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.
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".
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?