AP Statistics Flashcards: Biased And Unbiased Point Estimates

What this deck covers

This deck focuses on Biased And Unbiased Point Estimates, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Is an estimator with higher variance less reliable?

Answer: Yes, higher variance implies less reliability. Higher variance means more variability around target.

Flashcard 2: Find the bias in the estimate: sample variance $s^2 = \frac{\text{SS}}{n}$.

Answer: Bias: underestimates population variance. Dividing by $n$ instead of $n-1$ creates downward bias.

Flashcard 3: Is the median an unbiased estimator for mean?

Answer: No, median may not equal mean. Median only equals mean in symmetric distributions.

Flashcard 4: What is the bias in using sample median for population mean?

Answer: Median may systematically differ from mean. Creates bias except in symmetric populations.

Flashcard 5: State the formula for sample proportion.

Answer: $\frac{x}{n}$, where $x$ is the number of successes. Counts successes divided by total trials.

Flashcard 6: Is an estimator with higher variance less reliable?

Answer: Yes, higher variance implies less reliability. Higher variance means more variability around target.

Flashcard 7: Which condition ensures an unbiased estimate?

Answer: Expected value equals the true parameter. No systematic over or under estimation occurs.

Flashcard 8: Why is unbiasedness important in estimation?

Answer: Ensures estimates are centered around true parameter. Prevents systematic over or under estimation.

Flashcard 9: State the formula for sample proportion.

Answer: $\frac{x}{n}$, where $x$ is the number of successes. Counts successes divided by total trials.

Flashcard 10: Find and correct the bias in estimating $\theta$ with $\theta + 1$.

Answer: Correct by using estimator $\theta$. Subtract 1 to remove systematic overestimation.

Flashcard 11: Why is unbiasedness important in estimation?

Answer: Ensures estimates are centered around true parameter. Prevents systematic over or under estimation.

Flashcard 12: What is the bias of using $n$ in sample variance calculation?

Answer: Underestimates true variance, biased. Creates systematic underestimation of true variance.

Flashcard 13: What estimator property improves with increasing sample size?

Answer: Consistency. Estimator precision improves with larger samples.

Flashcard 14: Identify the unbiased estimate for population mean.

Answer: Sample mean $\bar{x}$. Expected value equals population mean $\mu$.

Flashcard 15: What is the bias of the sample mean for population mean?

Answer: Zero, sample mean is unbiased. Sample mean has no systematic error.

Flashcard 16: What measure do statisticians use to quantify bias?

Answer: Expected value of the estimator minus true parameter. Measures systematic deviation from true value.

Flashcard 17: Identify the unbiased estimator for population proportion.

Answer: Sample proportion $\frac{x}{n}$. Expected value equals true population proportion $p$.

Flashcard 18: What does it mean if an estimator is 'consistent'?

Answer: Estimator converges to true parameter as sample size increases. Approaches true value with infinite sample size.

Flashcard 19: Does increasing sample size always eliminate bias?

Answer: No, some biases persist regardless of sample size. Structural biases persist regardless of sample size.

Flashcard 20: What does it mean if an estimator is 'efficient'?

Answer: It has the smallest variance among unbiased estimators. Minimum variance unbiased estimator is optimal.

Flashcard 21: What is a point estimate?

Answer: A single value estimate for a population parameter. Provides best guess of unknown population parameter.

Flashcard 22: What is the relationship between bias and mean squared error?

Answer: MSE = Variance + Bias$^2$. Decomposition shows bias contributes to total error.

Flashcard 23: Are all consistent estimators also unbiased?

Answer: No, consistency does not imply lack of bias. Consistency and bias are independent properties.

Flashcard 24: Can a biased estimator converge to the true parameter?

Answer: Yes, if it is consistent. Asymptotic convergence overcomes initial bias.

Flashcard 25: Define unbiased point estimate.

Answer: An estimate whose expected value equals the true parameter. No systematic error in long run.

