The daily salaries of the 15 employees were as given below: 205, 190, 195, 218, 187, 168, 250, 168, 190, 168, 170, 175, 178, 175, 150. Calculate 1) Range and coefficient of range.... The daily salaries of the 15 employees were as given below: 205, 190, 195, 218, 187, 168, 250, 168, 190, 168, 170, 175, 178, 175, 150. Calculate 1) Range and coefficient of range. ii) Quartile deviation and coefficient of deviation. iii) Variance and coefficient of variation. Solve the above question using R language.
Understand the Problem
The question asks for statistical calculations involving the daily salaries of 15 employees. It requires the computation of range, coefficient of range, quartile deviation, coefficient of deviation, variance, and coefficient of variation, specifically using R language to perform these calculations.
Answer
To fully answer the problem, actual salary data are required to compute the individual statistics as outlined in the steps.
Answer for screen readers
To compute the answer, specific data for salaries is required. The calculations should include the range, coefficient of range, quartile deviation, coefficient of deviation, variance, and coefficient of variation based on those salaries.
Steps to Solve
- Calculate the Range
To find the range, subtract the minimum salary from the maximum salary.
If the minimum salary is $min$ and the maximum salary is $max$, then the range ($R$) is calculated as:
$$ R = max - min $$
- Calculate the Coefficient of Range
The coefficient of range is calculated by dividing the range by the sum of the maximum and minimum salaries:
$$ Coefficient\ of\ Range = \frac{R}{max + min} $$
- Calculate Quartiles
First, sort the salary data from lowest to highest. Then calculate the first quartile ($Q_1$) and the third quartile ($Q_3$).
Use the formula for quartile:
$$ Q_k = \frac{k(n + 1)}{4} $$
where $k$ is 1 for $Q_1$ and 3 for $Q_3$.
- Calculate the Quartile Deviation
The quartile deviation is half the difference between the first and third quartiles:
$$ Quartile\ Deviation = \frac{Q_3 - Q_1}{2} $$
- Calculate the Variance
To compute the variance, find the mean ($\mu$) of the salaries first. Then for each salary ($x_i$), calculate the squared difference from the mean, sum them up, and divide by the number of employees:
$$ Variance\ (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n} $$
- Calculate the Coefficient of Deviation
The coefficient of deviation is given by dividing the standard deviation ($\sigma$) by the mean ($\mu$) and multiplying by 100:
$$ Coefficient\ of\ Deviation = \frac{\sigma}{\mu} \times 100 $$
- Calculate Variance and Coefficient of Variation
After the variance is determined, the standard deviation ($\sigma$) is simply the square root of the variance. Coefficient of variation can be computed as:
$$ Coefficient\ of\ Variation = \frac{\sigma}{\mu} \times 100 $$
To compute the answer, specific data for salaries is required. The calculations should include the range, coefficient of range, quartile deviation, coefficient of deviation, variance, and coefficient of variation based on those salaries.
More Information
Statistical calculations like variance and quartile deviation help in understanding the spread and distribution of employee salaries. These metrics can inform better pay decisions and identify inequality.
Tips
- Forgetting to sort the salaries before calculating quartiles can lead to incorrect results.
- Incorrectly using bounds when calculating the range can yield an inaccurate result.
- Not following the standard procedures for calculating variance can lead to misunderstanding of the data variability.
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