Problem 176 - Applied Statistics and Probability for Engineers

Question

An article in Health and Population: Perspectives and Issues \((2000,\) Vol. \(23,\) pp. \(28-36)\) used the lognormal distribution to model blood pressure in humans. The mean systolic blood pressure (SBP) in males age 17 was \(120.87 \mathrm{~mm} \mathrm{Hg}\). If the co-efficient of variation \((100 \% \times\) Standard deviation/mean \()\) is \(9 \%,\) what are the parameter values of the lognormal distribution?

Step-by-Step Solution

Answer

Parameters are found using the mean \(\mu = \ln(120.87)\) and variance \(\sigma = \sqrt{\ln(1 + (10.8783/120.87)^2)}\).

Step 1: Understanding the Mean and the Coefficient of Variation

Given that the mean systolic blood pressure (SBP) for males age 17 is 120.87 mm Hg, and the coefficient of variation is 9%, we begin by recalling that the coefficient of variation is defined as \(100\% \times \frac{\sigma}{\mu}\), where \(\sigma\) is the standard deviation and \(\mu\) is the mean.

Step 2: Calculating the Standard Deviation

The coefficient of variation (CV) is 9%, which means:\[ CV = \frac{\sigma}{\mu} \times 100 = 9\% \]Substitute \(\mu = 120.87\) to find \(\sigma\):\[ \frac{\sigma}{120.87} = 0.09 \]\[ \sigma = 0.09 \times 120.87 = 10.8783 \]

Step 3: Identify the Conversion to Lognormal Parameters

In a lognormal distribution, the parameters are typically denoted as \(\mu\) (mean) and \(\sigma^2\) (variance) of the related normal distribution of the logarithm of the variable. The arithmetic mean of the untransformed (lognormal) variable is given, so we need to convert this to the parameters of its logarithm.

Step 4: Implementing Lognormal Parameter Formulas

For a lognormal distribution:- \(\mu_X = e^{\mu + \frac{\sigma^2}{2}}\)- \(\sigma_X^2 = (e^{\sigma^2} - 1)e^{2\mu + \sigma^2}\)Given \(\mu_X = 120.87\) and \(\sigma_X = 10.8783\), you substitute and solve for \(\mu\) and \(\sigma^2\) in these equations.

Step 5: Solving for Lognormal Parameters

We will use the above relationships step-by-step:1. \(\mu + \frac{\sigma^2}{2} = \ln(120.87)\).2. \(\sqrt{\ln(1 + \left(\frac{10.8783}{120.87}\right)^2)} = \sigma\).These would yield our parameters for the logarithm of the blood pressure.

Key Concepts

Systolic Blood Pressure

Systolic blood pressure (SBP) is a critical measure used in assessing cardiovascular health. It refers to the pressure in your arteries when your heart beats and fills them with blood. Understanding SBP is important as it provides insight into the force of blood against artery walls during the heartbeat, which can indicate various health conditions. Usually, SBP is measured in millimeters of mercury (mm Hg). In the context of the health study, the mean systolic blood pressure for 17-year-old males was identified as 120.87 mm Hg.
This measure is vital for modeling purposes, such as using distributions to estimate probabilities and variations around the mean. For individuals looking to maintain a healthy SBP, understanding its significance and how it can be modeled allows for better health monitoring and prediction of hypertension-related risks.

Mean and Standard Deviation

The mean and standard deviation are fundamental statistical concepts widely used to describe and understand data distributions. The mean is essentially an average value, representing the central tendency of a data set. For the systolic blood pressure, the mean was calculated as 120.87 mm Hg. This measure allows researchers to ascertain what is typical within a population.
Standard deviation, on the other hand, quantifies how much each number in a set of data varies from the mean. In the provided exercise, the standard deviation was calculated using the formula from the coefficient of variation. Knowing these two statistical metrics allows us to see both the central point of the data and how spread out, or how "variable," the data are from that mean point.
Utilizing these concepts, particularly in medical statistics, aids in developing an understanding of the typical ranges for medical parameters and detecting outliers that may indicate an anomaly or require further investigation.

Coefficient of Variation

The coefficient of variation (CV) is a normalized measure of dispersion of a probability distribution. It is particularly useful when we need to compare the degree of variation between datasets with different units or widely different means. In this exercise, it was used to connect the mean systolic blood pressure and its standard deviation.The formula for CV is given by:

• CV = \[ \frac{\sigma}{\mu} \times 100 \% \]

where \(\sigma\) is the standard deviation and \(\mu\) is the mean.In the discussed problem, using the mean of 120.87 mm Hg and a coefficient of variation of 9%, the standard deviation was calculated to be approximately 10.8783 mm Hg.
The CV here provides an understanding of how much variability exists in the SBP relative to the mean, offering a normalized perspective that pure standard deviation lacks.

Parameter Estimation

Parameter estimation in the context of a lognormal distribution involves converting known metrics of a data set into the parameters that define the distribution. The lognormal distribution is used here to model the systolic blood pressure data because this distribution is positively skewed, making it suitable for modeling such data. To estimate the parameters of a lognormal distribution, the arithmetic mean and standard deviation of the raw data are transformed into the parameters of its associated normal distribution (mean \(\mu\) and variance \(\sigma^2\) of the natural logarithm of data).The relationships used for parameter estimation in the lognormal distribution are:

• \( \mu + \frac{\sigma^2}{2} = \ln(\mu_X) \)
• With \( \mu_X = 120.87 \) and \( \sigma_X = 10.8783 \), these relationships guide the estimation process.
• \[ \sigma = \sqrt{\ln(1 + \left(\frac{10.8783}{120.87}\right)^2)} \]

Through these calculations, researchers convert typical statistical measures into the parameters necessary for accurately modeling data within the lognormal distribution.