What is Simple Arithmetic Mean?

Last Updated : 03 Jul, 2023

Arithmetic Mean is one approach to measure central tendency in statistics. This measure of central tendency involves the condensation of a huge amount of data to a single value. Arithmetic mean can be determined using two methods; viz., Simple Arithmetic Mean and Weighted Arithmetic Mean.

Meaning of Simple Arithmetic Mean

Simple Arithmetic Mean is the sum of a set of numbers divided by the total number of values. It is also referred to as the average. For instance, if there are four items in a series, i.e. 6, 7, 8, and 9. The simple arithmetic mean is (6 + 7 + 8 + 9) / 4 = 7.5.

The calculation of the arithmetic mean can be done in the following series:

Individual Series;
Discrete Series; and
Continuous Series

(1) Individual Series

The series in which the items are listed singly is known as Individual Series. In simple terms, a separate value of the measurement is given to each item. For example, if the weight of 10 students in a class is given individually, then the resultant series will be an individual series.

Arithmetic Mean in Individual Series

The mean is calculated by adding up all the observations and dividing it by the total number of observations in the set. There are three ways to determine the arithmetic mean of an individual series:

Direct Method;
Short-Cut Method; and
Step Deviation Method.

1. Direct Method

This method involves adding up all the units and dividing the total by the number of items. The resultant quotient becomes the arithmetic mean.

Steps of Direct Method:

1. Assume that there are X₁, X₂,………………X_n items (observations).

2. Determine the sum, or ΣX, by adding the values of all the items.

3. Determine N; i.e., the total number of items in the series.

4. Finally, to get the mean value, divide the total value of all items (ΣX) by the total number of items (N), i.e.

$\bar{X}=\frac{\sum{X}}{N}$

{Where, X̄ = Arithmetic Mean; ΣX = Sum of all the values of items; N = Total number of items}

Example: Calculate the arithmetic mean of the weight of 5 students in a class by using Direct Method.

Solution:

Total Weight (ΣX) = 225, Total number of students (N) = 5

$\bar{X}=\frac{\sum{X}}{N}$

$\bar{X}=\frac{225}{5}$

Arithmetic Mean = 45 kg

2. Short-Cut Method

This method involves assuming any figure of the data set as mean and calculating deviation on its basis. When there are a large number of observations or when it becomes difficult to calculate the arithmetic mean by the direct method, then the Short-Cut method is used. This method is also known as the Assumed Mean Method.

Steps of Short-Cut Method

1. Assume that there are X₁, X₂,………………X_n items (observations).

2. Select an item of the series as the Assumed Mean (A).

3. Determine the series’ deviations (d) from the assumed mean (A); i.e., subtract A from each item of the series to get X – A.

4. Write the total number of deviations and denote it as Σd.

5. Determine N, which implies the total number of items in the series.

6. Finally, use the formula below to calculate the mean value:

$\bar{X}=A+\frac{\sum{d}}{N}$

{X̄ = Arithmetic Mean; A = Assumed Mean; d = X – A; i.e., deviations of variables from assumed mean; Σd = Σ(X-A); i.e., a sum of deviations of variables from assumed mean; N =Total number of items}

Example: Calculate the arithmetic mean of the weight of the 5 students in the class by using the assumed mean method.

Solution:

$\bar{X}=A+\frac{\sum{d}}{N}$

$\bar{X}=40+\frac{25}{5}$

= 40 + 5

Arithmetic Mean = 45 kg

3. Step Deviation Method

The short-cut method was made simpler by the step deviation method. The deviations are calculated using this method by dividing deviations from the assumed mean by a common factor (C). Then, the value of the arithmetic mean is then calculated by using these step deviations.

Steps of the Step Deviation Method

1. Assume that the items (observations) are X₁, X₂,……………….X_n.

2. Select one item in the series as the assumed mean (A).

3. Subtract the assumed mean (A) from each item in the series to determine the deviations (d) of those items from that mean.

4. Determine the common factor (C) from d and use that value to calculate d′ (step deviations), which equals $\frac{d}{C}$ .

5. Denote the sum of step deviations (d′) as Σd′.

6. Determine N, which is the total number of items in the series.

7. Finally apply the following formula to get the value of the arithmetic mean:

$\bar{X}=A+\frac{\sum{d'}}{N}\times{C}$

{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d = X – A, i.e., Deviations of variables from Assumed Mean; $d'=\frac{X-A}{C}$ ; i.e., Step Deviations (deviations divided by common factor); Σd’ = Sum of Step Deviations; C = Common Factor; N = Total number of items}

Example: Calculate the arithmetic mean of the weight of the 5 students in the class by using the step deviation method.

Solution:

$\bar{X}=A+\frac{\sum{d'}}{N}\times{C}$

$\bar{X}=40+\frac{5}{5}\times{5}$

= 40 + 5

Arithmetic Mean = 45

(2) Discrete Series

In the case of discrete series (ungrouped frequency distribution), the values of variables represent the repetitions. It means that the frequencies are given corresponding to the different values of variables. The total number of observations in a discrete series, N, equals the sum of the frequencies, which is Σf.

Arithmetic mean in discrete series can be calculated by using:

Direct Method;
Short-Cut Method; and
Step Deviation Method.