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Difficulty: MediumData Types, Distributions, and Descriptive Statistics

A multicenter rheumatology research network is analyzing demographic data for a newly defined phenotype of large-vessel vasculitis. The registry includes 2,5002,500 confirmed cases. The investigators perform a descriptive analysis of the age of initial symptom onset. The resulting histogram demonstrates a normal distribution. According to the registry's statistical report, the 2.5th2.5^\text{th} percentile for age of symptom onset is 2929 years, and the 84th84^\text{th} percentile is 5353 years.

Based on the empirical characteristics of normal distributions, what is the mean age of symptom onset (in years) for this patient cohort?

Answer: 45 years

Answer

The mean age of symptom onset is 45 years.
Based on the 68-95-99.7 rule for normal distributions, the 84th84^\text{th} percentile corresponds to the mean plus one standard deviation (μ+1σ\mu + 1\sigma), because 50%50\% of the data is below the mean and 34%34\% is between the mean and +1σ+1\sigma. The 2.5th2.5^\text{th} percentile corresponds to the mean minus two standard deviations (μ−2σ\mu - 2\sigma), because 95%95\% of the data falls within two standard deviations, leaving 5%5\% in the tails, or 2.5%2.5\% in the lower tail. This creates the system of equations: μ+1σ=53\mu + 1\sigma = 53 and μ−2σ=29\mu - 2\sigma = 29. The difference between the two values (53−29=2453 - 29 = 24) represents 33 standard deviations. Thus, one standard deviation is 88 years (24/3=824 / 3 = 8). The mean is therefore 53−8=4553 - 8 = 45 years.

Step-by-Step Solution

1
Map the given percentiles to standard deviations (σ\sigma) from the mean (μ\mu) using the empirical (68-95-99.7) rule.
The 84th84^\text{th} percentile is 1σ1\sigma above the mean (50%+34%=84%50\% + 34\% = 84\%). The 2.5th2.5^\text{th} percentile is 2σ2\sigma below the mean (50%−47.5%=2.5%50\% - 47.5\% = 2.5\%).
The normal distribution curve is symmetric, and fixed percentages of the data fall within specific standard deviation intervals.
2
Translate the relationships into a system of equations.
Equation 1: μ+1σ=53\mu + 1\sigma = 53
Equation 2: μ−2σ=29\mu - 2\sigma = 29
This mathematically links the unknown mean and standard deviation to the provided age values.
3
Solve the system of equations to find the standard deviation (σ\sigma).
Subtracting Equation 2 from Equation 1 yields: 3σ=243\sigma = 24. Therefore, σ=8\sigma = 8.
Isolating one variable allows for the calculation of the other.
4
Substitute the standard deviation back into either equation to calculate the mean (μ\mu).
Using Equation 1: μ+8=53\mu + 8 = 53, which means μ=45\mu = 45.
This determines the final central tendency measure requested by the question.

Key Concept

Application of the empirical (68-95-99.7) rule to determine the mean and standard deviation of a normal distribution using specific percentiles.
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Topics

Data Types, Distributions, and Descriptive Statistics

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