Biased and unbiased estimators

Matilda was curious if sample interquartile range (IQR) was an unbiased estimator of population IQR. She placed ping pong balls numbered from 0 to 32 in a drum and mixed them well. Note that the IQR of the population is 16. She then took a random sample of 5 balls and calculated the IQR of the sample. She replaced the balls and repeated this process for a total of 50 trials. Her results are summarized in the dotplot below, where each dot represents the sample IQR from a sample of 5 balls. Based on these results, does sample IQR appear to be a biased or unbiased estimator of population IQR? Hint: The population IQR is 16.

Sample IQR appears to be a biased estimator since it consistently underestimated the population IQR. The sample IQRs appear to be centered below 16.

Isabelle was curious if sample maximum was an unbiased estimator of population maximum. She started with a large normally distributed population of test scores whose maximum was 99 points. She then took a random sample of 6 tests and calculated the maximum of the sample. She replaced those tests and repeated this process for a total of 40 trials. Her results are summarized in the dotplot below, where each dot represents the maximum score from a sample of 6 tests. Based on these results, does sample maximum appear to be a biased or unbiased estimator of population maximum? Hint: The population maximum is 99.

Sample maximum appears to be biased estimator since it consistently underestimates the population maximum. The sample maximum appear to be centered below 99.

Whitney was curious if sample mean was an unbiased estimator of population mean. She placed ping pong balls numbered from 0 to 32 in a drum and mixed them well. Note that the mean of the population is 16. She then took a random sample of 5 balls and calculated the mean of the sample. She replaced the balls and repeated this process for a total of 50 trials. Her results are summarized in the dotplot below, where each dot represents the sample mean from a sample of 5 balls. Based on these results, does sample mean appear to be a biased or unbiased estimator of population mean? Hint: The population mean is 16.

Sample mean appears to be an unbiased estimator of the population mean. The sample means appear to be centered near 16.

Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping pong balls numbered from 0 to 32 in a drum and mixed them well. Note that the median of the population is 16. He then took a random sample of 5 balls and calculated the median of the sample. He replaced the balls and repeated this process for a total of 50 trials. His results are summarized in the dotplot below, where each dot represents the sample median from a sample of 5 balls. Based on these results, does sample median appear to be a biased or unbiased estimator of population median? Hint: The population median is 16.

Sample median appears to be an unbiased estimator of the population median. The sample medians appear to be centered near 16.

Silas was curious if sample minimum was an unbiased estimator of population minimum. He started with a large normally distributed population of test scores whose minimum was 51 points. He then took a random sample of 6 tests and calculated the minimum of the sample. He replaced those tests and repeated this process for a total of 40 trials. His results are summarized in the dotplot below, where each dot represents the minimum score from a sample of 6 tests. Sample minimum Based on these results, does sample minimum appear to be a biased or unbiased estimator of population minimum?Hint: The population minimum is 51.

Sample minimum appears to be a biased estimator since it consistently overestimated the population minimum. The sample minimums appear to be centered above 51.

Ekene was curious if sample range was an unbiased estimator of population range. She placed ping pong balls numbered from 0 to 32 in a drum and mixed them well. Note that the range of the population is 32. She then took a random sample of 5 balls and calculated the range of the sample. She replaced the balls and repeated this process for a total of 50 trials. Her results are summarized in the dotplot below, where each dot represents the sample range from a sample of 5 balls. Based on these results, does sample range appear to be a biased or unbiased estimator of population range? Hint: The population range is 32.

Sample range appears to be a biased estimator since it consistently underestimated the population range. The sample ranges appear to be centered below 32.

Serge was curious if a sample third quartile (or Q3) was an unbiased estimator of a population third quartile. He started with a large normally distributed population of test scores whose third quartile was Q3=80 points. He then took a random sample of 666 tests and calculated the third quartile of the sample. He replaced those tests and repeated this process for a total of 40 trials. His results are summarized in the dotplot below, where each dot represents the third quartile from a sample of 6 tests. Based on these results, does sample third quartile appear to be a biased or unbiased estimator of population third quartile? Hint: The population third quartile is 80.

Sample third quartile appears to be an unbiased estimator of the population third quartile. The sample third quartiles appear to be centered near 80.

The dotplots below show an approximation to the sampling distribution for three different estimators of the same population parameter. If the actual value of the population parameter is 2, which dotplot displays the estimator with low bias and high variability?

Statistic B: Very spread out and the highest number of dots is on 2