Data Collection and Statistics

Understand data collection methods, sampling, observational studies vs experiments, and draw valid conclusions from statistical data.

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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Data Collection and Statistics

Measures of Center

Mean: Sum of all values divided by the number of values. Sensitive to outliers.
Median: Middle value when data is ordered. Resistant to outliers.
Mode: Most frequently occurring value.

When to Use Which

Situation	Best Measure
Symmetric data, no outliers	Mean
Skewed data or outliers	Median
Categorical data	Mode

Measures of Spread

Range: Maximum – Minimum
Interquartile Range (IQR): Q3 – Q1 (middle 50% of data)
Standard Deviation: Average distance from the mean

Data Collection Methods

Random sample: Every member of the population has an equal chance of being selected
Stratified sample: Population divided into groups, random sample from each
Convenience sample: Non-random, may introduce bias
Observational study: Researchers observe without intervention
Experiment: Researchers assign treatments to subjects

Bias Types

Selection bias: Sample not representative of population
Response bias: Question wording influences answers
Non-response bias: Some selected individuals don't participate

Box Plots and Histograms

Box plots show the five-number summary: min, Q1, median, Q3, max
Histograms show frequency distribution across intervals

SAT Tips

Outliers pull the mean toward them but don't affect the median
To find the median of even-numbered data: average the two middle values
Read study descriptions carefully to identify bias

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the median of: 12, 8, 15, 10, 9

💡 Show Solution

Solution:

Step 1: Put in order $8, 9, 10, 12, 15$

Step 2: Find middle value 5 numbers, so middle is 3rd value

Answer: 10

SAT Tip: ALWAYS order the data first when finding median!

2Problem 2medium

❓ Question:

A dataset has 5 values with a mean of 20. If a 6th value of 32 is added, what is the new mean?

💡 Show Solution

Solution:

Original sum: $5 \times 20 = 100$

Add new value: $100 + 32 = 132$

New mean: $\frac{132}{6} = 22$

Answer: 22

SAT Tip: Mean × count = sum. Use this to find totals!

3Problem 3hard

❓ Question:

A study found that students who eat breakfast tend to have higher test scores. Which conclusion is valid?

A) Eating breakfast causes higher test scores B) There is an association between eating breakfast and test scores C) Students should be required to eat breakfast D) Skipping breakfast lowers intelligence

💡 Show Solution

Solution:

Key word: "tend to" = correlation/association

Check each:

A) Causes - too strong! Correlation ≠ causation ❌
B) Association - this is what the data shows ✓
C) Should be required - policy decision, not data conclusion ❌
D) Lowers intelligence - causation + extreme claim ❌

Answer: B - There is an association

SAT Tip: Studies show correlation/association. Saying "causes" requires controlled experiments!

4Problem 4easy

❓ Question:

A histogram shows the following frequencies for test scores: 60-69: 3 students 70-79: 8 students 80-89: 12 students 90-100: 7 students

How many students scored below 80?

💡 Show Solution

Step 1: Add the frequencies for intervals below 80: $3 + 8 = 11$

Answer: 11 students scored below 80.

SAT Tip: On histograms, "below 80" includes the 60-69 and 70-79 intervals. Be careful about whether the boundary value is included.

5Problem 5easy

❓ Question:

A histogram shows the following frequencies for test scores: 60-69: 3 students 70-79: 8 students 80-89: 12 students 90-100: 7 students

How many students scored below 80?

💡 Show Solution

Step 1: Add the frequencies for intervals below 80: $3 + 8 = 11$

Answer: 11 students scored below 80.

SAT Tip: On histograms, "below 80" includes the 60-69 and 70-79 intervals. Be careful about whether the boundary value is included.

6Problem 6medium

❓ Question:

The mean of 6 numbers is 15. When a 7th number is added, the mean becomes 17. What is the 7th number?

💡 Show Solution

Step 1: Find the original sum: $6 \times 15 = 90$

Step 2: Find the new sum: $7 \times 17 = 119$

Step 3: The 7th number = $119 - 90 = 29$

Check: $(90 + 29) \div 7 = 119 \div 7 = 17$ ✓

Answer: The 7th number is 29.

7Problem 7medium

❓ Question:

The mean of 6 numbers is 15. When a 7th number is added, the mean becomes 17. What is the 7th number?

💡 Show Solution

Step 1: Find the original sum: $6 \times 15 = 90$

Step 2: Find the new sum: $7 \times 17 = 119$

Step 3: The 7th number = $119 - 90 = 29$

Check: $(90 + 29) \div 7 = 119 \div 7 = 17$ ✓

Answer: The 7th number is 29.

8Problem 8medium

❓ Question:

In a box plot, $Q_1 = 30$ , $\text{Median} = 45$ , $Q_3 = 60$ , $\text{Min} = 10$ , $\text{Max} = 85$ . What is the IQR, and which single value, if added, would most change the mean but least change the median?

