Which of the following statements is TRUE about the measures of variability? (Multiple selection is allowed)

Question

BlackTom AI · Accepted Answer

To evaluate the statements about variability, we consider what variability measures (like range, interquartile range, and standard deviation) tell us about how spread out the prices are for new versus old houses.

Option a: '75% of new houses are sold above $1.71 million' would imply a very high cutoff near the upper tail of the distribution. This would require the 25th percentile of new-house prices to be at or above $1.71 million and the 75th percentile to be even higher, which is uncommon unless the entire new-house price distribution is shifted very high. Without data showing such a tight, high-price distribution, this statement is unlikely to be universally true and contradicts typical variability patterns seen when comparing housing markets across vintages.

Option b: 'New house prices are more variable than old house prices.' This can be true if the spread (e.g., the interquartile range or standard deviation) of new-house prices is clearly larger than that of old houses. If the dataset shows a wider spread for new prices, this statement is supported. It aligns with a scenario where newer homes have a broader price range due to newer features, locations, or builder offerings, which increases variability.

Option c: 'The range of prices for the middle 50% of old houses is more spread out than new houses.' The middle 50% corresponds to the interquartile range (IQR). If the IQR for old houses is larger than the IQR for new houses, this statement would be true. However, if the dataset indicates that the old-house IQR is not larger, this claim would be false. Without the actual IQR values, this cannot be assumed.

Option d: '25% of old houses are sold below $0.9645 million.' This statement asserts a specific 25th percentile (Q1) value for old-house prices. If the data show that Q1 for old houses is indeed $0.9645 million, then this statement is true. It relies on the percentile being exactly that value in the distribution.

In summary, option b is true when the spread (variability) of new-house prices exceeds that of old-house prices, which matches a common pattern when newer properties exhibit a wider price range. Option d is true if the dataset’s first quartile (25th percentile) for old-house prices is $0.9645 million. Options a and c require specific percentile or range comparisons that would need the actual distribution data to confirm; without those data, they cannot be established as true.

Which of the following statements is TRUE about the measures of variability? (Multiple selection is allowed)多项选择题

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