A histogram for a sample of 200 home prices in a rural community shows that the distribution is nearly symmetric and has a single peak, except for 4 extremely large values that are at least $80,000 greater than the next lowest price. If it is decided to drop those 4 values from the data, which of the following would be impacted the most?

Question

BlackTom AI · Accepted Answer

The scenario describes a roughly symmetric, unimodal distribution with 4 extreme high values that are at least $80,000 above the next price.

Option 1: Median
- The median depends on the middle value(s) of the ordered data. With 200 observations, the median is the average of the 100th and 101st values. Removing four very large outliers will shift the ordered list, but since these outliers are all above the bulk of the data, the central position (the 100th and 101st values) is unlikely to move dramatically. Therefore, the median is relatively robust to this specific set of high outliers.

Option 2: IQR (Interquartile Range)
- The IQR is Q3 minus Q1, reflecting the spread of the middle 50% of the data. Dropping only the top outliers can alter Q3 if those top values were part of the upper quartile’s tail, but with 200 observations and only 4 extreme points, the quartile cutoffs are unlikely to change much. Hence, the IQR would be only minimally impacted compared to the mean.

Option 3: Mean
- The mean is sensitive to extreme values because it sums all values and divides by the count. Removing four values that are substantially larger than the rest removes a large amount of total value relative to the sample size, causing a noticeable downward shift in the mean. In contrast to the median and IQR, the mean reacts directly to the magnitude of outliers, so its value is most affected by removing these extreme highs.

Option 4: First quartile
- The first quartile (Q1) marks the 25th percentile. Since the data are largely driven by the more common values in the lower half, and because we removed only very high values (well above the bulk), Q1 is unlikely to shift substantially. The removal of top outliers typically leaves the lower portion intact, so Q1 remains largely unchanged.

In summary, among the given choices, the mean is the statistic that will be impacted the most by removing the four extreme high values, because it incorporates every value in the dataset and is therefore most sensitive to outliers. The median, IQR, and first quartile are more robust to this particular set of outliers and would experience only small changes (if any).

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