Step Deviation Method

Step deviation method is a method of obtaining the mean of grouped data when the values are large. In statistics, there are three types of mean - arithmetic mean, geometric mean, and harmonic mean. When the values of the data are large and the deviation of the class marks have common factors, the step deviation method is used. Let us learn more about the formula, the derivation of the step deviation method formula, and solve a few examples.

Step deviation method can be defined as the method used to obtain the mean of large values which is divisible by a common factor. These values of deviations are reduced to a smaller value by dividing all the values by a common factor. In other words, the step deviation method is used when the deviations of the class marks from the assumed mean are large and they all have a common factor. This method is also called a change of origin or scale method. The mean of grouped data is obtained by three different methods, direct method, assumed or short-cut method, and step deviation method. The step deviation method is considered as the extension of the assumed method as we use the deviation formula used in the assumed method.

Before we go to the step deviation formula, let us quickly look at both the direct method and assumed or short-cut method formulas.

Estimated or Direct Mean = ∑x\(_i\)f\(_i\) / ∑f\(_i\), where f\(_i\) is the frequency and x\(_i\) is the midpoint of the class interval.

Assumed Mean = A + ∑d\(_i\)f\(_i\) / ∑f\(_i\), where A is the assumed mean, f\(_i\) is the frequency, and deviation d\(_i\) = x\(_i\) - A.

Since this method is the extension of the assumed mean method, the formula is:

Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],

where,

• A is the assumed mean
• h is the class size
• u\(_i\) = d\(_i\)/h
• f\(_i\) is the frequency
• d\(_i\) = x\(_i\) - A
• x\(_i\) is the midpoint of the class interval

Let us derive the step deviation formula by using the direct method formula and the deviation process in the assumed mean method. Let us consider the class size to be h and the assumed mean to be A. Using the same formula used in assumed mean, d\(_i\) = x\(_i\) - A and calculating the value of u\(_i\) with the equation u\(_i\) = d\(_i\)/h, where h is the class width and d\(_i\) = x\(_i\) - A. Hence, we can derive the formula as:

x̄ = ∑d\(_i\)f\(_i\) / ∑f\(_i\)

x̄ = ∑[u\(_i\)h + A]f\(_i\) / ∑f\(_i\)

x̄ = ∑[u\(_i\)f\(_i\)h + Af\(_i\)] / ∑f\(_i\)

x̄ = ∑u\(_i\)f\(_i\)h + ∑Af\(_i\) / ∑f\(_i\)

x̄ = [∑u\(_i\)f\(_i\)h] / ∑f\(_i\) + ∑Af\(_i\) / ∑f\(_i\)

x̄ = h[∑u\(_i\)f\(_i\)] / ∑f\(_i\) + A∑f\(_i\) / ∑f\(_i\)

x̄ = A + h[∑u\(_i\)f\(_i\)] / ∑f\(_i\)

Steps to be followed while applying the step deviation method are given below,

• Create a table containing five columns as given below,
  Column 1- Class interval.
  Column 2- Classmarks (corresponding), denoted by x\(_i\). Take the central value from the class marks as the Assumed Mean and denote it as A.
  Column 3- Calculate the corresponding deviations given as, i.e. di = x\(_i\) - A
  Column 4- Calculate the values of u\(_i\) using the formula, u\(_i\) = d\(_i\)/h, where h is the class width.
  Column 5- Frequencies (f_i) (corresponding)
• Find the Mean of u\(_i\) = ∑x\(_i\)u\(_i\) / ∑u\(_i\)
• Finally, calculate the Mean by adding the assumed mean A to the product of class width(h) with mean of u\(_i\)

Let us look at an example to understand this better.

Example: Consider the following example to understand this method. Find the mean of the following using the step-deviation method.

Solution: To find the mean, we first have to find the class marks and decide A (assumed mean). Let A = 35 Here h (class width) = 10

Using mean formula:

x̄ = A + h × ∑x\(_i\)u\(_i\) / ∑u\(_i\) = 35 + (16/50) ×10 = 35 + 3.2 = 38.2

Mean = 38.

Related Topics

Listed below are a few topics related to the step deviation method, take a look.

• Mean of Grouped Data
• Range
• Mean, Median, Mode

What is Meant by Step Deviation Method?

Step deviation method is the extended method of the assumed or short-cut method of obtaining the mean of large values. These values of deviations are divisible by a common factor that is reduced to a smaller value. The step deviation method is also called a change of origin or scale method.

What is the Formula for Step Deviation Method?

To obtain the mean of large values, the formula of the step deviation method is applied. It is as follows:

Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],

where,

• A is the assumed mean
• h is the class size
• u\(_i\) = d\(_i\)/h
• f\(_i\) is the frequency
• d\(_i\) = x\(_i\) - A
• x\(_i\) is the midpoint of the class interval

When Should We Use Step Deviation Method?

To obtain the mean of data, there are three methods that can be used direct method, assumed method, and step deviation method. The direct method helps in obtaining the mean when the values are small, the assumed method helps in obtaining the mean when the values are in between, and the step deviation method is used for obtaining the mean when large values.

How Do You Calculate Step Deviation Method?

The following steps are used to calculate the mean using this method:

• Calculate the class marks of each class x\(_i\)
• Let A denote the assumed mean of the data.
• Find u\(_i\) =(x\(_i\)−A)/h, where h is the class size.
• Use the formula: A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]

What is the Common Factor in Step Deviation Method?

The common factor is the greatest positive number that divides the deviations. For instance, if deviations are –20, –10, 0, 40, and 60, then the common factor is 10 because 10 is the greatest positive number that divides the five numbers.

What is Step Deviation Method for Calculating Arithmetic Mean?

Arithmetic mean for grouped data can be calculated by using step deviation method when the values are large. The formula is:

Step Deviation of Mean = A + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],

where,

• A is the assumed mean
• h is the class size
• u\(_i\) = d\(_i\)/h
• f\(_i\) is the frequency
• d\(_i\) = x\(_i\) - A
• x\(_i\) is the midpoint of the class interval