Question

you catch 10 cockroaches in your bedroom and measure their lengths in centimeters. which of these sets of numerical descriptions are all measured in centimeters? (a) median length, variance of lengths, largest length (b) median length, first and third quartiles of lengths (c) mean length, standard deviation of lengths, median length (d) mean length, median length, variance of lengths. (e) both (b) and (c)

Explanation:

Step1: Recall units of statistical measures

• Median: Represents the middle value of a dataset, so its unit is the same as the original data (centimeters here).
• Quartiles (first and third): Also represent values from the dataset, so their unit is centimeters.
• Mean: Average of the data, unit is same as original data (centimeters).
• Variance: Calculated as \(\frac{\sum (x_i - \bar{x})^2}{n}\), so its unit is square of the original unit (square centimeters here).
• Standard deviation: Square root of variance, so its unit is same as original data (centimeters).

Step2: Analyze each option

• Option A: Variance has unit \(cm^2\), so not all in cm. Eliminate A.
• Option B: Median, first and third quartiles are all measures of position in the dataset, so their unit is cm. This option is valid.
• Option C: Mean, standard deviation, median: Mean and median are in cm, standard deviation is also in cm (since it's square root of variance, which makes its unit cm). This option is valid.
• Option D: Variance has unit \(cm^2\), so not all in cm. Eliminate D.
• Option E: Since both B and C are valid, this is the correct choice.

Answer:

E. both (B) and (C)