Calculating Mileage Within A Z-Score Of -1.5
In the realm of statistics, the z-score stands as a pivotal measure, offering a standardized way to gauge how far a particular data point deviates from the mean of a dataset. This concept finds practical applications across various fields, including human resources, where understanding employee commuting distances can be valuable for workforce management and policy development. Let's delve into a scenario where a supervisor grapples with analyzing employee commute distances using z-scores.
The Scenario: Analyzing Employee Commute Distances
Imagine a supervisor tasked with understanding the commuting patterns of employees within a specific department. The supervisor embarks on a data-gathering mission, collecting information on the distance each employee lives from the workplace. After meticulously compiling the data, the supervisor calculates the mean commuting distance, denoted as x̄, which turns out to be 29 miles. Furthermore, the supervisor determines the standard deviation, represented by s, which measures the spread or variability of the data around the mean, and finds it to be 3.6 miles.
Now, the supervisor poses a crucial question: Which mileage falls within a z-score of -1.5? This question seeks to identify the commuting distance that is 1.5 standard deviations below the average commute distance. To unravel this, we must understand the concept of z-scores and how they relate to the data distribution.
Z-Scores: A Standardized Measure of Deviation
A z-score, also known as a standard score, quantifies the number of standard deviations a particular data point lies above or below the mean of a dataset. A positive z-score indicates that the data point is above the mean, while a negative z-score signifies that it is below the mean. A z-score of zero corresponds to the mean itself.
The formula for calculating a z-score is given by:
z = (x - x̄) / s
where:
- z represents the z-score
- x denotes the individual data point
- x̄ is the mean of the dataset
- s is the standard deviation of the dataset
In our scenario, we are given a z-score of -1.5, the mean x̄ of 29 miles, and the standard deviation s of 3.6 miles. Our goal is to find the mileage x that corresponds to this z-score. To do so, we can rearrange the z-score formula to solve for x:
x = z * s + x̄
Substituting the given values, we get:
x = -1.5 * 3.6 + 29
x = -5.4 + 29
x = 23.6
Therefore, a z-score of -1.5 corresponds to a mileage of 23.6 miles. Among the given options (21 miles, 24 miles, 36 miles, and 41 miles), 24 miles is the closest to 23.6 miles. This implies that a commuting distance of 24 miles is approximately 1.5 standard deviations below the average commuting distance in the department.
Applying the Z-Score Concept to Mileage Calculation
Now that we've established the fundamental principles of z-scores and their application in determining distances relative to the mean, let's delve deeper into how this knowledge can be applied in practical scenarios. Understanding the distribution of employee commute distances can offer valuable insights for workforce management and policy development.
Option 1: 21 Miles
To determine if 21 miles falls within a z-score of -1.5, we can calculate its z-score using the formula:
z = (x - x̄) / s
Substituting the values, we get:
z = (21 - 29) / 3.6
z = -8 / 3.6
z ≈ -2.22
The z-score for 21 miles is approximately -2.22, which is significantly lower than -1.5. This indicates that 21 miles is more than 1.5 standard deviations below the mean. Thus, 21 miles is not within a z-score of -1.5.
Option 2: 24 Miles
Next, let's calculate the z-score for 24 miles:
z = (24 - 29) / 3.6
z = -5 / 3.6
z ≈ -1.39
The z-score for 24 miles is approximately -1.39, which is closer to -1.5 than the z-score for 21 miles. This suggests that 24 miles is within a z-score of -1.5.
Option 3: 36 Miles
Now, let's compute the z-score for 36 miles:
z = (36 - 29) / 3.6
z = 7 / 3.6
z ≈ 1.94
The z-score for 36 miles is approximately 1.94, which is a positive value and significantly greater than -1.5. This means that 36 miles is above the mean and not within a z-score of -1.5.
Option 4: 41 Miles
Finally, let's determine the z-score for 41 miles:
z = (41 - 29) / 3.6
z = 12 / 3.6
z ≈ 3.33
The z-score for 41 miles is approximately 3.33, which is a large positive value, indicating that 41 miles is far above the mean and not within a z-score of -1.5.
Conclusion: Identifying Mileage Within the Z-Score Threshold
Based on our calculations, the mileage that falls within a z-score of -1.5 is 24 miles. This means that a commuting distance of 24 miles is approximately 1.5 standard deviations below the average commute distance of 29 miles in the department. Understanding these statistical measures can help supervisors and HR professionals gain insights into employee commuting patterns, which can be valuable for various workplace considerations.
In summary, the concept of z-scores provides a standardized way to assess how individual data points deviate from the mean of a dataset. By applying this concept to employee commute distances, supervisors can gain valuable insights into the distribution of commuting patterns within their departments. This knowledge can be leveraged for informed decision-making in areas such as workforce management, policy development, and employee support programs. The ability to interpret z-scores and their implications empowers organizations to foster a more employee-centric and efficient work environment.
Z-score, standard deviation, mean, mileage, commuting distance, employee commute, statistical analysis, workforce management, policy development, data analysis, employee support, work environment, deviation from the mean, data distribution, HR professionals
