Statistical Bias and Sampling

Grade 10 · mathematics · 95 practice problems · read aloud

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📊 Statistical Bias and Sampling

What is Statistical Bias?

Statistical bias is a systematic error that makes survey or sample results different from the true population value. It's like a consistent tilt in your results. Understanding bias helps us collect data that truly represents the whole group we're studying, leading to more accurate conclusions.

How to Identify and Avoid Bias: A Step-by-Step Guide

  1. Identify the Population: Who are you trying to learn about? (e.g., all 10th-grade students in your school).
  2. Check the Sampling Method: Is every person in the population equally likely to be chosen? If not, bias is likely.
  3. Look for Bias Types:
    • Selection Bias: The sample isn't random.
    • Response Bias: Questions are worded to influence answers.
    • Non-response Bias: People who don't respond are different from those who do.
  4. Evaluate: Decide if the bias makes the sample unrepresentative.

Visual Examples

Example 1: The Cafeteria Survey

Scenario: A student surveys people in the cafeteria at 11:30 AM about their favorite school lunch.

Bias Identified: Selection Bias. This only surveys students who eat school lunch and are early to the cafeteria. It misses students who bring lunch, eat later, or skip lunch.

Example 2: The Leading Question

Scenario: A survey asks: "Don't you agree that the school's new policy is unfair to students?"

Bias Identified: Response Bias. The wording pushes people to agree. A better question is: "What is your opinion on the school's new policy?"

🚨 Common Mistakes

  • Confusing a Large Sample with a Good Sample: A survey of 500 students is still biased if they are all from the same soccer team. A representative sample is more important than a large one.
  • Thinking "Random" Means "Haphazard": Random sampling requires a method (like drawing names from a hat) to ensure everyone has an equal chance. Just asking people you see is not random.
  • Ignoring Non-Response: If 90% of people you survey don't answer, the 10% who did might have very strong (and different) opinions than the majority.

💡 Tips & Tricks

  • Use SRS: A Simple Random Sample is the gold standard for avoiding selection bias.
  • Question Your Questions: Read your survey questions out loud. Do they sound like they're pushing for an answer?
  • Think "Who's Missing?": The easiest way to spot bias is to ask yourself: "Which groups from the population are not included in my sample?"

Practice Suggestions

To master this, try these activities:

  1. Critique Real Surveys: Look at polls in the news or online. Identify the population and look for potential biases in how the sample was gathered or how questions were asked.
  2. Design a Fair Survey: Pick a topic (e.g., "favorite music genre at school"). Write unbiased questions and describe how you would get a simple random sample of 50 students.
  3. Spot the Flaw: Practice with problems that describe a sampling method. Your job is to name the type of bias present and explain how it affects the results.

Practice problems

6 of the 95, worked through step by step — try them before opening the answer.

1 A circle is inscribed in a right triangle with legs of length 6 cm and 8 cm. The circle touches all three sides of the triangle. What is the radius of the inscribed circle?

Hint: Consider how the inscribed circle touches each side of a right triangle. The distances from the vertices to the points of tangency create segments that can be related to the triangle's side lengths. For a different triangle with legs 3 and 4, the radius would be found using the area and semiperimeter relationship.

Show the answer

Answer: 2

  1. Understand the problem. We have a right triangle with legs 6 cm and 8 cm. A circle is inscribed inside it, touching all three sides. The radius of this inscribed circle is what we need to find.
  2. Recall the formula for the inradius of a triangle. For any triangle, the inradius r can be found using the formula: r = (Area of triangle) / (Semiperimeter of triangle) That is, r = A / s, where s = (a + b + c) / 2.
  3. Find the hypotenuse of the triangle. Since it's a right triangle with legs 6 and 8, the hypotenuse is: c = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm.
  4. Calculate the area of the triangle. Area A = (1/2) * base * height = (1/2) * 6 * 8 = 24 cm².
  5. Calculate the semiperimeter. The sides are a = 6, b = 8, c = 10. Semiperimeter s = (6 + 8 + 10) / 2 = 24 / 2 = 12 cm.
  6. Apply the inradius formula. r = A / s = 24 / 12 = 2 cm.
  7. Final answer. The radius of the inscribed circle is 2 cm.

2 Matiu wants to estimate the average height of students at his school. He surveys 150 students from the basketball team. Identify the type of sampling bias present and explain why it leads to an inaccurate estimate.

Hint: Think about whether the group surveyed is similar to the whole school population. What characteristic might basketball players share that differs from other students?

Show the answer

Answer: Selection bias (undercoverage bias); the sample is not representative of the entire student population because basketball players tend to be taller than average, so the estimate will be too high.

