Question

Ten patients with high blood pressure participated in a study to evaluate the effectiveness of the drug Timolol in reducing their blood pressure. The accompanying table shows systolic blood pressure measurements taken before and after 2 weeks of treatment with Timolol. \({ }^{40}\) Calculate the mean and SD of the change in blood pressure (note that some values are negative). $$ \begin{array}{|cccc|} \hline & & {\text { Blood pressure (mm HG) }} \\ \hline \text { Patient } & \text { Before } & \text { After } & \text { Change } \\ \hline 1 & 172 & 159 & -13 \\ 2 & 186 & 157 & -29 \\ 3 & 170 & 163 & -7 \\ 4 & 205 & 207 & 2 \\ 5 & 174 & 164 & -10 \\ 6 & 184 & 141 & -43 \\ 7 & 178 & 182 & 4 \\ 8 & 156 & 171 & 15 \\ 9 & 190 & 177 & -13 \\ 10 & 168 & 138 & -30 \\ \hline \end{array} $$

Short answer

The mean change in blood pressure is \textbf{-12.4 mm Hg} and the standard deviation (SD) is \textbf{18.33 mm Hg}

Step-by-step solution

Calculate the Mean of the Change in Blood Pressure

First, sum all the changes in blood pressure for the ten patients. Then, divide by the number of patients (10) to find the mean change in blood pressure.

Calculate the Summation of Squared Deviations

For each patient, calculate the squared difference between the change in blood pressure and the mean change found in Step 1. Sum all these squared differences together.

Calculate the Standard Deviation (SD)

Take the sum calculated in Step 2 and divide it by the number of patients minus one (N-1, which is 9 for 10 patients). Take the square root of the resulting value to get the standard deviation of the change in blood pressure.

Key concepts

Mean Change Calculation

Understanding mean change is vital when analyzing data sets to comprehend average variations over time or between conditions. In the context of the blood pressure study, calculating the mean change involves assessing the average reduction or increase in patients' blood pressure after using Timolol. To achieve this, we first sum the individual changes in blood pressure for all patients. For instance, if a patient's blood pressure dropped from 172 to 159 mm Hg, the change would be -13 mm Hg.

Once the total change is calculated by adding up each patient's change, we divide this sum by the number of patients, which provides us with the average change across the group. The formula for the mean change \( \bar{X} \) is given by:
\[ \bar{X} = \frac{\sum_{i=1}^{n} (X_{\text{after}} - X_{\text{before}})}{n} \]
where \( X_{\text{after}} \) and \( X_{\text{before}} \) are the measurements after and before treatment, respectively, and \( n \) is the total number of patients. This calculation is the cornerstone for further statistical analysis, such as assessing variability and testing hypotheses about treatment effects.

Standard Deviation Calculation

Standard deviation (SD) is a measure that tells us how much the individual values in a data set deviate from the mean value. In simpler terms, it gives us an insight into the spread or dispersion of the data. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation suggests a wide range of values.

In our blood pressure study example, after calculating the mean change, we need to determine the variation of changes from this mean. To compute the standard deviation, we follow a few steps:

• Subtract the mean change from each individual patient's blood pressure change to determine the deviation of each change.
• Square each deviation to make them positive values.
• Add all squared deviations to get the total sum of squared deviations.
• Divide this sum by the number of observations minus one (this is known as the 'degrees of freedom' in statistics).
• Finally, take the square root of the result to return to the original units (mm Hg in this case), yielding the standard deviation.

The formula for standard deviation \( SD \) is:
\[ SD = \sqrt{\frac{\sum_{i=1}^{n} (X_{i} - \bar{X})^2}{n-1}} \]
Where \( X_{i} \) represents each individual change, and \( \bar{X} \) represents the mean change. Understanding SD is crucial for interpreting how consistent the effect of Timolol is across the patients in the study.

Paired Sample t-test

A paired sample t-test is a statistical procedure used to compare two related means — typically, the same group measured twice, as is the case with pre-treatment and post-treatment blood pressure levels. It helps determine if the mean difference between the paired observations is statistically significant, that is, unlikely to be due to chance alone.

The choice of a paired sample t-test for the Timolol study is appropriate because each patient's before and after treatment measurements are related and, thus, 'paired'. This test takes into account that individual variations in blood pressure are natural and that the 'paired' aspect helps control for these differences when evaluating the treatment effect.

• The null hypothesis (H0) in such tests usually states that there is no mean difference between the two sets of observations.
• The alternative hypothesis (H1) would argue that a significant difference does exist.

The t-value is calculated using the mean change and its standard deviation, along with the number of paired observations. A t-distribution table or software is then used to find the p-value, which if below a chosen significance level (commonly 0.05) indicates that the null hypothesis can be rejected, lending support to the effectiveness of the treatment.

This statistical method ensures that the conclusions drawn from the study are not just due to random fluctuations in blood pressure but are attributable to the intervention being assessed — in our case, the drug Timolol.