Arithmetic Mean

 The arithmetic mean, also known as the average, is a measure of central tendency that represents the typical or average value of a set of data. It is calculated by adding up all the values in a data set and dividing the total by the number of values. The formula for the arithmetic mean for a data set with n values is:

x̄ = (x₁ + x₂ + ... + xn) / n

where x̄ represents the arithmetic mean, and x₁, x₂, ..., xn represent the individual values in the data set.

For ungrouped data, the formula for the arithmetic mean is simply the sum of all the values in the data set divided by the number of values.

Example of arithmetic mean for ungrouped data: Consider the following set of data: 5, 6, 3, 8, 7 The arithmetic mean is: x̄ = (5 + 6 + 3 + 8 + 7) / 5 = 29 / 5 = 5.8

For grouped data, the formula for the arithmetic mean is:

x̄ = ∑ (fi * xi) / ∑ fi

where x̄ represents the arithmetic mean, fi represents the frequency of each class interval, and xi represents the midpoint of each class interval.

Example of arithmetic mean for grouped data: Consider the following frequency distribution table:

Class Interval	Frequency
0-10	5
10-20	10
20-30	15
30-40	8
40-50	2

The midpoint of each class interval can be calculated as follows:

Class Interval	Frequency	Midpoint
0-10	5	5
10-20	10	15
20-30	15	25
30-40	8	35
40-50	2	45

The arithmetic mean can be calculated as: x̄ = ((55) + (1015) + (1525) + (835) + (2*45)) / (5 + 10 + 15 + 8 + 2) = 690 / 40 = 17.25

Therefore, the arithmetic mean for the given frequency distribution table is 17.25.