Before you read

Answer first — the lesson below explains it.

A student's average across 5 math quizzes is 82. If four of the quiz scores are 78, 85, 90, and 76, what is the fifth quiz score?

A
79
B
81
C
82
D
83

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A researcher records response times (in seconds): 10, 15, 20, 20, 25, 30, 30, 30, 40. Which statement about the mean, median, and mode is true?

A
Mean > Median > Mode
B
Mean = Median = Mode
C
Mean < Median < Mode
D
Median < Mean < Mode

Check your answer

Answer:

Theory

Measures of Center

Mean (average): Add all values and divide by the count.
$\text{Mean} = \frac{\sum x_i}{n}$

Median: The middle value when data is ordered. For an even number of values, average the two middle values.

Data set: 3, 7, 8, 12, 15 — median = 8

Mode: The most frequent value. A data set can have no mode, one mode, or multiple modes.

Mode = 8 (most frequent value)

Example 1

Find the mean, median, and mode of: 3, 7, 7, 12, 15

Mean $= \frac{3 + 7 + 7 + 12 + 15}{5} = \frac{44}{5} = 8.8$

Median $= 7$ (middle value of 5 sorted numbers)

Mode $= 7$ (appears most often)

Mean = 8.8, Median = 7, Mode = 7

Example 2

Find the median of: 4, 8, 15, 22

Even number of values: average the two middle values

Median $= \frac{8 + 15}{2} = 11.5$

Median = 11.5

Quick check

You just read it. Can you apply it?

A data analyst has the data set {12, 14, 15, 18, 21}. Adding a value of 100 would most affect which measure of central tendency?

A
Mean
B
Median
C
Mode
D
Range and Mean equally

Check your answer

Answer:

Original mean = 16, with 100 added: mean $= \frac{180}{6} = 30$ , a change of 14. Original median = 15, new median $= \frac{15 + 18}{2} = 16.5$ , a change of 1.5. The mean is most affected by outliers.

Theory

Range and Outliers

Range $= \text{Maximum} - \text{Minimum}$

The range measures spread but is sensitive to outliers (extreme values).

Effect of outliers:
- Mean: strongly affected (pulled toward the outlier)
- Median: resistant (barely changes)
- Mode: not affected

The SAT loves asking which measure is most affected by an outlier — it's always the mean.

Example 1

Data: 10, 12, 13, 14, 50. How does removing 50 affect the mean and median?

With 50: Mean $= \frac{99}{5} = 19.8$ , Median $= 13$

Without 50: Mean $= \frac{49}{4} = 12.25$ , Median $= \frac{12+13}{2} = 12.5$

Mean changed by 7.55; Median changed by only 0.5

The mean is much more affected by the outlier.

Quick check

You just read it. Can you apply it?

In a school, a class of 20 students scored an average of 75 on a standardized test, while another class of 30 students scored an average of 85. What is the combined mean of all 50 students?

A
79
B
80
C
81
D
82

Check your answer

Answer:

Combined mean $= \frac{20 \times 75 + 30 \times 85}{50} = \frac{1500 + 2550}{50} = \frac{4050}{50} = 81$ .

Common Mistakes

Common Mistakes to Avoid

Forgetting to sort before finding the median

Always order the data first!

Remember: avoid this error.

Wrong middle value

For $n$ values, the median is at position $\frac{n+1}{2}$ .

Remember: avoid this error.

Weighted mean errors

When told '5 tests averaging 80 and 3 tests averaging 90', the overall mean is NOT 85. It's $\frac{5(80)+3(90)}{8} = \frac{670}{8} = 83.75$ .

Remember: weighted mean errors — When told '5 tests averaging 80 and 3 tests averaging 90', the overall mean is N...

Quick check

You just read it. Can you apply it?

A survey records the number of hours students spent studying last week:\n\n| Hours | 0 | 1 | 2 | 3 | 4 | 5 |\n|---|---|---|---|---|---|---|\n| Students | 2 | 5 | 8 | 6 | 3 | 1 |\n\nWhat is the median number of hours studied?

A
1
B
2
C
2.5
D
3

Check your answer

Answer:

Total students: $2 + 5 + 8 + 6 + 3 + 1 = 25$ . Median is the 13th value. Cumulative: 0 $\to$ 2, 1 $\to$ 7, 2$\to$15. The 13th value falls in the '2 hours' group. Median = 2.

Strategy

SAT Pro Tip

Shortcut for finding missing values: If the mean of $n$ numbers is $\bar{x}$ , the total is $n \times \bar{x}$ . So if you know the mean and all values except one, the missing value is $n\bar{x} - \text{sum of known values}$ .

After reading

Same question as before. What do you say now?

A student's average across 5 math quizzes is 82. If four of the quiz scores are 78, 85, 90, and 76, what is the fifth quiz score?

A
79
B
81
C
82
D
83

Check your answer

Answer:

Total needed: $5 \times 82 = 410$ . Sum of four: $78 + 85 + 90 + 76 = 329$ . Fifth score: $410 - 329 = 81$ .

After reading

Same question as before. What do you say now?

A researcher records response times (in seconds): 10, 15, 20, 20, 25, 30, 30, 30, 40. Which statement about the mean, median, and mode is true?

A
Mean > Median > Mode
B
Mean = Median = Mode
C
Mean < Median < Mode
D
Median < Mean < Mode

Check your answer

Answer:

Sum $= 10 + 15 + 20 + 20 + 25 + 30 + 30 + 30 + 40 = 220$ .
Mean $= \frac{220}{9} \approx 24.4$ .
Median $= 25$ (the 5th value out of 9 ordered values).
Mode $= 30$ (appears 3 times, more than any other value).
Since $24.4 < 25 < 30$ : Mean < Median < Mode.
Answer: C.

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