Types of Assumed Mean Method
Quantitative Data can be categorized into 2 categories:
- Grouped data are those created by grouping individual observations of a variable so that a frequency distribution table of these groups offers a practical way to summarise or analyze the data.
- Ungrouped data is the data you initially collect from an experiment or study. The information is unorganized—that is, it hasn’t been classified, categorized or in any other way grouped. An ungrouped set of data is basically a list of numbers.
Assumed Mean Method for Grouped Data
Grouped data are data formed by aggregating individual observations of a variable into groups so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data. Grouped data is of the form \(1-10, 11-20, 21, -30\) and so on. Hence, we find its class mark. The midpoint of each class interval is the definition of a class mark. Calculating a class mark is as follows:
\(\text{Class Mark}= \frac{\text{Upper Limit}+\text{Lower Limit}}{2}\)
Thus, assumed mean formula becomes,
\(d_i = x_i – x_0\)
where \(x_i\) is class marks.
\(A = \sum_{i=1}^N f_i d_i\)
Where \(f_i\) is class frequency. The frequency of a class interval is the number of observations that occur in a particular predefined interval. So, for example, if 30 people of weight 55 to 60 appear in our study’s data, the frequency for the 55 – 60 interval is 30.
\(D = \frac{A}{N}\)
Finally,
\(\bar{x} = x_0 + D\)
Assumed Mean Method for Ungrouped Data
Let’s take a look at a set of N data whose mean we need to determine. Each value is denoted by \(x_i\). We make a guess and choose an approximate mean, \(x_0, roughly in the middle of the data. We refer to this as the assumed mean. Let’s now follow the steps we learned earlier. Then, we calculate deviations from the assumed mean, defined as:
We now add up these deviations over all data values.
\(A = \sum_{i=1}^N d_i\)
This is proportionally divided by N, the overall number of observations. The difference between the actual mean and the assumed mean, denoted by the letter \(D\), is the value.
\(D = \frac{A}{N}\)
Finally, the actual mean is obtained as follows.
\(\bar{x} = x_0 + D\)
Learn more about Sum of Harmonic Progression
Difference Between Assumed Mean Method and Step Deviation Method
Step deviation method is the extended version of the shortcut or assumed method for calculating the mean of large values. These deviation values can be divided by a common factor that has been scaled down to a smaller amount. Change of origin or scale method is another name for the step deviation method.
Assumed mean method is used when the data values are small. When the data values are large, the step-deviation method is used to find the mean. The formula is given by: Mean \((\overline{x})=a+h\frac{\sum f_iu_i}{\sum f_i}\). In statistics the assumed mean is a method for calculating the arithmetic mean and standard deviation of a data set. It simplifies calculating accurate values by hand. Sometimes, during the application of the short-cut method for finding the mean we use the step deviation method, the deviation d, is divisible by a common number \(‘h’\).
Learn about Rolle’s Theorem