  1. Identify the sampling method. Matiu uses a convenience sample by only surveying basketball team members, not randomly selecting from all students.
  2. Recognize the bias type. This is selection bias, specifically undercoverage bias, because certain groups (non-athletes, shorter students) are systematically excluded.
  3. Explain the effect. Basketball players are generally taller than the average student. Therefore, the sample's average height will be higher than the true school average, leading to an overestimate.
  4. Conclusion. The estimate is biased and not reliable for the whole school.

3 Mason wants to estimate the average number of hours students at his school spend on homework each week. He surveys 72 students who are members of the chess club. Identify the type of sampling bias present and explain why it is a problem.

Hint: Think about whether the group Mason surveyed represents all students fairly. Consider what kind of students join a chess club and how their homework time might differ from the general student body.

Show the answer

Answer: Selection bias (undercoverage bias) because the sample is not representative of the entire student population; chess club members may have different homework habits than non-members.

  1. Identify the sampling method. Mason used a convenience sample by only surveying chess club members.
  2. Determine the bias. This is selection bias, specifically undercoverage bias, because the sample excludes students who are not in the chess club.
  3. Explain the issue. Chess club members may be more academically inclined or have different schedules, leading to a systematic overestimate or underestimate of the true average homework hours for all students. The sample is not representative, so the estimate is likely biased.

4 Noah wants to estimate the average height of students at his school. He surveys 80 students from the basketball team. Identify the type of sampling bias present and explain why this sample is not representative of the entire school population.

Hint: Think about which group of students is being surveyed and how their height might differ from the rest of the school. Consider whether every student had an equal chance of being selected.

Show the answer

Answer: Selection bias (undercoverage bias); the sample only includes basketball players, who tend to be taller than average, so the estimate will be too high.

  1. Identify the sampling method. Noah selects 80 students from the basketball team only. This is a convenience sample or a purposive sample, not a random sample.
  2. Recognize the bias. Because he only surveys basketball players, the sample systematically excludes students who are not on the basketball team. Basketball players are generally taller than the average student due to the nature of the sport.
  3. Explain the impact. The average height calculated from this sample will be higher than the true average height of all students at the school. This is an example of selection bias, specifically undercoverage bias, because a large portion of the population (non-basketball players) is not represented.
  4. Conclusion. The sample is biased and cannot be used to accurately estimate the average height of the entire school.

The answer is selection bias (undercoverage bias).

5 Liam wants to estimate the average height of students at his school. He surveys 50 students from the basketball team. Identify the type of sampling bias present and explain why this sample is not representative of the entire student population.

Hint: Think about which group of students is being surveyed and how their height might differ from the rest of the school. Consider whether every student had an equal chance of being selected.

Show the answer

Answer: Selection bias (specifically, undercoverage bias). The sample only includes basketball players, who tend to be taller than the average student, so the sample overrepresents tall students and underrepresents shorter students.

  1. Identify the sampling method. Liam uses a convenience sample by only surveying members of the basketball team.
  2. Recognize the bias. This is selection bias, specifically undercoverage bias, because the sample excludes all students who are not on the basketball team.
  3. Explain why it is not representative. Basketball players are generally taller than the average student due to the nature of the sport. Therefore, the sample will overestimate the average height of all students at the school.
  4. Conclusion. The sample is biased and cannot be used to accurately estimate the population mean height.

The correct answer is selection bias (undercoverage bias).

6 Hana wants to estimate the average height of all students at her school. She stands outside the gymnasium during a basketball game and surveys the first 50 students who enter. Identify the type of sampling bias present and explain why it is a problem.

Hint: Think about where and when the sample is collected. Are the students at a basketball game likely to have something in common that affects height? Consider how the selection method might exclude certain groups.

Show the answer

Answer: Convenience sampling bias; the sample is not representative of all students because it only includes those attending a basketball game, who may be taller on average than the general student population.

  1. Identify the sampling method. Hana is using a convenience sample by surveying the first 50 students entering a specific location (gymnasium) during a specific event (basketball game).
  2. Recognize the bias. This is convenience sampling bias because the sample is chosen based on ease of access, not random selection.
  3. Explain the problem. Students attending a basketball game may be more likely to be athletes or taller individuals, so the sample overrepresents taller students and underrepresents shorter students or those not interested in basketball. This means the average height estimate will likely be higher than the true average height of all students at the school.
  4. Conclusion. The sample is not representative, leading to systematic error (bias) in the estimate.

The answer is: Convenience sampling bias; the sample is not representative of all students because it only includes those attending a basketball game, who may be taller on average than the general student population.

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