Sarah collected sample grades data from her French class and Daniel collected sample grades data from his German class. Sarah’s value for range was the same as Daniel’s but her data was positively skewed and Daniel’s data was negatively skewed. Both distributions were unimodal.

Which of the following must be TRUE about Sarah’s data compared to Daniel’s data?

Question

Sarah collected sample grades data from her French class and Daniel collected sample grades data from his German class. Sarah’s value for range was the same as Daniel’s but her data was positively skewed and Daniel’s data was negatively skewed. Both distributions were unimodal.

Which of the following must be TRUE about Sarah’s data compared to Daniel’s data?

BlackTom AI · Accepted Answer

We start by restating the setup to ground the analysis: Sarah has a sample of grades in French with the same range as Daniel's German grades, but Sarah’s data are positively skewed while Daniel’s are negatively skewed, and both distributions are unimodal.

Option 1: None of the options must be true.
- This claim would be true only if none of the other statements could be deduced from the given information. However, the fact that one distribution is positively skewed and the other negatively skewed while both are unimodal implies a clear structural difference from symmetry about the center. Since a normal distribution is symmetric and typically exhibits little skew, the given skewness directly challenges the possibility that either distribution is normal. Therefore, this option cannot be correct because there is at least one statement that must be true.

Option 2: Sarah’s standard deviation was higher than Daniel’s standard deviation.
- Range being equal does not fix or compare standard deviations. Standard deviation measures spread around the mean, while range measures the extreme spread. It is perfectly possible for Sarah to have a larger, smaller, or equal SD despite equal ranges, depending on how the data are distributed within that range. So this is not a must-have conclusion.

Option 3: Sarah’s highest score was the same as Daniel’s highest score.
- The highest (maximum) value of a dataset is not determined by skewness direction or unimodality alone, and equal ranges do not ensure equal maxima. Two datasets can share the same range yet have different maximum values if their minimums differ or if the range equality arises from different pairings of min and max. Therefore this statement is not required.

Option 4: Neither distribution was normally distributed.
- This is the statement that must be true given the setup. A normal distribution is symmetric and has near-zero skewness. Here, Sarah is positively skewed and unimodal, while Daniel is negatively skewed and unimodal. The presence of positive skewness in one distribution and negative skewness in the other makes both distributions depart from symmetry, hence neither can be normal. This conclusion follows directly from the definitions of skewness and normality.

Option 5: Sarah’s mean is higher than Daniel’s mean.
- Skewness affects the relationship between mean and median, not necessarily the ordering of the means between two independent samples with related range constraints. There is no information tying the mean of Sarah’s French grades to Daniel’s German grades in a way that guarantees one mean exceeds the other. Therefore, this statement is not must-true.

In summary, the key logical point is that normal distributions are symmetric, whereas the given data sets exhibit opposite directions of skewness (one positive, one negative) while both are unimodal. This directly rules out normality for both data sets, making Option 4 the must-true statement, while the other options fail to be guaranteed by the information provided.

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