💡 Show Solution

IQR: $Q_3 - Q_1 = 60 - 30 = 30$

Effect of adding an extreme value: Adding a very large value (e.g., 200) would:

Mean: Increase significantly (the mean is sensitive to extremes)
Median: Change very little (the median is resistant to outliers — it only depends on the middle value)

Answer: IQR = 30. An extreme outlier (very large or very small) would most change the mean but least change the median.

9Problem 9medium

❓ Question:

💡 Show Solution

IQR: $Q_3 - Q_1 = 60 - 30 = 30$

Effect of adding an extreme value: Adding a very large value (e.g., 200) would:

Mean: Increase significantly (the mean is sensitive to extremes)
Median: Change very little (the median is resistant to outliers — it only depends on the middle value)

Answer: IQR = 30. An extreme outlier (very large or very small) would most change the mean but least change the median.

10Problem 10hard

❓ Question:

A researcher wants to determine if a new teaching method improves test scores. She randomly assigns 50 students to use the new method and 50 to use the traditional method. The new method group has a mean score of 82, while the traditional group has a mean of 78. Can she conclude the new method CAUSES higher scores?

💡 Show Solution

Answer: YES — with appropriate caveats.

Why: This is a randomized controlled experiment, not just an observational study.

Key features that allow a causal conclusion:

Random assignment to treatment groups — this controls for confounding variables
Control group (traditional method) for comparison
Same number in each group

However, she should also consider:

Statistical significance — is the 4-point difference large enough to not be due to chance? (She'd need a p-value or confidence interval.)
Practical significance — is a 4-point difference meaningful in practice?

SAT Rule:

Randomized experiment → CAN conclude causation
Observational study → can only conclude ASSOCIATION

11Problem 11hard

❓ Question:

💡 Show Solution

Answer: YES — with appropriate caveats.

Why: This is a randomized controlled experiment, not just an observational study.

Key features that allow a causal conclusion:

Random assignment to treatment groups — this controls for confounding variables
Control group (traditional method) for comparison
Same number in each group

However, she should also consider:

Statistical significance — is the 4-point difference large enough to not be due to chance? (She'd need a p-value or confidence interval.)
Practical significance — is a 4-point difference meaningful in practice?

SAT Rule:

Randomized experiment → CAN conclude causation
Observational study → can only conclude ASSOCIATION

12Problem 12expert

❓ Question:

Set A has values {10, 12, 14, 16, 18} and Set B has values {12, 13, 14, 15, 16}. Without calculating, which set has the larger standard deviation? Explain.

💡 Show Solution

Set A has the larger standard deviation.

Reasoning: Both sets have the same mean:

Set A: $\frac{10+12+14+16+18}{5} = 14$
Set B: $\frac{12+13+14+15+16}{5} = 14$

Set A has values that are more spread out from 14 (ranging from 10 to 18, each value 2 units apart from the next).

Set B has values that are more tightly clustered around 14 (ranging from 12 to 16, each value only 1 unit apart).

Since standard deviation measures how far values are from the mean on average, Set A has the larger standard deviation.

Answer: Set A

SAT Tip: You don't need to calculate SD on the SAT — just understand that wider spread = larger SD.

13Problem 13expert

❓ Question:

Set A has values {10, 12, 14, 16, 18} and Set B has values {12, 13, 14, 15, 16}. Without calculating, which set has the larger standard deviation? Explain.

💡 Show Solution

Set A has the larger standard deviation.

Reasoning: Both sets have the same mean:

Set A: $\frac{10+12+14+16+18}{5} = 14$
Set B: $\frac{12+13+14+15+16}{5} = 14$

Set A has values that are more spread out from 14 (ranging from 10 to 18, each value 2 units apart from the next).

Set B has values that are more tightly clustered around 14 (ranging from 12 to 16, each value only 1 unit apart).

Since standard deviation measures how far values are from the mean on average, Set A has the larger standard deviation.

Answer: Set A

SAT Tip: You don't need to calculate SD on the SAT — just understand that wider spread = larger SD.

Explain using:

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What is Data Collection and Statistics?▾

Understand data collection methods, sampling, observational studies vs experiments, and draw valid conclusions from statistical data.

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Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 13 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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Yes, this page includes 13 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Data Collection and Statistics

Try the Interactive Version!

Data Collection and Statistics

Measures of Center

When to Use Which

Measures of Spread

Data Collection Methods

Bias Types

Box Plots and Histograms

SAT Tips

📚 Practice Problems

1Problem 1easy

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2Problem 2medium

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3Problem 3hard

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4Problem 4easy

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5Problem 5easy

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6Problem 6medium

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7Problem 7medium

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8Problem 8medium

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9Problem 9medium

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10Problem 10hard

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11Problem 11hard

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12Problem 12expert

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13Problem 13expert

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