Question

As the head of a social services agency, you believe that your staff of 20 social workers is very much overworked compared to 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? What measure of central tendency is most appropriate to answer this question? Why? $$\begin{array}{cccc} {2000} & & {2010} \\ \hline 52 & 55 & 42 & 82 \\ 50 & 49 & 75 & 50 \\ 57 & 50 & 69 & 52 \\ 49 & 52 & 65 & 50 \\ 45 & 59 & 58 & 55 \\ 65 & 60 & 64 & 65 \\ 60 & 65 & 69 & 60 \\ 55 & 68 & 60 & 60 \\ 42 & 60 & 50 & 60 \\ 50 & 42 & 60 & 60 \end{array}$$

Short answer

The mean caseload increased from 51.7 in 2000 to 57.8 in 2010. The most appropriate measure of central tendency is the mean.

Step-by-step solution

Calculate the Mean Caseload for 2000

To find the average caseload for the year 2000, add up all the caseload numbers for that year and then divide by the number of data points. The caseloads for the year 2000 are: 52, 42, 50, 57, 49, 45, 65, 60, 55, 42.

Sum of Caseloads for 2000

Add the caseloads: 52 + 42 + 50 + 57 + 49 + 45 + 65 + 60 + 55 + 42 = 517.

Mean Caseload for 2000

Divide the sum by the number of social workers: \( \frac{517}{10} = 51.7 \). The mean caseload for the year 2000 is 51.7.

Calculate the Mean Caseload for 2010

To find the average caseload for the year 2010, add up all the caseload numbers for that year and then divide by the number of data points. The caseloads for the year 2010 are: 55, 82, 49, 50, 50, 52, 55, 65, 60, 60.

Sum of Caseloads for 2010

Add the caseloads: 55 + 82 + 49 + 50 + 50 + 52 + 55 + 65 + 60 + 60 = 578.

Mean Caseload for 2010

Divide the sum by the number of social workers: \( \frac{578}{10} = 57.8 \). The mean caseload for the year 2010 is 57.8.

Compare the Means

Compare the mean caseloads for the two years: The mean caseload for 2000 is 51.7 and for 2010 is 57.8. Since 57.8 is greater than 51.7, the average caseload has increased.

Determine the Appropriate Measure of Central Tendency

The most appropriate measure of central tendency for this scenario is the mean because it takes into account all the data points and provides an average number of cases per social worker, which is useful for comparing the overall workload between the two years.

Key concepts

central tendency

Central tendency is a statistical measure that identifies a single value as representative of an entire dataset. This value aims to describe a set of data by identifying the central position within that set. The three most common measures of central tendency are the mean, median, and mode. Each of these measures provides different insights and is useful in various scenarios.

• Mean: The average of all data points.
• Median: The middle value when data points are arranged in order.
• Mode: The most frequently occurring value in the dataset.

By understanding central tendency, you can make informed decisions about data, particularly when you want to summarize and describe the data as a single, central value.

mean calculation

The mean, or average, is a crucial measure of central tendency. To find the mean, you sum up all the values in a dataset and divide by the number of values. This gives an overall idea of the dataset's central value. Let's go through the steps to calculate the mean caseload for 2000 and 2010.

• First, sum all the caseload values for the year 2000: \( 52 + 42 + 50 + 57 + 49 + 45 + 65 + 60 + 55 + 42 = 517 \).
• Next, divide the sum by the number of data points, which is 10: \( \frac{517}{10} = 51.7 \).
• Repeat the process for the year 2010: \( 55 + 82 + 49 + 50 + 50 + 52 + 55 + 65 + 60 + 60 = 578 \).
• Divide by the number of data points: \( \frac{578}{10} = 57.8 \).

The mean caseloads are 51.7 for 2000 and 57.8 for 2010. This calculation shows how the average caseload can be summarized and easily compared between different years.

workload assessment

Workload assessment involves evaluating the amount of work assigned to individuals or groups over a specific period. In this context, caseload refers to the number of cases handled by each social worker in a given year. Effectively assessing workload helps in understanding if the resources are adequately allocated and if staff are overburdened.
To assess the workload:

• Calculate the mean caseload for each year.
• Analyze changes in average workloads over time.
• Compare the workloads to identify any significant increases or decreases.

It was found that the mean caseload increased from 51.7 in 2000 to 57.8 in 2010. This rise indicates that social workers are handling more cases on average over the decade, signaling an increase in workload.

data analysis

Data analysis involves systematically applying statistical techniques to describe, illustrate, and evaluate data. In this exercise, we analyze caseload numbers to determine trends and patterns in workload over time. Here’s a simple guide to analyzing data like this:

• Gather your data and organize it clearly (e.g., caseload numbers for each year).
• Apply statistical measures (mean, median, mode) to summarize the data.
• Visualize your findings using charts or graphs for better understanding.
• Interpret the results to derive meaningful insights, such as changes in workload.

Data analysis in this scenario revealed that the average caseload per social worker increased, pointing to a higher demand for social services and possibly an understaffed workforce. Such analysis helps in making informed management decisions.

statistical comparison

Statistical comparison is comparing two or more datasets to identify similarities and differences. It's vital for understanding changes over time and making data-driven decisions.
To compare the 2000 and 2010 caseloads:

• Calculate the mean for both years.
• Observe the differences between these means.
• Determine if the difference is significant or just a minor variation.

In this case, the mean caseload for the year 2000 was 51.7, while for 2010, it was 57.8. The comparison shows a clear increase, indicating that the workload has grown over the decade. Such insights can prompt actions like hiring more staff or redistributing cases to balance the workload more effectively, ensuring that social workers are not overburdened.