Overview
Statistical Measures of Center and Spread is a medium-difficulty topic tested consistently in the Problem-Solving and Data Analysis domain on the digital SAT. Questions typically ask students to compute or interpret mean, median, mode, range, IQR, and standard deviation, and to reason about how adding, removing, or changing a data value affects these statistics. Expect 1–2 questions per Math module from this topic area. These are calculator-active questions.
Key Points
1. Measures of Center
| Measure | Definition | Sensitivity to Outliers |
|---|---|---|
| Mean | Sum of all values ÷ count | Highly sensitive — pulled toward outliers |
| Median | Middle value when data sorted in order | Resistant — barely affected by outliers |
| Mode | Most frequent value | Resistant — determined by frequency, not magnitude |
Calculating mean: mean = (sum of all values) / n
Calculating median:
- Odd count: middle value after sorting
- Even count: average of the two middle values after sorting
- Always sort first — this is the most common computational error
Useful relationship: If mean > median → distribution is likely right-skewed (outlier on the high end). If mean < median → likely left-skewed.
2. Effect of Changing a Data Set
| Action | Effect on Mean | Effect on Median |
|---|---|---|
| Add a value equal to the mean | Mean unchanged | May shift slightly |
| Add a high outlier | Mean increases | Median barely changes |
| Remove a high outlier | Mean decreases | Median barely changes |
| Add a value above median | Median may shift by one position | Mean increases |
| Multiply every value by k | Mean multiplies by k | Median multiplies by k |
| Add k to every value | Mean increases by k | Median increases by k |
3. Measures of Spread
Range = Maximum − Minimum (simple but affected by outliers)
IQR (Interquartile Range) = Q3 − Q1
- Represents the spread of the middle 50% of the data
- Resistant to outliers
- Used in box plots — the “box” spans from Q1 to Q3
Outlier boundary: A value is a potential outlier if it is more than 1.5 × IQR below Q1 or above Q3.
Standard deviation (SD): Measures the average distance of each data point from the mean.
- Larger SD → data more spread out
- Smaller SD → data more clustered around the mean
- The SAT tests SD conceptually — you will not need to calculate it using the formula
- Two data sets can have the same mean but different SDs; the SAT tests this distinction
4. Weighted Average
When combining groups of different sizes:
Example: 20 students averaged 80 and 30 students averaged 90 → combined mean = (20×80 + 30×90) / 50 = (1600+2700)/50 = 86
5. Identifying Skew from Center Statistics
| Relationship | Shape | Tail Direction |
|---|---|---|
| Mean > Median | Right-skewed | Tail points right |
| Mean < Median | Left-skewed | Tail points left |
| Mean ≈ Median | Symmetric | No clear tail |
Pitfalls and Common Mistakes
Mistake 1: Forgetting to sort data before finding the median. Students pick the middle number from an unsorted list. Fix: Always write out the values in ascending order before identifying the median position.
Mistake 2: Confusing mean with median in the question stem. The SAT frequently uses both words in the same problem. Students compute the mean when asked for the median. Fix: Circle or underline the measure you are asked for before solving.
Mistake 3: Assuming a high outlier does not affect the median. While the median is resistant, adding multiple extreme values or changing the data structure can shift it. Fix: Track the position of the median (by count) rather than assuming it never changes.
Mistake 4: Computing the weighted average by averaging the two group means without weighting by group size. Example: (80 + 90) / 2 = 85, which is wrong when group sizes differ. Fix: Use the weighted formula: (n₁ × mean₁ + n₂ × mean₂) / (n₁ + n₂).
Mistake 5: Confusing range with IQR. A question asking “what is the spread of the middle 50%” is asking for the IQR, not the range. Fix: “Middle 50%” → IQR; “entire data set” → range.
Related Entries
Quick Reference Card
| Statistic | Formula / Rule |
|---|---|
| Mean | sum / n |
| Median | Middle value after sorting |
| IQR | Q3 − Q1 |
| Range | Max − Min |
| Outlier threshold | Below Q1 − 1.5×IQR or above Q3 + 1.5×IQR |
| Weighted mean | (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂) |
| Effect of outlier | Mean shifts; Median stays nearly fixed |
| SD (conceptual) | Larger spread = larger SD; no calculation needed |