Applied Math - Units - Health

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Introduction

We can compare numbers from different groups by standardizing the variable. One way of standardizing is by calculating a z-score:

\[z = \frac{x - \mu}{\sigma}\]

where \(x\) is the number you are converting to a z-score, \(\mu\) is the mean of the group, and \(\sigma\) is the standard deviation. We can convert two different health statistic numbers to z-scores in order to compare them to see who is more extraordinary.

Example:

1.

Assume that people aged 60 to 64 have a mean systolic blood pressure of 134 with a standard deviation of 5. Also assume that people aged 80-84 have a mean blood pressure of 149 with a standard deviation of 4.

A doctor has two patients with high blood pressure. The first patient is 63 years old and has a systolic blood pressure reading of 113. The second patient is 83 years old and has a systolic blood pressure reading of 162. Which patient has the more extraordinary blood pressure?

The \(83\) year old patient has a higher blood pressure, but it might not be as extraordinary as people of this age have a different mean and standard deviation. We can compare the two blood pressures by converting both to a z-score.

The 63 year old’s z-score is

\[z = \frac{113 - 134}{5} = \frac{-21}{5} = -4.2\]

We subtract the blood pressure of the 63 year old by the mean for this age group, 134, then divide by the standard deviation of this group.

The 83 year old’s z-score is

\[z = \frac{162 - 149}{4} = \frac{13}{4} = 3.25\]

Relative to other people their age, the 63 year old’s blood pressure is more extraordinary as it has a z-score further from zero, as zero is "average".

Practice Problems

1. Assume the mean heart rate for 80 year olds is 80 beats per minute with a standard deviation of 20. Assume the mean resting heart rate of 65 year olds is 72 with a standard deviation of 10.

Two patients in the E.R. need the attention of one doctor. One patient is 80 years old, looks unwell, and has a heart rate of 105 beats per minute. Another patient is 65 years old, looks unwell, and has a heart rate of 60 beats per minute. Which patient has a more extraordinary heart rate?



2. Blood tests measure Leukocytes. Leukocytes are white blood cells. Too little Leukocytes or too many Leukocytes indicates a potential issue with a person’s health. Leukocytes with z-scores less than -2 or greater than 2 are asked to follow up with their healthcare provider.

The mean value for Leukocytes is 7.1 thou/uL, with a standard deviation of 1.5.

Patient A has a Leukocyte level of 4.7 thou/uL. Patient B has a Leukocyte level of 10.3. Are either persons going to be asked to follow up with their healthcare provider for their Leukocyte levels? Which patient has the more extraordinary Leukocyte level?

Theory Questions