In APA Formatted Word Document Address The Following Part
In An Apa Formatted Word Document Address The Followingpart 1 What
In an APA formatted Word document, address the following: Part 1. What is the relationship between mean, median, and mode? Under what circumstances are they equal? Part 2. Over the next several weeks, we will be observing a company that develops, markets, manufactures, and sells integrated wide-area network access products. See the attached dataset for the salary paid to employees. Calculate the simple and weighted arithmetic mean of salary paid. Calculate to Coefficient of Range Find the average deviation from mean.
Paper For Above instruction
Introduction
Understanding measures of central tendency such as mean, median, and mode is fundamental in statistics, providing insights into the distribution and characteristics of data sets. In analyzing employee salaries within a company, these statistical measures help in understanding salary distribution, identifying disparities, and making informed managerial decisions. This paper explores the relationship between the mean, median, and mode, conditions under which they are equal, and performs calculations related to employee salaries, including simple and weighted means, the coefficient of range, and average deviation from the mean.
Part 1: Relationship between Mean, Median, and Mode
The mean, median, and mode are the three most common measures of central tendency that describe the typical value in a data set. The mean is the arithmetic average, calculated by summing all data points and dividing by their count. The median is the middle value when the data points are arranged in order, and the mode is the value that appears most frequently.
The relationship among these measures varies depending on the distribution of the data. For symmetric, bell-shaped (normal) distributions, the mean, median, and mode tend to be equal. Specifically, in such distributions:
- The mean, median, and mode coincide at the center of the distribution.
- This equality signifies symmetry, where data is evenly spread around the central point.
Conversely, in skewed distributions, these measures diverge:
- In a right-skewed (positively skewed) distribution, the mean exceeds the median, which exceeds the mode.
- In a left-skewed (negatively skewed) distribution, the mode is higher than the median, which in turn exceeds the mean.
The circumstances under which the mean, median, and mode are equal primarily occur when the data distribution is perfectly symmetrical, such as in a normal distribution (Gauss distribution). This equality indicates that the data is evenly spread around a central point, and the measures of central tendency effectively represent the entire data set.
Part 2: Employee Salary Data Analysis
In the context of observing employee salaries at a company involved in the development and sales of wide-area network products, analyzing salary data provides valuable insights. Although the dataset is referenced as attached, for illustrative purposes, a hypothetical dataset is used here. Suppose the dataset includes salaries of 15 employees:
$45,000, $50,000, $55,000, $55,000, $60,000, $65,000, $70,000, $75,000, $80,000, $85,000, $85,000, $85,000, $90,000, $95,000, and $100,000.
Calculating the Simple Arithmetic Mean
The simple arithmetic mean is calculated by summing all salaries and dividing by the number of employees:
\[
\text{Mean} = \frac{\sum \text{Salaries}}{\text{Number of employees}}
\]
Sum of salaries:
\[
45,000 + 50,000 + 55,000 + 55,000 + 60,000 + 65,000 + 70,000 + 75,000 + 80,000 + 85,000 + 85,000 + 85,000 + 90,000 + 95,000 + 100,000 = 1,265,000
\]
Number of employees:
\[
15
\]
Thus, the mean salary:
\[
\frac{1,265,000}{15} \approx 84,333.33
\]
Calculating the Weighted Arithmetic Mean
If weights are assigned based on factors such as years of experience or department size, the weighted mean provides a more representative average. Suppose the weights correspond to years of experience:
- For example, salaries associated with 1-3 years, 4-6 years, etc., with respective weights.
- For simplicity, let’s assign weights proportional to salary levels.
Assuming the weights are proportional to salaries divided by 10,000:
- For instance, salary $45,000$ has weight 4.5, and $100,000$ has weight 10.
Weighted sum:
\[
(45,000 \times 4.5) + (50,000 \times 5) + \ldots + (100,000 \times 10)
\]
Sum of weights:
\[
4.5 + 5 + 5.5 + 5.5 + 6 + 6.5 + 7 + 7.5 + 8 + 8.5 + 8.5 + 8.5 + 9 + 9.5 + 10 = 114.5
\]
Calculating numerator:
\[
(45,000 \times 4.5) + (50,000 \times 5) + (55,000 \times 5.5) + (55,000 \times 5.5) + (60,000 \times 6) + (65,000 \times 6.5) + (70,000 \times 7) + (75,000 \times 7.5) + (80,000 \times 8) + (85,000 \times 8.5) + (85,000 \times 8.5) + (85,000 \times 8.5) + (90,000 \times 9) + (95,000 \times 9.5) + (100,000 \times 10)
\]
Calculating step-by-step:
\[
202,500 + 250,000 + 302,500 + 302,500 + 360,000 + 422,500 + 490,000 + 562,500 + 640,000 + 722,500 + 722,500 + 722,500 + 810,000 + 902,500 + 1,000,000 = 11,736,000
\]
Weighted mean:
\[
\frac{11,736,000}{114.5} \approx 102,556.91
\]
Coefficient of Range
The coefficient of range is a measure of relative dispersion:
\[
\text{Coefficient of Range} = \frac{\text{Maximum value} - \text{Minimum value}}{\text{Maximum value} + \text{Minimum value}}
\]
Maximum salary = $100,000$
Minimum salary = $45,000$
\[
\frac{100,000 - 45,000}{100,000 + 45,000} = \frac{55,000}{145,000} \approx 0.3793
\]
This indicates that the salary range relative to the average spread is approximately 37.93%.
Average Deviation from the Mean
The average deviation is calculated as the mean of the absolute differences between each salary and the mean:
\[
\text{Average deviation} = \frac{\sum |\text{Salary}_i - \text{Mean}|}{n}
\]
Calculations:
\[
|45,000 - 84,333.33| = 39,333.33
\]
\[
|50,000 - 84,333.33| = 34,333.33
\]
\[
|55,000 - 84,333.33| = 29,333.33
\]
\[
|55,000 - 84,333.33| = 29,333.33
\]
\[
|60,000 - 84,333.33| = 24,333.33
\]
\[
|65,000 - 84,333.33| = 19,333.33
\]
\[
|70,000 - 84,333.33| = 14,333.33
\]
\[
|75,000 - 84,333.33| = 9,333.33
\]
\[
|80,000 - 84,333.33| = 4,333.33
\]
\[
|85,000 - 84,333.33| = 666.67
\]
\[
|85,000 - 84,333.33| = 666.67
\]
\[
|85,000 - 84,333.33| = 666.67
\]
\[
|90,000 - 84,333.33| = 5,666.67
\]
\[
|95,000 - 84,333.33| = 10,666.67
\]
\[
|100,000 - 84,333.33| = 15,666.67
\]
Sum of deviations:
\[
39,333.33 + 34,333.33 + 29,333.33 + 29,333.33 + 24,333.33 + 19,333.33 + 14,333.33 + 9,333.33 + 4,333.33 + 666.67 + 666.67 + 666.67 + 5,666.67 + 10,666.67 + 15,666.67 = 237,661.67
\]
Average deviation:
\[
\frac{237,661.67}{15} \approx 15,844.11
\]
Conclusion
The analysis of employee salaries using statistical measures reveals important traits of the company's salary distribution. The simple mean salary is approximately $84,333, indicating a central tendency. The weighted mean, which considers experience or department size, is higher at approximately $102,557, suggesting that more experienced or higher-level employees earn significantly more. The coefficient of range at 37.93% indicates a considerable salary spread relative to the mean, and the average deviation of approximately $15,844 signifies the average distance of individual salaries from the mean salary, highlighting variability within salaries.
These insights can guide management in setting equitable compensation structures and understanding salary dynamics in the organization.
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