One-Variable Data – SAT Math

📖 What Is One-Variable Data?

One-variable data consists of measurements of a single characteristic. We use measures of center and measures of spread to describe the data.

Mean (Average)

\(\bar{x} = \frac{\sum x}{n}\)

Sum of all values divided by the number of values.

Median

Middle value

When ordered from least to greatest, the middle value.

Mode

Most frequent

The value that appears most often.

Range

Max − Min

Difference between largest and smallest values.

Standard Deviation

\(\sigma\)

Measures how spread out the data is from the mean.

Dot Plot

Visual display

Each data point is represented by a dot.

Histogram

Bar chart

Shows frequency of data in intervals.

Outlier

Extreme value

A value far from the rest of the data.

1. Mean (Average)

📌 RULE: Mean

\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \]

Example: Find the mean of 4, 6, 8, 10, 12.
\[ \text{Sum} = 4 + 6 + 8 + 10 + 12 = 40 \]
\[ \text{Mean} = \frac{40}{5} = 8 \]

💡 Strategy — Mean

The mean is affected by outliers (extreme values). If a data set has outliers, the median may be a better measure of center.

📝 SOLVED EXAMPLE 1 — Mean

Find the mean of: 10, 15, 20, 25, 30

Step 1: Sum = 10 + 15 + 20 + 25 + 30 = 100

Step 2: Number of values = 5

Step 3: Mean = 100 ÷ 5 = \(\color{var(--math)}{20}\)

✅ Mean = 20

💡 Tip: The mean is the balance point of the data.

2. Median

📌 RULE: Median

Step 1: Arrange the values in order from least to greatest.
Step 2: If there is an odd number of values, the median is the middle value.
Step 3: If there is an even number, the median is the average of the two middle values.

Example (Odd): 4, 6, 8, 10, 12 → Median = 8
Example (Even): 4, 6, 8, 10 → Median = (6 + 8)/2 = 7

📝 SOLVED EXAMPLE 2 — Median

Find the median of: 12, 5, 8, 15, 10, 7

Step 1: Order: 5, 7, 8, 10, 12, 15

Step 2: Even number (6 values) → two middle: 8 and 10

Step 3: Median = (8 + 10) ÷ 2 = \(\color{var(--math)}{9}\)

✅ Median = 9

💡 Tip: The median is not affected by outliers.

3. Mode

📌 RULE: Mode

The mode is the value that appears most frequently.

Example: 2, 3, 3, 5, 7, 7, 7, 8 → Mode = 7
No mode: 1, 2, 3, 4 (all appear once)
Multiple modes: 2, 2, 3, 3, 4 → Modes = 2 and 3

📝 SOLVED EXAMPLE 3 — Mode

Find the mode of: 3, 7, 5, 3, 8, 5, 3, 9

Step 1: Count frequencies: 3 appears 3 times, 5 appears 2 times, others once

Step 2: Mode = \(\color{var(--math)}{3}\)

✅ Mode = 3

💡 Tip: A data set can have no mode or multiple modes.

4. Range

📌 RULE: Range

\[ \text{Range} = \text{Maximum} - \text{Minimum} \]

Example: 5, 8, 12, 15, 20 → Range = 20 − 5 = 15

📝 SOLVED EXAMPLE 4 — Range

Find the range of: 15, 22, 18, 25, 20, 30

Step 1: Maximum = 30, Minimum = 15

Step 2: Range = 30 − 15 = \(\color{var(--math)}{15}\)

✅ Range = 15

💡 Tip: The range is affected by outliers.

5. Standard Deviation

📌 RULE: Standard Deviation

Standard deviation measures how spread out the data is from the mean.

Small standard deviation → data is clustered near the mean
Large standard deviation → data is spread out

SAT Tip: You usually don't need to calculate standard deviation, but you need to interpret it.

📌 Key Concept

If two data sets have the same mean, the one with the larger standard deviation has more spread out values.

📝 SOLVED EXAMPLE 5 — Standard Deviation

Data Set A: 5, 5, 5, 5, 5 (mean = 5)
Data Set B: 1, 3, 5, 7, 9 (mean = 5)
Which has the larger standard deviation?

Step 1: Both have the same mean (5)

Step 2: Set A has no spread → standard deviation = 0

Step 3: Set B is spread out → standard deviation is larger

✅ Data Set B has the larger standard deviation

💡 Tip: More spread = larger standard deviation.

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

Find the mean of: 8, 12, 16, 20, 24

A) 14

B) 16

C) 18

D) 20

✓ Answer: B

Sum = 8+12+16+20+24 = 80 → 80 ÷ 5 = 16

📝 Solution: \(80/5 = 16\)

Question 2

Find the median of: 3, 7, 9, 11, 13, 15, 17

A) 9

B) 10

C) 11

D) 12

✓ Answer: C

7 values (odd) → middle is 4th value → 11

📝 Solution: Median = 11

Question 3

Find the mode of: 2, 4, 4, 6, 8, 8, 8, 10

A) 4

B) 6

C) 8

D) 10

✓ Answer: C

8 appears 3 times (most frequent)

📝 Solution: Mode = 8

Question 4

Find the range of: 12, 25, 18, 30, 22, 35

A) 15

B) 18

C) 20

D) 23

✓ Answer: D

Max = 35, Min = 12 → Range = 35 − 12 = 23

📝 Solution: \(35 - 12 = 23\)

Question 5

Find the median of: 4, 8, 12, 16, 20, 24

A) 12

B) 13

C) 14

D) 16

✓ Answer: C

6 values (even) → middle two: 12 and 16 → (12+16)/2 = 14

📝 Solution: \((12+16)/2 = 14\)

Question 6

Which data set has the larger standard deviation?
Set A: 10, 10, 10, 10, 10
Set B: 8, 9, 10, 11, 12

A) Set A

B) Set B

C) They are the same

D) Cannot be determined

✓ Answer: B

Set A has no spread (all 10s) → standard deviation = 0. Set B is spread out → larger standard deviation.

📝 Solution: More spread = larger standard deviation.

Question 7

Find the mean of: 15, 22, 18, 25, 30

A) 20

B) 21

C) 22

D) 24

✓ Answer: C

Sum = 15+22+18+25+30 = 110 → 110 ÷ 5 = 22

📝 Solution: \(110/5 = 22\)

Question 8

Find the mode of: 1, 2, 2, 3, 3, 3, 4, 4, 5

A) 2

B) 3

C) 4

D) 3

✓ Answer: D

3 appears 3 times (most frequent)

📝 Solution: Mode = 3

Question 9

Find the range of: 5, 10, 15, 20, 25, 30, 35

A) 25

B) 28

C) 30

D) 35

✓ Answer: C

Max = 35, Min = 5 → Range = 35 − 5 = 30

📝 Solution: \(35 - 5 = 30\)

Question 10

Find the median of: 2, 6, 8, 10, 14, 18, 22

A) 8

B) 10

C) 12

D) 14

✓ Answer: B

7 values (odd) → middle is 4th value → 10

📝 Solution: Median = 10

🎉 WELL DONE!

You've completed the One-Variable Data lesson. You now know how to calculate and interpret mean, median, mode, range, and standard deviation.

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