Question

As the head of a social services agency, you believe that your staff of 20 social workers is very much overworked compared with 10 years ago. The case loads for each worker are reported below for each of the two years in question. Has the average caseload increased? Which measure of central tendency is most appropriate to answer this question? Why? \(\begin{array}{llll} \hline {2002} & & {2012} \\ \hline 52 & 55 & 42 & 82 \\ 50 & 49 & 75 & 50 \\ 57 & 50 & 69 & 52 \\ 49 & 52 & 65 & 50 \\ 45 & 59 & 58 & 55 \\ 65 & 60 & 64 & 65 \\ 60 & 65 & 69 & 60 \\ 55 & 68 & 60 & 60 \\ 42 & 60 & 50 & 60 \\ 50 & 42 & 60 & 60 \\ \hline \end{array}\)

Short answer

The average caseload has increased. The mean is the most appropriate measure because it provides a comprehensive overview of the data.

Step-by-step solution

- Organize the Data

Organize the given caseload data for the years 2002 and 2012.

- Calculate the Mean for 2002

Calculate the mean (average) for the caseload of 2002. The mean is calculated by adding all the numbers and dividing by the total count of numbers: \( \text{Mean}_{2002} = \frac{52 + 50 + 57 + 49 + 45 + 65 + 60 + 55 + 42 + 50}{10} \).

- Calculate the Mean for 2012

Calculate the mean (average) for the caseload of 2012. The mean is calculated by adding all the numbers and dividing by the total count of numbers: \( \text{Mean}_{2012} = \frac{55 + 49 + 50 + 52 + 59 + 60 + 65 + 68 + 60 + 42}{10} \).

- Compare the Means

Compare the means of 2002 and 2012. The average caseload has increased if \( \text{Mean}_{2012} \) is greater than \( \text{Mean}_{2002} \).

- Determine the Appropriate Measure of Central Tendency

The mean is the most appropriate measure of central tendency for this question because it takes into account all data points and provides a single value that represents the overall caseload of the workers.

Key concepts

Mean Calculation

The mean, also known as the average, is a fundamental concept in statistics. It's used to find the central value of a data set. To calculate a mean, you sum up all the numbers in the data set, and then divide by the number of values.
For example, for the caseloads in 2002:
\[ \text{Mean}_{2002} = \frac{52 + 50 + 57 + 49 + 45 + 65 + 60 + 55 + 42 + 50}{10} = \frac{525}{10} = 52.5 \]
Similarly, for the caseloads in 2012:
\[ \text{Mean}_{2012} = \frac{55 + 49 + 50 + 52 + 59 + 60 + 65 + 68 + 60 + 42}{10} = \frac{560}{10} = 56.0 \] The mean gives us a clear picture of the central tendency of the caseload distribution for each year. By comparing these means, we can determine if the workload has increased or decreased over time.

Data Comparison

Once you've calculated the means for both years, the next step is to compare them. In our example, the mean caseload for 2002 was 52.5, and for 2012 it was 56.0. Since 56.0 is greater than 52.5, we can conclude that the average caseload has increased over the 10-year period.
It's crucial to compare means to see trends over time. This can help in identifying whether certain interventions or changes (like increased support or policy changes) might be necessary.
When comparing data:

• Look for increases or decreases between the compared values.
• Consider factors that might have influenced the changes.
• Use clear metrics, like the mean, to make informed decisions.

Comparing the two means helps us understand the trend and make data-driven decisions about workload management.

Workload Assessment

Assessing workload is an important task for any organization. For social services agencies, understanding the caseload for each worker is vital in ensuring efficiency and avoiding burnout.
The increase in mean caseload from 2002 to 2012 indicates that social workers are handling more cases on average, suggesting an increase in overall workload. Regular assessment can help manage resource allocation and ensure workers are not overwhelmed.
Key steps in workload assessment include:

• Regularly collecting data on the task loads of workers.
• Computing the mean to identify central trends.
• Comparing these trends over different time periods.
• Implementing changes based on data insights to improve work conditions.

By continually monitoring and assessing the workload, agencies can take proactive steps to maintain a balanced and manageable distribution of tasks among their staff.