What is Assumed Mean Method? What is its Formula?

What is Assumed Mean Method?

The Assumed Mean Method, also known as the Step Deviation Method, is a statistical technique used to calculate the mean, variance, and standard deviation of a dataset. It’s commonly employed when dealing with grouped frequency distributions, where the data is presented in the form of classes or intervals along with their corresponding frequencies.

Here’s how the Assumed Mean Method works:

• Assumption of Mean: In this method, you assume a preliminary or assumed mean value (often denoted by ‘A’ or ‘a’) that lies within the range of the data. The assumed mean should be easy to work with mathematically.
• Deviation: Calculate the deviations of each class (interval) midpoint from the assumed mean. The deviation for each class is the difference between the midpoint of the class and the assumed mean.
• Step Deviation: Divide each deviation by the width of the class interval. This step is known as “step deviation.” It’s done to normalize the deviations and make them comparable across different class intervals.
• Calculate: Multiply the step deviations by their respective frequencies and sum up these products. This sum gives you the total sum of deviations.
• Mean Calculation: Add the total sum of deviations to the assumed mean to obtain the true mean of the dataset.
• Variance and Standard Deviation: Once you have the true mean, you can use it to calculate the variance and standard deviation of the dataset using appropriate formulas.

The Assumed Mean Method is a simplified approach that can be used when the exact individual data points are not available but only the grouped frequency distribution is given. It provides an estimate of the mean, variance, and standard deviation based on the assumption of the mean value. While it may not be as accurate as other methods for calculating these statistics, it can be useful when detailed data is not accessible.

Please note that this method might not be as commonly used as more advanced statistical techniques, especially with the availability of modern software and tools for data analysis.

What is the Assumed Mean Method Formula?

The Assumed Mean Method, also known as the Step Deviation Method, is a statistical method used for finding the mean of a set of data when the values are given along with their respective frequencies. This method is particularly useful when dealing with large datasets. The basic idea behind this method is to assume a preliminary or assumed mean, then calculate the deviations of the values from this assumed mean, and finally use these deviations to find the actual mean.

Here’s the formula for the Assumed Mean Method:

Calculate the assumed mean (A):

where Σ(fx) is the sum of the products of frequencies (f) and corresponding values (x), and Σf is the sum of frequencies.

Calculate the deviations (d) of each value from the assumed mean:

Calculate the products of frequencies (f) and corresponding deviations (d):

Calculate the actual mean (M):

In this formula:

• A: Assumed mean
• f: Frequency of each value
• x: Value from the dataset
• d: Deviation of each value from the assumed mean
• Σ: Summation symbol (Σf represents the sum of frequencies, and Σ(fd) represents the sum of products of frequencies and deviations)

Keep in mind that the Assumed Mean Method is just one of several methods used to calculate the mean for grouped data. It is important to ensure that the assumed mean is reasonably close to the actual mean to obtain accurate results. If the initial assumption is far from the actual mean, the calculations may lead to less accurate results.

Assumed Mean Method Examples

The Assumed Mean Method, also known as the Direct Method, is a statistical technique used to calculate the mean of a data set when the values of the data are grouped into intervals and frequencies are provided for each interval. This method assumes that the data values within each interval are uniformly distributed.

Here are a couple of examples of how to use the Assumed Mean Method to calculate the mean of a grouped data set:

Example 1:

Suppose we have the following grouped data for the weights (in kg) of a sample of individuals:

Let’s assume the assumed mean (midpoint) for each class interval is the value in the middle of the interval.

• Midpoint of 50 – 60: (50 + 60) / 2 = 55
• Midpoint of 60 – 70: (60 + 70) / 2 = 65
• Midpoint of 70 – 80: (70 + 80) / 2 = 75
• Midpoint of 80 – 90: (80 + 90) / 2 = 85
• Midpoint of 90 – 100: (90 + 100) / 2 = 95

Now, we can calculate the mean using the Assumed Mean Method:

Mean = A + (∑f * h) / N

Where:

• A = Assumed Mean (a chosen midpoint, often the midpoint of the entire data range)
• ∑f = Sum of the frequencies
• h = Size of the class interval
• N = Total number of observations

Let’s assume the Assumed Mean (A) is 75 (midpoint of the assumed central interval).

Mean = 75 + ((4 * 10) + (10 * 10) + (15 * 10) + (12 * 10) + (8 * 10)) / (4 + 10 + 15 + 12 + 8) Mean ≈ 80.95 kg

Example 2:

Suppose we have the following grouped data for the heights (in cm) of a sample of students:

Assuming the Assumed Mean (A) is 175 (midpoint of the assumed central interval), we can calculate the mean:

Mean = 175 + ((6 * 10) + (12 * 10) + (18 * 10) + (10 * 10) + (4 * 10)) / (6 + 12 + 18 + 10 + 4) Mean = 178.33 cm

Remember that the Assumed Mean Method assumes a uniform distribution of data within each interval, which may not always hold true for all data sets. Additionally, this method is not appropriate when there are extreme values or outliers present in the data.

