Arithmetic Means

Introduction

The arithmetic mean is sometimes known as the mean or the arithmetic average. It’s determined by multiplying the total number of items in a data collection by the total number of numbers in that set. The arithmetic mean (AM) equals the middle number for uniformly distributed numbers. Furthermore, the AM is estimated using various approaches based on the amount of data and how it is distributed.

Consider an instance where the arithmetic mean is used in the study material. Because 6 + 8 + 10 = 24 and 24 divided by 3 [there are three numbers] equals 8, the mean of the digits 6, 8, and 10 is 8. The arithmetic mean continues to play a role in determining a stock’s average closing price for a given month.

Arithmetic Mean Formula

Use the following formula to get the arithmetic mean of a set of data:

The mean (x) is calculated by dividing the total number of observations by the number.

The symbol for it is x. (read as x bar). Data can be presented in a number of different ways. We sum the grades in each subject and divide the total by five because there are five topics when we receive raw data, such as a student’s grades in five subjects.

Consider the case where we have a large amount of information, such as the heights of 40 students in a class or the number of people that attended an amusement park on each of the seven days of the week.

Is the approach for calculating the arithmetic mean described above feasible? The answer is no.  So, how can we figure out what the meaning is? We organise the data in a significant and easy way to comprehend. Let’s look at how to calculate the arithmetic average in such cases.

Properties of Arithmetic Mean

If we have n observations represented by x₁, x₂, x₃, ….,xₙ and x̄ is their arithmetic mean, then:

1. If all of the observations in a data set have the same value, say’m,’ then the arithmetic mean of the data set is also ‘m’. Consider the data having 5 observations: 15,15,15,15,15. So, their total = 15+15+15+15+15= 15 × 5 = 75; n = 5. Now, arithmetic mean = total/n = 75/5 = 15
2. The algebraic sum of a set of observations’ departures from their arithmetic mean is zero. (x₁−x̄)+(x₂−x̄)+(x₃−x̄)+…+(xₙ−x̄) = 0. For discrete data, ∑(xi−x̄) = 0. For grouped frequency distribution, ∑f(xi−∑x̄) = 0
3. If each value in the data grows or drops by a certain amount, the mean will similarly increase or decrease by that amount. Let the mean of x₁, x₂, x₃ ……xₙ be X̄, then the mean of x₁+k, x₂+k, x₃ +k ……xₙ+k will be X̄+k.

Advantages of Arithmetic Mean

The arithmetic mean is utilised in experimental science, economics, sociology, and a variety of other academic areas, in addition to statistics and mathematics. Some of the key advantages of the arithmetic mean are listed below.

• Because the formula for calculating the arithmetic mean is fixed, the outcome remains constant. It is unaffected by the position of the value in the data set, unlike the median.
• Each value in the data set is taken into account.
• It’s a good measure of central tendency because it tends to produce useful results even when there are a lot of numbers in a group.
• Unlike mode and median, it can be subjected to various algebraic manipulations. The mean of two or more series, for example, can be calculated using the mean of the individual series.
• The arithmetic mean is also commonly employed in geometry. The arithmetic means of the coordinates of the vertices, for example, is the coordinates of the “centroid” of a triangle (or any other object limited by line segments).

Having gone through the Permutations and Combinations study material, the following are some tips and tricks that you should keep in mind:

• When there are fewer classes and the data has smaller magnitudes, the direct technique is chosen over the other two methods for calculating the arithmetic mean.
• When we have a clustered frequency distribution with a high number of class intervals and the width of each class interval remains constant, step deviation works well.

Conclusion

Arithmetic mean is a widely used method of determining a mean. It could be as quick as adding up a bunch of numbers and then dividing that total by the number of numbers in the series. The arithmetic mean is a term that refers to the average or mean.