30, 35
Solution:
Step 1: Data is already ordered (9 values)
12, 15, 18, 20, 22, 25, 28, 30, 35
Step 2: Find Q2 (median)
Middle value (5th position): Q2 = 22
Step 3: Find Q1 (median of lower half)
Lower half: 12, 15, 18, 20
Q1 = \(\frac{15 + 18}{2} = 16.5\)
Step 4: Find Q3 (median of upper half)
Upper half: 25, 28, 30, 35
Q3 = \(\frac{28 + 30}{2} = 29\)
Step 5: Calculate IQR
\(\text{IQR} = Q3 - Q1 = 29 - 16.5 = 12.5\)
Answer: IQR = 12.5
The middle 50% of data spans 12.5 units
A data set has a mean of 75 and a median of 82. What can you conclude about the shape of the distribution?
Solution:
Step 1: Compare mean and median
Mean = 75, Median = 82
Mean < Median
Step 2: Apply the rule
When Mean < Median:
• The mean is pulled down by low values
• Distribution is left-skewed (negatively skewed)
• Long tail extends toward lower values
Quick Reference:
Mean > Median → Right-skewed
Mean < Median → Left-skewed
Mean ≈ Median → Symmetric
Answer: The distribution is left-skewed (negatively skewed)
Two classes took the same test. Class A had scores clustered around 85, while Class B had scores ranging widely from 50 to 100. Both classes have the same mean of 85. Which class has a larger standard deviation?
Solution:
Understanding standard deviation:
Standard deviation measures spread (variability) around the mean
• Larger SD = data more spread out
• Smaller SD = data more clustered
Analyzing the classes:
Class A: Scores clustered around 85
→ Values close to mean → SMALL standard deviation
Class B: Scores range 50 to 100
→ Values far from mean → LARGE standard deviation
Answer: Class B has a larger standard deviation
Greater variability means larger SD
A data set has five values with a mean of 20. If a sixth value of 32 is added, what is the new mean?
Solution:
Step 1: Find original sum
If mean of 5 values = 20:
\(\text{Sum} = 20 \times 5 = 100\)
Step 2: Add the new value
\(\text{New sum} = 100 + 32 = 132\)
Now have 6 values
Step 3: Calculate new mean
\(\text{New mean} = \frac{132}{6} = 22\)
Observation:
The new value (32) is above the original mean (20)
So the new mean (22) increased from the original (20)
Answer: New mean = 22
A box plot shows: Minimum = 10, Q1 = 20, Median = 30, Q3 = 50, Maximum = 90. What is the IQR? What does this tell us about the data?
Solution:
Calculate IQR:
\(\text{IQR} = Q3 - Q1 = 50 - 20 = 30\)
Interpret the box plot:
• Range = 90 - 10 = 80
• IQR = 30 (middle 50% spans 30 units)
• Lower 50% (10 to 30) spans 20 units
• Upper 50% (30 to 90) spans 60 units
Distribution Shape:
Upper half is more spread out than lower half (60 vs. 20)
This suggests a right-skewed distribution
Long tail extending toward higher values
Answer:
IQR = 30
Distribution appears right-skewed
Quick Reference Summary
SAT Statistics Checklist
For Center
- Mean: sum ÷ count
- Median: order first, find middle
- Use median for skewed data
- Check mean vs. median for shape
For Spread
- Range = max - min
- IQR = Q3 - Q1
- IQR resistant to outliers
- Larger SD = more variability
For Shape
- Mean > Median → Right-skewed
- Mean < Median → Left-skewed
- Mean ≈ Median → Symmetric
- Tail points toward skew direction
Common Errors
- Don't forget to order data
- Don't confuse skew direction
- Watch for outlier effects
- Check units/context
Distributions: Understanding Data Patterns
The ability to analyze center, spread, and shape of distributions is fundamental to data literacy in the modern world. Every time you encounter survey results, medical studies, economic reports, or scientific findings, someone has summarized data using these statistical measures. Understanding when to use mean versus median, how outliers distort averages, what standard deviation reveals about consistency, and how skewness affects interpretation makes you a critical consumer of quantitative information. The SAT tests these concepts because they represent genuine statistical reasoning—the foundation for fields ranging from public health and social science to business analytics and machine learning. Master these measures not just for test success, but to become someone who can read data, question claims, and make evidence-based decisions in a world overflowing with statistics.