Mean Formula: Direct, Assumed Mean & Step-deviation Method

Mean of a data set can be calculated by dividing the sum of all data points with the total number of data points in the set. To put it simply, the mean is the arithmetic average of all data points. It is one of the most popular methods used in statistical analysis of data. Mean refers to a quantitative notion that has a lot of meaning in finance and is utilized in various financial areas and company valuation. There are three statistical measures of the central tendency of data, which are mean, median, and mode.

What is Mean in Statistics?

Mean of a data set is derived by dividing all data points by the number of data points in the data set. It's a data point in a collection that represents the aggregate of all the pieces of data in the set. The mean is the most popular and widely used statistical approach for determining the center of a data set. It is a basic but necessary aspect of data analysis of statistics. For example, the population mean is the result which we obtain as the average value of a population set. Occasionally, population statistics are so huge that we cannot do any kind of analysis on it. As a result, in that situation, we collect samples and aggregate them. That sample symbolizes the community, and the sample mean the average of this subset of the data.

Mean = Sum of observations / Number of observations

Note: The average value of a data collection is the value that lies between the lower and upper limits. The average value will not be the same as the data set's value, but it's readings are often the same.

Properties of Mean

• When all elements have the same gap (and the quantity of elements is odd), the mean equals the mid-term. The mean of the beginning and last terms is likewise the average of the sequence's terms.
• When x is multiplied by all of the components, the mean rises by x.
• When you subtract x from all of the things, the mean falls by x.
• If each item is multiplied by x, the mean is multiplied by x as well.
• If each item is divided by x, the mean is divided by x as well.

Real-Life Instances of Mean

In actual life, an example of mean is –

• The mean of a cricketer's runs scored in test series
• Real estate brokers assess the average property prices in a given area.

Mean Formula for Ungrouped Data

The method of calculating the average of data that are ungrouped is as follows:

If there are n samples in a data collection with x1, x2, x3, …, xn, the average of these numbers are: = 

∑xi / n

Here,

n = Number of observations

xi = ith observation, 1 ≤ i ≤ n

∑xi = Sum of observations

Mean Formula for Grouped Data

Depending on the quantity of the data, there are three methods for calculating the mean for grouped data. They are as follows:

• Direct Method
• Assumed Mean Method
• Step-deviation Method

Direct Method

Direct method is the most basic way to calculate the mean of grouped data. The procedures for utilizing the direct technique to obtain the mean for grouped data are outlined below.

Make a table with four columns, as shown below.

• Column 1 - Interval between classes.
• Column 2 - Corresponding class marks, marked by xi.
• Column 3 - Frequencies (fi) in (corresponding)
• Column 4 - xifi (corresponding product of column 2 and column 3)

Determine the mean using the Formula = ∑xifi/∑fi

Assume that x1, x2, x3...., xn are n measurements, each with a frequency of f1, f2, f3,...., fn. This means that x1 happens f1 times, x2 happens f2 times, x3 happens f3 times, and so on. As a result, the direct method's mean calculation formula is:

x = (f1x1 + f2x2 + f3x3 + …. + fnxn) / (f1+f2+f3+…. +fn)

Or

x = (∑n i = 1 fixi) / (∑n i = 1 fi)

Here,

• ∑fixi = total number of samples
• ∑fi = Frequency or observation total

When the number of samples is small, the direct mean method is utilized.

Assumed Mean Method

When the direct technique becomes too time-consuming, we use the assumed mean method to calculate the average of a set of grouped data. We can use the assumed mean method to detect and calculate the mean by following the procedures listed below.

Make a table using the five columns listed below.

• Column 1: Interval between classes.
• Column 2: Classmarks (correlating), represented by xi. Assumed Mean A: Pick the median value from the class marks and indicate it as A.
• Column 3: Determine the relevant deviations using the formula di = xi - A.
• Column 4: Frequencies (fi) (corresponding)
• Column 5: Mean of di = ∑xidi/∑di

Finally, calculate the Mean by adding the assumed mean to the average of the di.

We usually assume a value as the average in this manner (namely, a). This value is used to calculate the deviations that the formula is based on. In addition, the information will be presented as a frequency distribution table with classifications. As a result, the formula for calculating the mean using the assumed mean technique is:

Mean (x) = A + ∑fidi / ∑fi

Here,

• fi = ith class’s frequency
• A = assumed mean
• Σfi = N = Total number of samples

• di = xi – A = ith class’s deviation

• xi = class mark = (higher class limit + lower class limit)/2

The video below explains this:

Assumed Mean Method Detailed Video Explanation:

Step-Deviation Method

The shift of origin and scale method are other names for step deviation. When calculating the mean for grouped data, we use the step deviation approach to simplify the calculations. The following are the steps to take while using the step deviation method:

Make a table with five columns, as shown below.

• Column 1: Interval between classes.
• Column 2: Interval between classes. Classmarks (corresponding), represented by xi. Assumed Mean A: Take the median value from the class marks and indicate it as A.
• Column 3: Determine the corresponding deviations using the formula di = xi – A.
• Column 4: Use the formula ui = di/h to calculate the values of ui, where h is the class width.
• Column 5: Frequencies (fi) (corresponding)

Lastly, add the assumed mean A to the product of class width(h) and mean of ui to get the Mean.

The step-deviation method can be used to find the mean when the data values are large. The formula is as follows:

Mean (¯x) = A + (h∑fiui / ∑fi)

Here,

• fi = ith class’s frequency
• A = assumed mean
• ui = (xi – a)/h
• xi – a = ith class’s deviation
• xi = class mark = (higher class limit + lower class limit)/2
• Σfi = N = Total number of samples