Flashcard 26: Identify an unbiased estimator for population parameter.

Answer: Estimator with expected value equal to the parameter. Expected value must match the true parameter.

Flashcard 27: State the formula for sample mean.

Answer: $\bar{x} = \frac{\textstyle \text{Sum of all sample values}}{n}$. Sum divided by sample size gives average.

Flashcard 28: Is the sample mode an unbiased estimator for population mean?

Answer: No, mode does not equate to mean. Mode represents most frequent value, not average.

Flashcard 29: What is the impact of sample size on bias?

Answer: Larger samples may reduce bias, not eliminate it. Bias is property of estimator, not sample size.

Flashcard 30: What is a common source of bias in surveys?

Answer: Non-response bias. Missing responses skew results systematically.

Flashcard 31: Is an estimator unbiased if its bias is zero?

Answer: Yes, zero bias means unbiased. Zero bias is definition of unbiased estimator.

Flashcard 32: State the formula for sample mean.

Answer: $\bar{x} = \frac{\textstyle \text{Sum of all sample values}}{n}$. Sum divided by sample size gives average.

Flashcard 33: What characterizes a biased point estimate?

Answer: An estimate systematically different from the parameter. Contains systematic error, not centered on true value.

Flashcard 34: Name a situation where an estimator can be unbiased but inefficient.

Answer: Estimator has high variance. Large variability reduces precision despite unbiasedness.

Flashcard 35: Which estimate is unbiased for population variance?

Answer: Sample variance $s^2 = \frac{\textstyle \text{SS}}{n-1}$. Dividing by $n-1$ corrects for degrees of freedom.

Flashcard 36: Why is $s^2 = \frac{\text{SS}}{n-1}$ unbiased?

Answer: Accounts for degrees of freedom, matches true variance. Corrects for lost degree of freedom from estimating mean.

Flashcard 37: Identify the unbiased estimator for population proportion.

Answer: Sample proportion $\frac{x}{n}$. Expected value equals true population proportion $p$.

Flashcard 38: Is the sample mode an unbiased estimator for population mean?

Answer: No, mode does not equate to mean. Mode represents most frequent value, not average.

Flashcard 39: Find the bias in the estimate: sample variance $s^2 = \frac{\text{SS}}{n}$.

Answer: Bias: underestimates population variance. Dividing by $n$ instead of $n-1$ creates downward bias.

Flashcard 40: What is a point estimate?

Answer: A single value estimate for a population parameter. Provides best guess of unknown population parameter.

Flashcard 41: What measure do statisticians use to quantify bias?

Answer: Expected value of the estimator minus true parameter. Measures systematic deviation from true value.

Flashcard 42: What is the role of sample size in unbiased estimation?

Answer: Larger samples improve estimate accuracy, not bias. Size affects precision, not bias direction.

Flashcard 43: Name a situation where an estimator can be unbiased but inefficient.

Answer: Estimator has high variance. Large variability reduces precision despite unbiasedness.

Flashcard 44: What is the bias of the sample mean for population mean?

Answer: Zero, sample mean is unbiased. Sample mean has no systematic error.

Flashcard 45: Can a biased estimator converge to the true parameter?

Answer: Yes, if it is consistent. Asymptotic convergence overcomes initial bias.

Flashcard 46: Determine if sample range is an unbiased estimator for range.

Answer: No, sample range underestimates population range. Sample cannot capture full population range.

Flashcard 47: Which condition ensures an unbiased estimate?

Answer: Expected value equals the true parameter. No systematic over or under estimation occurs.

Flashcard 48: What is the relationship between bias and mean squared error?

Answer: MSE = Variance + Bias$^2$. Decomposition shows bias contributes to total error.

Flashcard 49: Is the sample mean consistent for estimating population mean?

Answer: Yes, the sample mean is consistent. Converges to $\mu$ by Law of Large Numbers.

