Question

A small warehouse employs a supervisor at $$\$ 1200$$ a week, an inventory manager at $$\$ 700$$ a week, six stock boys at $$\$ 400$$ a week, and four drivers at \(\$ 500\) a week. a) Find the mean and median wage. b) How many employees earn more than the mean wage? c) Which measure of center best describes a typical wage at this company: the mean or the median? d) Which measure of spread would best describe the payroll: the range, the IQR, or the standard deviation? Why?

Short answer

Mean wage is 525, median is 450. Two employees earn more than the mean. Median best describes typical wade, standard deviation best describes spread.

Step-by-step solution

Calculate Total Wages

First, calculate the total wage for each job quantity. Supervisor: \(1 \times 1200 = 1200\), Inventory Manager: \(1 \times 700 = 700\), Stock Boys: \(6 \times 400 = 2400\), Drivers: \(4 \times 500 = 2000\). The total wage is \(1200 + 700 + 2400 + 2000 = 6300\).

Calculate Mean Wage

To calculate the mean wage, divide the total wages by the number of employees. The total number of employees is \(12\), so the mean wage is \(\frac{6300}{12} = 525\).

Determine Median Wage

List all individual wages: \([400, 400, 400, 400, 400, 400, 500, 500, 500, 500, 700, 1200]\). With \(12\) numbers, the median is the average of the 6th and 7th wages: \(\frac{400 + 500}{2} = 450\).

Find Employees Earning Above Mean Wage

Count the number of employees earning more than the mean wage of \(525\): inventory manager (\(700\)) and supervisor (\(1200\)) — totaling \(2\) employees.

Compare Mean and Median Wages

The median wage is \(450\), lower than the mean wage of \(525\). The median is less affected by the high wage of the supervisor, suggesting it better represents a typical wage.

Determine Measure of Spread

Consider measures of spread: range (\(1200 - 400 = 800)\), IQR (between stock boys and drivers is \(500-400 = 100\)), or standard deviation (reflects all variation). Range highlights max and min, but IQR or standard deviation better describe typical payroll variation. Standard deviation captures overall spread better.

Key concepts

Mean Wage

The mean wage is a fundamental statistical concept that helps in understanding the average earnings of a group of employees. To calculate the mean wage, you need to divide the total wages paid by the number of employees.
For example, in our warehouse scenario, we calculated the total wage to be \( \$6300 \) for 12 employees. Thus, the mean wage is \((\frac{6300}{12} = 525) \) dollars per week.
The mean wage provides a simple average, which is useful for getting a quick sense of the overall wage distribution. However, it doesn't always account for outliers or variations within the data.
Understanding the mean wage can help businesses evaluate their payroll budget, make adjustments to retain talent, and assess their financial standing in terms of labor costs.

Median Wage

The median wage is another measure of central tendency that offers a better representation of the typical wage for groups with income disparity.
Unlike the mean, the median is not affected by extreme high or low wages. It is simply the middle point in a sorted list of wages.
In our scenario, we list the individual wages like this: \([400, 400, 400, 400, 400, 400, 500, 500, 500, 500, 700, 1200]\) and the median is the average of the 6th and 7th wage. \((\frac{400 + 500}{2} = 450) \) dollars per week.

• This measure is particularly useful when the data set includes outliers.
• In our warehouse example, the supervisor's significantly higher wage skews the mean, making the median a better indicator of a typical wage.

Using the median wage can help companies understand the most common wage level and identify wage inequality among employees.

Measures of Center

Measures of center, such as the mean and median, offer different perspectives on wage data, helping to summarize and interpret it effectively.
These measures each play an important role:

• Mean: A useful average that considers all data points but can be influenced by outliers.
• Median: Represents the middle ground when data is ordered, thus unaffected by extremely high or low values.

Choosing between mean and median as the best measure of center depends on the data's nature:

• When data is normally distributed, the mean usually suffices.
• If the data contains outliers or is skewed, the median often provides a clearer insight into typical values.

In practice, understanding both measures allows for a more rounded picture of employee wages, aiding in comprehensive data-driven decision-making for efficient payroll management.