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Difficulty: MediumData Types, Distributions, and Descriptive Statistics

A research group is analyzing the baseline characteristics of patients enrolled in a clinical trial for a novel acute pancreatitis therapy. They measure admission serum levels of a prognostic biomarker, Protein-Z (measured in mg/dL\text{mg/dL}), in 350350 subjects. The frequency distribution of the biomarker concentrations reveals that the majority of values are concentrated between 1.51.5 and 4.0 mg/dL4.0\text{ mg/dL}, while a small number of patients have extremely high values, creating a long tail extending up to 22.0 mg/dL22.0\text{ mg/dL}.

Which of the following is the most appropriate pair of descriptive statistics to summarize these biomarker levels, and what is the expected relationship between the mean and the median in this dataset?

  1. A
    Mean and standard deviation; mean is less than the median
  2. B
    Median and interquartile range; mean is less than the median
  3. Median and interquartile range; mean is greater than the medianAnswer
  4. D
    Mean and standard deviation; mean is greater than the median
  5. E
    Mode and range; mean and median are approximately equal

Answer

Median and interquartile range; mean is greater than the median
The clinical data described exhibits a positive (right) skew, characterized by a cluster of lower values and a long tail of extreme high values. In highly skewed distributions, the mean and standard deviation are disproportionately influenced by the extreme outliers, making them poor representations of the typical patient. Instead, non-parametric descriptive statistics—specifically the median and interquartile range (IQR)—are the most appropriate measures for central tendency and dispersion. Additionally, because the few extremely high biomarker values heavily influence the arithmetic average, the mean will be mathematically pulled to the right, making it greater than the median.

Step-by-Step Solution

1
Identify the shape of the data distribution based on the clinical vignette.
The text describes a concentration of lower values with a 'long tail extending up' to extremely high values. This indicates a positive (right) skew.
Understanding the shape of the distribution is necessary to select the correct measures of central tendency and dispersion.
2
Determine the most robust descriptive statistics for summarizing this type of continuous distribution.
Because the data is heavily skewed by outliers, non-parametric measures such as the median and interquartile range (IQR) should be used.
The mean and standard deviation are heavily influenced by extreme values and would misrepresent the typical patient's biomarker level.
3
Determine the mathematical relationship between the mean and the median in this specific distribution.
In a right-skewed distribution, the extreme high values in the long tail exert a disproportionate mathematical pull on the arithmetic average, causing the mean to be greater than the median.
This relationship (Mean > Median) is a fundamental characteristic of positively skewed data.

Key Concept

Selection of Descriptive Statistics for Skewed Distributions
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Topics

Data Types, Distributions, and Descriptive Statistics

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