How to Calculate Mean using Assumed Mean Method?

The Assumed Mean Method is a statistical technique used to calculate the mean of a dataset when the values of the data are given along with their respective frequencies. This method involves assuming a preliminary or assumed mean value and then making adjustments based on the deviations from this assumed mean.

Here are the steps to calculate the mean using the Assumed Mean Method:

Organize Data: Arrange the data in a tabular form, showing the values, their corresponding frequencies, and the product of value and frequency.

• Assume a Preliminary Mean (A): Choose an initial assumed mean (A) for the calculation. This value is usually a convenient estimate that lies within the range of the dataset.
• Calculate Deviations: Calculate the deviations (d) of each value from the assumed mean (A) using the formula: d = X – A.

Calculate Adjustments: Calculate the product of the deviations (d) and their corresponding frequencies (f) and sum them up. This step is crucial as it allows you to determine how much the preliminary mean needs to be adjusted.

Sum of (d * f) = Σ(d * f)

• Calculate the Adjusted Mean: Adjust the assumed mean (A) based on the sum of (d * f) calculated in the previous step. The adjusted mean (M) is given by:
  M = A + (Σ(d * f) / N)
  Where N is the total number of observations (sum of frequencies: N = f1 + f2 + … + fn).
• Final Step: The adjusted mean (M) obtained in the previous step is the desired mean of the dataset.

Keep in mind that the Assumed Mean Method is iterative. You may need to repeat steps 3 to 5 using the adjusted mean as the new assumed mean until the difference between successive assumed means becomes negligible.

Note: This method is more suitable for hand calculations when dealing with large datasets and requires iterative adjustments. For larger datasets, using software or statistical tools can provide a more efficient and accurate way to calculate the mean.

Types of Assumed Mean Method

Both the Assumed Mean Method and the Step Deviation Method are techniques used in statistical calculations, particularly in the context of finding the mean of a given set of data. These methods are used when the data is presented in the form of a frequency distribution table, where the data values are grouped into intervals or classes along with their respective frequencies.

Here’s a comparison between the two methods in tabular form:

In both methods, the assumed mean is used to simplify the calculations by reducing the number of arithmetic operations. The step deviation method, however, considers the class widths (h) which can provide a more accurate estimate of the mean, especially for distributions with varying class widths or skewed data.

Remember that the choice of method depends on the nature of the data and the assumptions that can be reasonably made about the distribution. If possible, it’s a good practice to compare the results obtained using both methods and choose the one that aligns better with the characteristics of the data.

Solved Examples on Assumed Mean Method

The Assumed Mean Method is a technique used in statistics to calculate the mean of a data set when the individual data points are not given, but the deviations from an assumed mean are provided. Let’s go through a couple of solved examples using normal text explanations and tables.

Example 1: Suppose we have the following data representing the deviations from an assumed mean of 50 for a set of observations:

Using the Assumed Mean Method, we can find the mean of the data.

Step 1: Calculate the total frequency (N): N = 4 + 8 + 12 + 10 + 6 + 5 + 5 = 50

Step 2: Calculate the sum of (f * d): Sum of (f * d) = (-6 * 4) + (-4 * 8) + (-2 * 12) + (0 * 10) + (2 * 6) + (4 * 5) + (6 * 5) = -24 – 32 – 24 + 0 + 12 + 20 + 30 = -18

Step 3: Calculate the assumed mean (A): A = Assumed Mean + (Sum of (f * d) / N) = 50 + (-18 / 50) = 50 – 0.36 = 49.64

So, the calculated mean using the Assumed Mean Method is approximately 49.64.

Example 2: Let’s consider another set of data deviations from an assumed mean of 75:

Using the Assumed Mean Method, we can find the mean of the data.

Step 1: Calculate the total frequency (N): N = 5 + 10 + 15 + 20 + 18 + 15 + 12 = 95

Step 2: Calculate the sum of (f * d): Sum of (f * d) = (-8 * 5) + (-6 * 10) + (-4 * 15) + (-2 * 20) + (0 * 18) + (2 * 15) + (4 * 12) = -40 – 60 – 60 – 40 + 0 + 30 + 48 = -82

Step 3: Calculate the assumed mean (A): A = Assumed Mean + (Sum of (f * d) / N) = 75 + (-82 / 95) = 75 – 0.863 = 74.137

The calculated mean using the Assumed Mean Method is approximately 74.137.

These examples illustrate how to use the Assumed Mean Method to calculate the mean of a data set when only deviations from an assumed mean are provided. The method involves finding the total frequency, calculating the sum of (f * d), and then using these values to determine the assumed mean.