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Normal distribution is a fundamental concept in statistics, often represented as a bell-shaped curve. This distribution describes how values, like BMI changes, are spread across data. In the context of the Diabetes Prevention Trial (DPT), the normal distribution allows us to understand how different individual BMI changes are distributed around their mean. The key characteristics of normal distribution include symmetry around the mean, which means half of the values lie above the mean and half below. This aids in visualizing performance and variability within a group. For instance, when saying a BMI reduction is normally distributed, it implies that most individuals will have BMI changes close to the average value, with fewer individuals having extreme changes.

Standard deviation is a critical concept in statistics that measures how much individual data points deviate from the mean. In our exercise, the standard deviation of the BMI change in the active intervention group was given as 6.7 kg/m². This high standard deviation indicates that there is a significant variance in how effective the intervention was across different individuals. The standard deviation informs us about the spread and dispersion of the data. For example:

• A small standard deviation means that the BMI changes are clustered closely around the mean.
• A large standard deviation, like 6.7 kg/m², shows that the individual changes varied widely, indicating inconsistent outcomes of the intervention.

This helps in identifying whether the intervention impact is predictably homogeneous or highly varied.

The Z-score is a way of quantifying how many standard deviations a data point is from the mean. In the exercise, calculating the Z-score for a BMI reduction of 1 unit involves the formula:\[Z = \frac{X - \mu}{\sigma}\]where:

• \(X\) is the BMI change we are interested in studying (here, 1).
• \(\mu\) is the mean BMI reduction (1.9 kg/m²).
• \(\sigma\) is the standard deviation (6.7 kg/m²).

By applying the formula, we find a Z-score of approximately -0.1343 for a BMI change of 1 unit. This Z-score tells us that a change of 1 unit is slightly less than the average expected change when considering the standard deviation. Z-scores are particularly useful because they allow us to standardize results and compare them across different data sets or populations, providing a measure of how "unusual" or "common" a particular result is.

Statistical interpretation involves understanding and communicating the results obtained from statistical calculations. In this exercise, after we calculate the Z-score, we use it to find the corresponding probability from the standard normal distribution. The Z-score of -0.1343 corresponds to a percentile of approximately 0.4467, meaning that 44.67% of individuals have a BMI change less than or equal to 1 unit.
To find the probability that an individual lost at least 1 unit of BMI, we compute the complementary probability, which is:\[P(X > 1) = 1 - 0.4467 = 0.5533\]This means there is a 55.33% chance that a randomly selected individual from the active group would reduce more than 1 BMI unit over 24 months. These interpretations are vital for assessing the success of interventions like those in the DPT, providing insights into how effectively strategies might work in broader population scenarios.