Flashcard 50: What estimator property improves with increasing sample size?

Answer: Consistency. Estimator precision improves with larger samples.

Flashcard 51: What does it mean if an estimator is 'consistent'?

Answer: Estimator converges to true parameter as sample size increases. Approaches true value with infinite sample size.

Flashcard 52: Define unbiased point estimate.

Answer: An estimate whose expected value equals the true parameter. No systematic error in long run.

Flashcard 53: Why is $s^2 = \frac{\text{SS}}{n-1}$ unbiased?

Answer: Accounts for degrees of freedom, matches true variance. Corrects for lost degree of freedom from estimating mean.

Flashcard 54: What is the bias in using sample median for population mean?

Answer: Median may systematically differ from mean. Creates bias except in symmetric populations.

Flashcard 55: What is the impact of sample size on bias?

Answer: Larger samples may reduce bias, not eliminate it. Bias is property of estimator, not sample size.

Flashcard 56: Is the sample mean consistent for estimating population mean?

Answer: Yes, the sample mean is consistent. Converges to $\mu$ by Law of Large Numbers.

Flashcard 57: What is the role of sample size in unbiased estimation?

Answer: Larger samples improve estimate accuracy, not bias. Size affects precision, not bias direction.

Flashcard 58: Are all consistent estimators also unbiased?

Answer: No, consistency does not imply lack of bias. Consistency and bias are independent properties.

Flashcard 59: Which estimate is unbiased for population variance?

Answer: Sample variance $s^2 = \frac{\textstyle \text{SS}}{n-1}$. Dividing by $n-1$ corrects for degrees of freedom.

Flashcard 60: What is the bias of using $n$ in sample variance calculation?

Answer: Underestimates true variance, biased. Creates systematic underestimation of true variance.

Flashcard 61: Why might a point estimate be biased?

Answer: Due to systematic error or sample selection issues. Flawed methodology or non-random sampling causes bias.

Flashcard 62: What is a common source of bias in surveys?

Answer: Non-response bias. Missing responses skew results systematically.

Flashcard 63: Find and correct the bias in estimating $\theta$ with $\theta + 1$.

Answer: Correct by using estimator $\theta$. Subtract 1 to remove systematic overestimation.

Flashcard 64: Is the median an unbiased estimator for mean?

Answer: No, median may not equal mean. Median only equals mean in symmetric distributions.

Flashcard 65: What characterizes a biased point estimate?

Answer: An estimate systematically different from the parameter. Contains systematic error, not centered on true value.

Flashcard 66: Determine if sample range is an unbiased estimator for range.

Answer: No, sample range underestimates population range. Sample cannot capture full population range.

Flashcard 67: Is an estimator unbiased if its bias is zero?

Answer: Yes, zero bias means unbiased. Zero bias is definition of unbiased estimator.

Flashcard 68: Why might a point estimate be biased?

Answer: Due to systematic error or sample selection issues. Flawed methodology or non-random sampling causes bias.

Flashcard 69: Identify an unbiased estimator for population parameter.

Answer: Estimator with expected value equal to the parameter. Expected value must match the true parameter.

Flashcard 70: State the formula for sample variance.

Answer: $s^2 = \frac{\text{SS}}{n-1}$. Uses $n-1$ to correct for degrees of freedom.

Flashcard 71: What does it mean if an estimator is 'efficient'?

Answer: It has the smallest variance among unbiased estimators. Minimum variance unbiased estimator is optimal.

Flashcard 72: Does increasing sample size always eliminate bias?

Answer: No, some biases persist regardless of sample size. Structural biases persist regardless of sample size.

Flashcard 73: State the formula for sample variance.

Answer: $s^2 = \frac{\text{SS}}{n-1}$. Uses $n-1$ to correct for degrees of freedom.

Flashcard 74: Identify the unbiased estimate for population mean.

Answer: Sample mean $\bar{x}$. Expected value equals population mean $\mu$.