Mean (Average): Sum of all values ÷ number of values > {3, 7, 9, 11, 15} → Mean = (3 + 7 + 9 + 11 + 15)/5 = 45/5 = 9
Median: The middle value when data is ordered. If there's an even number of values, average the two middle ones. > {3, 7, 9, 11, 15} → Median = 9 (middle value) > {3, 7, 9, 11} → Median = (7 + 9)/2 = 8
Mode: The most frequently occurring value.
When to use which:
Range: Maximum − Minimum > {3, 7, 9, 11, 15} → Range = 15 − 3 = 12
Standard deviation: Measures how spread out values are from the mean. The SAT tests conceptual understanding, not calculation.
The SAT tests your ability to evaluate the validity of statistical conclusions.
Representative sample: A sample that reflects the population's characteristics. Random sampling produces representative samples.
Margin of error: The uncertainty range around a sample estimate. A survey result of "60% ± 3%" means the true value is likely between 57% and 63%.
Bias: When a sample systematically misrepresents the population.
Inference principle: You can generalize results to the population only if the sample was randomly selected from that population.
The SAT often presents tables or stats and asks about the meaning.
Example: A class of 30 students took a test. The average score was 74. 5 students scored above 90. What does this tell you?
Real-world example: A school surveys 100 randomly selected students and finds 72% prefer a later school start time. Can you conclude that the majority of all students in the school prefer later start times? Yes — because the sample was randomly selected, you can generalize to the larger population (with some margin of error).
---
---
Quiz Questions:
Q1. Five test scores are: 82, 90, 74, 96, 88. What is the mean score?
A) 88 B) 86 C) 84 D) 90
Answer: B — Sum = 82 + 90 + 74 + 96 + 88 = 430. Mean = 430/5 = 86.
---
Q2. A real estate agent reports the average (mean) home sale price in a neighborhood as $650,000. One sale was an unusually expensive mansion at $3,000,000. A buyer asks if this average reflects typical prices. What should the agent say?
A) Yes, the mean accurately represents typical prices in all situations B) The mean is heavily influenced by the outlier (the mansion), so the median would better represent the typical home price C) The mean is only valid for populations over 100 homes D) The mean is accurate because it includes all data points
Answer: B — The mean is pulled upward by extremely high values (outliers). For home prices with extreme outliers, the median is a better measure of the "typical" price.
---
Q3. Two datasets have the same mean. Dataset A has a standard deviation of 2; Dataset B has a standard deviation of 15. Which statement is correct?
A) Dataset A and Dataset B have identical distributions B) Dataset B has values more spread out from the mean than Dataset A C) Dataset A has values more spread out from the mean than Dataset B D) Standard deviation cannot be compared across different datasets
Answer: B — A larger standard deviation means greater spread from the mean. Dataset B (SD = 15) has values spread much further from the mean than Dataset A (SD = 2). Both have the same center (mean) but different spreads.
---
Q4. A researcher surveys 50 randomly selected customers from a store's database of 10,000 customers. 64% say they are satisfied. Which conclusion is best supported?
A) Exactly 6,400 customers are satisfied B) It is likely that approximately 64% of all 10,000 customers are satisfied, with some margin of error C) The survey result is invalid because the sample size is too small D) The conclusion only applies to the 50 surveyed customers
Answer: B — Because the sample was randomly selected, you can generalize to the population — but with some uncertainty (margin of error). Choice A claims exact precision the sample doesn't provide. Choice C is wrong — samples don't need to be huge if they're random. Choice D ignores the purpose of statistical inference.
---
Q5. A school surveys only students who volunteer to participate in a study about phone use. The results show 90% use their phones over 4 hours daily. What is a concern about this study?
A) The sample size might be too large B) Voluntary response bias — students who volunteer for phone surveys may be more heavy users, making the sample unrepresentative of all students C) The result proves that 90% of all students use phones over 4 hours daily D) The survey is valid because voluntary participation ensures honest answers
Answer: B — Voluntary response sampling overrepresents people with strong opinions or a stake in the topic (here, heavy phone users might be more likely to participate). This is voluntary response bias, and the results cannot be generalized to all students without this caveat.