Question

During contract negotiations, a company seeks to change the number of sick days employees may take, saying that the annual "average" is 7 days of absence per employee. The union negotiators counter that the "average" employee misses only 3 days of work each year. Explain how both sides might be correct, identifying the measure of center you think each side is using and why the difference might exist.

Short answer

The company likely uses the mean, while the union uses the median. The difference exists due to the influence of outliers on the mean, but not on the median.

Step-by-step solution

Understanding the Concept of Average

In statistics, the term "average" can refer to different measures of central tendency: mean, median, and mode. Each measure provides a different perspective on data, potentially leading to different statements about an average.

Exploring the Company's Perspective

The company claims the "average" number of sick days is 7 days per employee. It is likely that the company is using the mean, which is calculated by adding all the sick days taken by employees and dividing by the total number of employees. The mean can be influenced by outliers.

Exploring the Union's Perspective

The union claims the "average" number of sick days is only 3 days per employee. It is likely that the union is using the median, which represents the middle value when all employees' sick days are lined up in order. The median is less affected by outliers and may provide a different perspective.

Understanding Potential Data Disparity

The difference between the mean and median might exist due to the distribution of sick days among employees. A small number of employees taking significantly more sick leaves can skew the mean higher, creating a disparity between mean and median.

Justifying the Measures Used

The company might favor the mean as it reflects the overall impact on operations, particularly if a few employees take excessive leaves. The union might prefer the median to highlight that most employees take fewer days off, suggesting less frequent sick leave.

Key concepts

Mean

The mean is a common measure of central tendency that you might hear referred to as the "average." To calculate it, you add up all the numbers in your data set and then divide the total by the number of items in the set. For example, if five employees took 0, 0, 3, 10, and 12 sick days, the mean would be calculated as follows: \[ \text{Mean} = \frac{0 + 0 + 3 + 10 + 12}{5} = \frac{25}{5} = 5 \]The mean gives you an overall idea of the data, but it's important to note that it can be heavily influenced by outliers. In the context of sick days, if a few employees are taking many more days off than the rest, they could skew the mean higher, making it seem like all employees are taking lots of time off.

• Mean is useful for understanding overall resource use.
• Can be skewed by extreme values or outliers.
• Ideal for data without extreme variations.

Median

The median is another key measure of central tendency, and it represents the middle value in a data set when the numbers are arranged in ascending or descending order. If you have an odd number of values, the median is the middle one; if even, it is the average of the two middle numbers. For instance, with sick days of 0, 0, 3, 10, and 12, the median is 3 because it is the middle value in this ordered set. The median is useful because it is not affected by extreme values or outliers as the mean is. This characteristic makes it a reliable indicator of the "central" part of your data, particularly when your data set might include outliers. In the case of the sick days argument, the union's use of median can highlight that the majority of employees take few sick days, which can seem like a fairer representation of the typical employee's experience.

• Resistant to outliers, providing a true central value.
• Best used when data includes extremes.
• Gives an intuitive sense of where "most" data lies.

Mode

The mode is often the least utilized measure of central tendency but can be quite useful in determining the most frequently occurring data point in a set. Unlike the mean or median, the mode does not need arithmetic for its determination and can be directly observed. If employees have sick days of 0, 0, 3, 10, and 12, the mode is 0 because it appears most frequently in the data set. The mode is especially helpful in situations where the most common item is of interest. Though not directly applicable in our exercise, understanding the mode could still provide insights, such as identifying a situational trend (for example, if 0 is the mode for sick days, most employees may not be taking any sick days at all). This information could be valuable in scenarios of resource planning or evaluating employee wellness programs.

• Indicates the most frequent occurrence.
• Simple to identify, requiring no complex calculations.
• Useful in categorical data analysis.