Calculating the Coefficient of Range

The question asks us to calculate the coefficient of range for a given frequency distribution. The coefficient of range is a measure of dispersion, which tells us about the spread of the data. It is calculated using the highest and lowest values in the dataset.

Understanding Range and Coefficient of Range

The range is the simplest measure of dispersion. It is defined as the difference between the highest value and the lowest value in a dataset.

Range \( = \text{Highest Value} - \text{Lowest Value} \)

The coefficient of range is a relative measure of dispersion. It is useful for comparing the dispersion of different datasets that might have different units or significantly different magnitudes. It is calculated using the following formula:

\[ \text{Coefficient of Range} = \frac{\text{Highest Value} - \text{Lowest Value}}{\text{Highest Value} + \text{Lowest Value}} \]

In a frequency distribution, the 'Items' represent the values of the variable, and the 'Frequency' tells us how many times each value occurs. For calculating the range and coefficient of range, we only need the maximum and minimum values of the variable (the Items).

Analyzing the Given Series Data

The given series is presented as a frequency distribution:

From the 'Item' column, we can identify the highest and lowest values:

• Lowest Value (S) = 10
• Highest Value (L) = 22

Step-by-Step Calculation of Coefficient of Range

Now we can use the formula for the coefficient of range with the identified highest and lowest values:

Coefficient of Range \( = \frac{L - S}{L + S} \)

Substitute the values L = 22 and S = 10 into the formula:

Coefficient of Range \( = \frac{22 - 10}{22 + 10} \)

First, calculate the difference in the numerator:

\( 22 - 10 = 12 \)

Next, calculate the sum in the denominator:

\( 22 + 10 = 32 \)

Now, divide the result of the numerator by the result of the denominator:

Coefficient of Range \( = \frac{12}{32} \)

To simplify the fraction, divide both the numerator and denominator by their greatest common divisor, which is 4:

\( \frac{12 \div 4}{32 \div 4} = \frac{3}{8} \)

Finally, convert the fraction to a decimal:

\( \frac{3}{8} = 0.375 \)

Comparing this result with the given options, 0.375 is closest to 0.37.

Comparing with Options

Let's compare our calculated coefficient of range (0.375) with the provided options:

• Option 1: 0.17
• Option 2: 0.27
• Option 3: 0.37
• Option 4: 0.47

The calculated value 0.375 is approximately 0.37 when rounded to two decimal places. Therefore, option 3 is the correct answer.

Revision Table: Key Statistical Measures

Additional Information on Dispersion

Measures of dispersion are crucial in statistics as they quantify the spread or variability of data. While measures of central tendency (like mean, median, mode) tell us about the center of the data, measures of dispersion tell us how spread out the data points are from the center or from each other.

There are two main types of dispersion measures:

• Absolute Measures: These are expressed in the same units as the data. Examples include Range, Quartile Deviation, Mean Deviation, and Standard Deviation.
• Relative Measures: These are unitless and are ratios. They are used to compare the dispersion of datasets with different units or scales. Examples include Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, and Coefficient of Variation. The Coefficient of Range calculated here is a relative measure.

The coefficient of range is simple to calculate but is highly affected by extreme values. It uses only the two most extreme values and ignores all other data points.