A gastroenterology research team is evaluating colonic transit times in a cohort of health… — USMLE Step 3

A gastroenterology research team is evaluating colonic transit times in a cohort of healthy volunteers using a radiopaque marker technique. The distribution of transit times in this population is determined to be normally distributed. The researchers report a mean transit time of $36 \text{ hours}$ and a variance of $16 \text{ hours}^2$ . Based on these descriptive statistics, a volunteer whose colonic transit time sits exactly at the $84^{\text{th}}$ percentile has a transit time of how many hours?

Answer: 40 hours

Answer

40 hours

The data follows a normal distribution. To identify a specific percentile based on the empirical rule, the variance (

16 \text{ hours}^2

) must first be converted into a standard deviation by calculating its square root (

\sqrt{16} = 4 \text{ hours}

). The empirical (68-95-99.7) rule dictates that approximately 68% of observations in a normal distribution lie within 1 standard deviation of the mean. Because the distribution is symmetrical, 34% of the population lies exactly between the mean (

50^{\text{th}}

percentile) and

+1

standard deviation. Consequently, the

84^{\text{th}}

percentile (

50\% + 34\%

) is located exactly 1 standard deviation above the mean. Adding 1 standard deviation (

4 \text{ hours}

) to the mean (

36 \text{ hours}

) results in a colonic transit time of

40 \text{ hours}

Step-by-Step Solution

Calculate the standard deviation from the given variance.

Standard deviation =

\sqrt{16} = 4 \text{ hours}

The empirical rule requires standard deviations, not variance, to determine data distribution intervals.

Determine the Z-score that corresponds to the

84^{\text{th}}

percentile.

Z-score =

+1

In a perfectly normal distribution, 50% of values fall below the mean. The empirical rule states 68% of values fall within

\pm1

SD. Half of this (34%) falls between the mean and

+1

SD.

50\% + 34\% = 84\%

, so the

84^{\text{th}}

percentile sits at exactly

+1

SD.

Calculate the exact transit time at

+1

standard deviation.

36 \text{ hours} + (1 \times 4 \text{ hours}) = 40 \text{ hours}

This translates the calculated Z-score back into the original clinical units of measurement.

Key Concept

Application of the empirical (68-95-99.7) rule and conversion of variance to standard deviation in a normal distribution.

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