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. Read more

Steps to Solve

1. Determine the Range To find the range, subtract the smallest salary from the largest salary.

If the salaries are sorted in ascending order as follows:

$$ \text{salaries} = [s_1, s_2, ..., s_{15}] $$

Then,

$$ \text{Range} = s_{15} - s_1 $$

2. Calculate Quartiles First, find the first quartile ($Q_1$), median ($Q_2$), and third quartile ($Q_3$).

• To find $Q_1$, locate the median of the first half of the data (salaries):

If there are 15 salaries, $Q_1$ is the median of the first 7 salaries.

• For $Q_2$, locate the median of all salaries. Since there are 15 values, $Q_2$ is the 8th value.

• To find $Q_3$, locate the median of the last half of the data (salaries):

$Q_3$ is the median of the last 7 salaries.

3. Calculate the Quartile Deviation The quartile deviation is calculated as:

$$ \text{Quartile Deviation} = \frac{Q_3 - Q_1}{2} $$

4. Calculate Variance Variance measures how data points differ from the mean. The formula for variance ($\sigma^2$) is:

$$ \sigma^2 = \frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n} $$

Where $\mu$ is the mean of the salaries and $n$ is the number of salaries.

5. Calculate Coefficient of Variation The coefficient of variation (CV) is given by:

$$ \text{CV} = \frac{\sigma}{\mu} \times 100 $$

Where $\sigma$ is the standard deviation (the square root of the variance) and $\mu$ is the mean.