Question

Listed in increasing order are the serum creatine phosphokinase (CK) levels (U/I) of 36 healthy men (these are the data of Example 2.2 .6 ): \(\begin{array}{lllrll}25 & 62 & 82 & 95 & 110 & 139 \\ 42 & 64 & 83 & 95 & 113 & 145 \\ 48 & 67 & 84 & 100 & 118 & 151 \\ 57 & 68 & 92 & 101 & 119 & 163 \\ 58 & 70 & 93 & 104 & 121 & 201 \\ 60 & 78 & 94 & 110 & 123 & 203\end{array}\) The sample mean CK level is \(98.3 \mathrm{U} / \mathrm{l}\) and the \(\mathrm{SD}\) is \(40.4 \mathrm{U} / \mathrm{I} .\) What percentage of the observations are within (a) \(1 \mathrm{SD}\) of the mean? (b) 2 SDs of the mean? (c) 3 SDs of the mean?

Short answer

Approximately 83.3% of observations are within 1 SD of the mean, 97.2% are within 2 SDs of the mean, and 100% are within 3 SDs of the mean. (Note: The percentage for 3 SDs assumes the lowest CK level is greater than 0.)

Step-by-step solution

Determine the range for each standard deviation interval

To determine the percentage of observations within each standard deviation (SD) of the sample mean, first calculate the range for each interval. For 1 SD, this is the mean minus 1 SD to the mean plus 1 SD. For 2 SDs, it is the mean minus 2 SDs to the mean plus 2 SDs, and similarly for 3 SDs.

Calculate the ranges for 1 SD, 2 SDs, and 3 SDs

For 1 SD: \(57.9 \text{U/I} \text{to} 138.7 \text{U/I}\) (since \(98.3 - 40.4 = 57.9\) and \(98.3 + 40.4 = 138.7\)). For 2 SDs: \(17.5 \text{U/I} \text{to} 179.1 \text{U/I}\) (since \(98.3 - 2\times40.4 = 17.5\) and \(98.3 + 2\times40.4 = 179.1\)). For 3 SDs: \(\text{-}22.9 \text{U/I} \text{to} 219.5 \text{U/I}\) (since \(98.3 - 3\times40.4 = -22.9\) which is not possible, so we take 0 as the lower boundary, and \(98.3 + 3\times40.4 = 219.5\)).

Count the number of observations within each range

Observe the data to see how many values fall within the calculated ranges for each SD interval and then count them.

Calculate the percentages for each SD interval

Divide the counts from Step 3 by the total number of observations and multiply by 100 to convert to percentage.

Key concepts

Serum Creatine Phosphokinase Levels

Serum creatine phosphokinase (CK) levels provide crucial insight into muscle health. This enzyme, found in the heart, brain, and skeletal muscles, becomes elevated in the blood following muscle damage. By analyzing CK levels in individuals, medical professionals can assess muscular disorders or damage.

Measurements of serum CK are presented in units per liter (U/L). In a group of individuals, the serum CK levels can vary widely depending on factors such as exercise intensity, muscle mass, and overall health. A dataset representing a healthy population’s CK levels can help establish a reference range, which is essential for identifying abnormal CK levels that might indicate conditions like myocardial infarction or muscular dystrophy.

Sample Mean Calculation

The sample mean is a fundamental statistical measure that represents the average value of a sample. To calculate the sample mean of CK levels, you sum up all the observed CK values and divide by the number of observations.

In the given exercise, the sample mean CK level is calculated as 98.3 U/L. This is the central point around which other statistical calculations, such as deviations and variances, are based. Understanding the sample mean allows us to compare individual CK levels and determine whether they're above or below the average for the group studied.

Normal Distribution

The concept of normal distribution is foundational in statistics and represents how the values of a dataset are distributed about the mean. In a perfectly normal distribution, most of the data clusters around the mean, with fewer values observed as you move away in either direction.

Standard deviation (SD) intervals play a critical role in defining the spread in a normally distributed dataset. One SD from the mean includes about 68% of the data, two SDs include about 95%, and three SDs include about 99.7%. These percentages are commonly referred to as the Empirical Rule or 68-95-99.7 rule. It's important to note that real-world data might not follow a perfect normal distribution, and actual percentages can differ from these theoretical values.

Statistical Data Analysis

Statistical data analysis involves examining and interpreting data to uncover underlying patterns and trends. In the context of serum CK levels, statistical analysis provides an opportunity to understand the range and distribution of these values within a sample population.

After calculating the sample mean and standard deviation, the analysis involves determining the percentage of observations within certain ranges of the mean. This process, demonstrated in the step-by-step solution, allows researchers and medical professionals to quantify the variability of CK levels in a population and identify outliers that may warrant further examination. Data analysis techniques such as these are cornerstone in ensuring accurate assessments and informed decision-making in a variety of scientific and medical applications.