🎯 Learning Objectives
- Definitions of mean, median, mode, range, and standard deviation.
- How an outlier shifts the mean but barely moves the median.
- How to use the mean shortcut: sum = mean × count.
- Standard deviation: what it measures and how to compare two data sets.
- 10 SAT-style practice questions.
1. The Big Five Statistics
Statistic | Definition | Quick Way to Remember |
Mean | Sum ÷ count | The 'average' |
Median | Middle value when sorted | The 'middle child' |
Mode | Most frequent value | The most popular |
Range | Max − Min | Width of data |
Std. Dev. | How spread out the data is | Higher = more scattered |
2. The Mean: and Its Shortcut
The most useful rearrangement
Mean = Sum / Count ⇒ Sum = Mean × Count
On the SAT, you'll often be told the mean and asked to find a missing value. The fastest path is: compute the total sum (mean × count), then subtract the values you already have.
📝 Worked Example
A student has scores 78, 82, 90, and 85 on four tests. What must she score on the 5th to have a mean of 86?
Required total = 86 × 5 = 430.
Current total = 78 + 82 + 90 + 85 = 335.
Needed score = 430 − 335 = 95.
3. Median: Sort First, Always
To find the median, sort the data from smallest to largest. The median is the middle value if there's an odd number of data points, or the average of the two middle values if there's an even number.
⚡ Strategy
If you forget to sort, you'll get the wrong middle value. Make sorting your reflex action whenever you see the word 'median'.
4. Mode and Range
Mode is the value that appears most often. A data set can have no mode (if all values are unique), one mode, or multiple modes.
Range is simply the largest value minus the smallest value. It tells you how 'wide' the data is, but it ignores everything in between.
5. Mean vs. Median: How Outliers Affect Each
❓ Are there extreme values (outliers)? | ||
✅ YES ▼ | ❌ NO ▼ | |
Mean shifts toward the outlier; median barely moves | Mean and median are roughly equal | |
💡 Pro Tip: When in Doubt, Use the Median
Real-world data with outliers (like income or house prices) is usually summarized with the median, because it isn't pulled by a few extreme values.
6. Standard Deviation: A Visual Intuition
Standard deviation measures how far values are from the mean, on average. A small standard deviation means the data is tightly clustered around the mean. A large standard deviation means the values are spread out.
💡 SAT Standard Deviation Rules
- For comparison questions, examine how far values lie from the mean. When a numerical standard deviation is requested, use the specified population or sample convention.
- You only need to COMPARE two data sets and decide which has a larger or smaller spread.
- A flat, spread-out histogram has high standard deviation. A tall, narrow one has low standard deviation.
- Adding a value far from the mean INCREASES standard deviation. Adding one near the mean DECREASES it.
7. Strategies
⚡ Use Sum = Mean × Count
Anytime a question gives you the mean and asks about a missing value, this rearrangement saves the day.
⚡ For weighted averages, use total / total
Combined mean = (Σ values) / (total count). Don't average the means directly.
⚡ When asked 'which has higher std dev', look for spread
Compare the visual spread or the size of the gaps: no calculation needed.
⚡ Median is unchanged when extremes shift further out
If the smallest goes down and the largest goes up, the median doesn't budge.
⚡ For symmetric data, mean ≈ median
Use this to spot answer-choice traps.
8. Summary
📌 Key Takeaways
- Mean: Sum / Count. Strategy: Sum = Mean × Count.
- Median: Middle of sorted list.
- Mode: Most common value.
- Range: Max − Min.
- Std. Dev.: How spread out: compare distances from the mean; calculate when requested.
- Outliers: Pull the mean, leave the median.
9. Practice: 10 Questions
Q1. The mean of 5 numbers is 12. If four of the numbers are 10, 14, 8, and 15, what is the fifth?
A) 11 B) 12 C) 13 D) 14
Hint: Sum needed = 60. Current sum = 47. Fifth = 13.
Q2. What is the median of 7, 3, 9, 1, 5?
A) 3 B) 5 C) 7 D) 9
Hint: Sort: 1, 3, 5, 7, 9. Middle is 5.
Q3. What is the mode of 4, 7, 4, 8, 2, 4, 6?
A) 2 B) 4 C) 6 D) 7
Hint: 4 appears three times.
Q4. The range of a data set is 18. If the smallest value is 7, what is the largest?
A) 11 B) 18 C) 25 D) 32
Hint: Largest − 7 = 18 → Largest = 25.
Q5. Which set has the larger standard deviation: A) {10, 10, 10, 10}, or B) {2, 8, 12, 18}?
A B same cannot tell
Hint: B is more spread out → larger std dev.
Q6. Five scores have a mean of 80. If a 6th score of 92 is added, what is the new mean?
A) 81 B) 82 C) 83 D) 84
Hint: Old sum = 400. New sum = 492. 492/6 = 82.
Q7. What is the median of {11, 14, 16, 19, 22, 26}?
A) 16 B) 17 C) 17.5 D) 19
Hint: Even count: avg of 16 and 19 → 17.5.
Q8. Add the value 100 to the set {2, 4, 6, 8}. What happens to the mean and median?
A) Both increase B) Mean ↑, median same C) Median ↑, mean same D) Both same
Hint: Mean jumps a lot; median moves slightly.
Q9. The mean of x, x+2, x+4 is 12. What is x?
A) 8 B) 10 C) 11 D) 12
Hint: (3x+6)/3 = 12 → x = 10.
Q10. Class A scores: {78, 80, 82, 84, 86}. Class B: {60, 70, 80, 90, 100}. Which has greater std dev?
A) A B) B C) Same D) Cannot tell
Hint: B is more spread out → larger std dev.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | C) 13 | Sum = 60; missing = 60 − 47 = 13. |
Q2. | B) 5 | Sorted middle is 5. |
Q3. | B) 4 | Most frequent value. |
Q4. | C) 25 | Range = max − min. |
Q5. | B) B | B is spread out; A has zero spread. |
Q6. | B) 82 | (400 + 92) / 6 = 82. |
Q7. | C) 17.5 | Average of 16 and 19. |
Q8. | B) Mean ↑, median ≈ same | Outliers shift mean far more. |
Q9. | B) 10 | (3x + 6)/3 = 12 → x = 10. |
Q10. | B) B | B has wider spread. |