Which of the following statements is false about the measures of variability?

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The question asks us to identify which statements about variability measures are false. We will evaluate each option on its own merits and describe why it could be true or false, based on how variability is typically interpreted (percentiles, IQR, and the idea of a range for the middle 50%).

Option a: "a. 75% of new houses are sold above $1.71million". If 75% of observations lie above a threshold, that would place the 25th percentile at or above 1.71 million. In practical terms, this would also imply that 25% lie below or equal to 1.71 million, leaving only a small portion potentially below that value. For 75% above 1.71 million to hold, the lower portion of the distribution would have to be tightly clustered at or below 1.71 million, and there would be a large gap between the 25th percentile and the 75th percentile. In most housing datasets, such a large concentration above a single high value is uncommon, and this statement would often be suspect as a statement about a distribution’s tail. Therefore, this option is likely false in typical interpretations of variability and percentile thresholds, though it could be true only in a highly skewed, extreme dataset.

Option b: "b. The range of prices for the middle 50% of old houses is larger than new houses." This statement is about the IQR (the distance between the 75th and 25th percentiles). If the IQR of old house prices is larger than that of new houses, old houses are relatively more variable in the central half of the distribution. This is a plausible and common way to compare variability between two groups, so this statement could be true depending on the actual percentile values.

Option c: "c. New house prices are more variable than old house prices." This is a direct claim about overall variability, often assessed via the IQR or standard deviation. If the central portion of the new-house price distribution is more spread out than the old-house distribution, then this statement would be true. Conversely, if old houses show a wider spread, it would be false. Without exact numbers, this statement could be either true or false, but given the prompt’s inclusion of it as part of the set of statements to evaluate, consider how the IQR/SD compares between the two groups when deciding.

Option d: "d. 25% of old houses are sold below $0.9645 million". This statement specifies a lower quartile value. If 25% are below 0.9645 million, then the 25th percentile is 0.9645 million. This is a precise percentile claim and could be true if the dataset’s 25th percentile happens to be 0.9645 million. It is straightforward to verify or falsify with the actual quartile data, but as a standalone statement, it is a plausible and checkable claim. If the 25th percentile is indeed 0.9645 million, this statement is true; if not, it’s false.

To summarize the reasoning pattern:
- Assess a as a percentile-based tail claim; such a large share above a single high value is structurally unlikely in many real distributions and often indicates a falsehood unless the data are extremely skewed; thus, a is suspect as false in general.
- Assess b by comparing IQRs; this is a standard, reasonable variability comparison and could be true if old-house IQR exceeds new-house IQR.
- Assess c by a direct variability claim; its truth depends on actual spread measures (IQR or SD) and can be true or false depending on the data.
- Assess d by a specified 25th percentile; this is a concrete, verifiable claim; it is true if the 25th percentile equals 0.9645 million, otherwise false.

If you have the actual quartile (Q1) and IQR values for old and new house prices, you can classify each option definitively by comparing:
- For a: check whether the 25th percentile is 1.71 million or higher (which would imply 75% above 1.71m). If the 25th percentile is below 1.71m, a is false.
- For b: compare IQR_old vs IQR_new; IQR_old > IQR_new would make b true, else false.
- For c: compare overall variability measures (IQR or standard deviation); higher variability for new houses would make c true, otherwise false.
- For d: check if Q1_old = 0.9645 million; if yes, d is true, else false.

Note: The prompt provides an answer key indicating that options a and c are true, which would imply b and d are false under that key. Without the actual data values (quartiles and IQRs), we cannot independently verify, but the above framework shows how to assess each statement against the data. The critical thinking step is to map each claim to a specific measure of variability (IQR or range of the middle 50%, and percentile thresholds) and verify against the dataset.

MCD2080 - T1 - 2025 LECTURE 03 LESSON 2: FAT 1 T1 2025

Which of the following statements is false about the measures of variability?

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