Understanding the Arithmetic Mean Formula using the Step Deviation Method

The formula \( \overline X = A + \frac{\Sigma f_i \mu_i}{\Sigma f_i} \times h \) is used to calculate the Arithmetic Mean for Grouped Data. This particular approach is known as the Step Deviation Method, building upon the Assumed Mean method. It is particularly useful for simplifying calculations involving Frequency Distribution when class intervals are equal.

Components of the Step Deviation Formula

Let's define the terms in this Mean Formula:

• \( \overline X \): The Arithmetic Mean.
• A: The Assumed Mean, usually the midpoint of a class interval.
• \( f_i \): The frequency of the \( i^{\text{th}} \) class interval.
• \( \Sigma f_i \): The total sum of all frequencies.
• h: The class size (width of the class interval).
• \( \mu_i \): The step deviation for the \( i^{\text{th}} \) class.

Defining μi: Standard Practice vs. Given Option

In the standard statistical practice for the Step Deviation Method, the value \( \mu_i \) is calculated using the formula based on the deviation of the class mark \( x_i \) from the assumed mean \( A \), divided by the class size \( h \). The standard definition is:

\[ \text{Standard } \mu_i = \frac{x_i - A}{h} \]

This definition is used in the Mean Formula provided to simplify calculations. The multiplication by \( h \) outside the summation compensates for the division by \( h \) in the definition of \( \mu_i \).

However, the question asks for the definition of \( \mu_i \) based on the options presented. Let's look at the given options:

• Option 1: \( \frac{x_i + A}{n} \)
• Option 2: \( h(x_i - A) \)
• Option 3: \( \frac{x_i - A}{n} \)
• Option 4: \( \frac{A + x_i}{n} \)

Identifying μi from the Provided Choices

Comparing these options with the standard definition, we see that option 3, \( \frac{x_i - A}{n} \), represents the deviation \( (x_i - A) \) divided by a quantity 'n'. While the standard Step Deviation Method divides by the class size \( h \), the question requires selecting from the provided options. Based on the structure of the deviation and the format of the options, the expression \( \frac{x_i - A}{n} \) is the intended definition of \( \mu_i \) in this specific context as presented in the options.

Thus, among the given choices for the formula \( \overline X=A+\frac{\Sigma fiμ i}{\Sigma f}\times h \), \( \mu_i \) is defined as \( \frac{x_i - A}{n} \) according to the options provided for calculating the Arithmetic Mean of Grouped Data and their corresponding Frequency Distribution.