Question

Five subjects were weighed before and after an 8 -week exercise program. What is the average amount of weight lost in pounds for all five subjects, rounded to the nearest pound? $$ \begin{array}{|c|c|c|} \hline \text { Subject } & \begin{array}{c} \text { Starting } \\ \text { Weight } \\ \text { (pounds) } \end{array} & \begin{array}{c} \text { Final } \\ \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 1 & 184 & 176 \\ \hline 2 & 200 & 190 \\ \hline 3 & 221 & 225 \\ \hline 4 & 235 & 208 \\ \hline 5 & 244 & 225 \\ \hline \end{array} $$ (A) 12 pounds (B) 13 pounds (C) 14 pounds (D) 15 pounds

Short answer

The average amount of weight lost in pounds for all five subjects is \(12\) pounds.

Step-by-step solution

Calculate the weight change for each subject

To find the weight change for each subject, subtract the final weight from the starting weight. Do this for all 5 subjects: Subject 1: \(184-176=8\) Subject 2: \(200-190=10\) Subject 3: \(221-225=-4\) (This subject gained weight instead of losing it) Subject 4: \(235-208=27\) Subject 5: \(244-225=19\)

Calculate the total weight change

Add the weight changes for each subject to find the total weight change: Total weight change: \(8+10-4+27+19=60\) pounds

Calculate the average weight change

Divide the total weight change by the number of subjects (5) to find the average weight change: Average weight change: \(\frac{60}{5}=12\) pounds

Round the average weight change and find the answer choice

Since the average weight change is already a whole number (12 pounds), there's no need to round it. The average amount of weight lost in pounds for all five subjects is 12 pounds. This corresponds to answer choice (A).

Key concepts

Data Interpretation

Data interpretation is a critical skill when analyzing results from studies or experiments, particularly when dealing with changes like weight loss or gain. It involves reviewing raw data to uncover patterns or insights that are not immediately obvious. In our exercise, the data presented is in a table form, showing each subject's weight before and after an exercise program. To interpret this data, one must be attentive to details, such as the negative weight change indicating weight gain rather than loss.

It's vital to understand that each figure in the dataset represents a unique piece of information. In the context of our problem, incorrect interpretation, like overlooking the fact that one subject gained weight, could lead to a wrong calculation of the average weight loss. Effective data interpretation allows for more accurate and meaningful outcomes, which is precisely what we aim for when assessing the effectiveness of the exercise program in our example.

Arithmetic Mean

The arithmetic mean, or simple average, is a fundamental statistical measure used to find the 'central tendency' or typical value of a set of numbers. It is calculated by adding together all the values, and then dividing the total by the number of values. In the weight change example, each subject's weight change represents a value in our set. To find the average weight change, we add these values together and divide by five, the number of subjects.

The arithmetic mean is sensitive to extreme values, which is evident from the range of weight changes (-4 to 27 pounds). An outlier can skew the mean, making it essential to consider all data points carefully. The mean provides a single number that summarizes the entire dataset, which in this case helps to understand the general effectiveness of the weight loss program across all individuals.

Weight Change Analysis

Weight change analysis involves evaluating the difference in weight measurements taken at different times. This type of analysis often aims to determine the impact of a specific intervention, like an exercise program. For accurate analysis, it's important to calculate the change for each individual separately before looking at the broader group data.

To determine whether the program was successful, we calculate the weight lost by each subject. If we notice that a subject gained weight, it's essential to interpret this as a negative number, which will affect the group average. This meticulous approach ensures that we factor in both weight loss and weight gains, providing a realistic assessment of the program's outcome. Through careful weight change analysis, we can conclude not just the average effectiveness of the program, but also observe the range of individual responses.