Arithmetic Mean
According to statistics, the arithmetic mean is the ratio of all observations in a data set to the total number of observations. For instance, the average rainfall of a place and the average salary of an organization's employees can be used. A lot of time, we hear statements like "the average family income is ₹15,000 or there is about 1000 mm of rain every month in a certain area.". Average is also referred to as Arithmetic Mean.
What Does Arithmetic Mean?
In mathematics, the arithmetic mean is usually referred to as the mean or as an average. An arithmetic mean is calculated by adding all the numbers in a set of data and then dividing by the number of items included in the set. As a result, the middle number is the arithmetic mean for equally distributed numbers. As an added bonus, there are numerous ways to calculate the arithmetic mean, which is based on the amount of data and the distribution of that data.
Let's take a look at an example where the arithmetic mean is used. Since 6 + 8 + 10 = 24, and 24 divided by 3 (there are three numbers) is 8, the mean of the numbers 6, 8, 10 is 8. During a particular month, the arithmetic mean continues to be used to calculate the average closing price of a stock. In the example, suppose there are 24 trading days in a month. Where would we find the mean? The arithmetic mean can be calculated by adding all the prices together and dividing by 24.
Calculation of Arithmetic Mean
The arithmetic mean is derived using the mathematical formula below.
Where,
A- arithmetic mean or average
n- number of items or terms being averaged
x1- value of every single item in the list of numbers being averaged
A= \[\frac {1} {n}\] × n ∑i=0 xi
Here is a more understandable version of the arithmetic mean formula.
A= \[\frac {S} {N}\]
Where,
A- Arithmetic mean or average.
n- number of items or terms being averaged.
S- The sum total of the numbers in the set of interest being averaged.
Arithmetic Mean for Ungrouped Data
Using the formula below, we calculate the arithmetic mean:
Mean x̄ = Sum of all observations / Number of observations
Example: Calculate the arithmetic mean of the first six odd, natural numbers.
The first six odd, natural numbers are: 1, 3, 5, 7, 9, 11.
The formula \[\frac {(1+3+5+7+9+11)} {6}\] = \[\frac {36} {6}\] = 6.
As a result, the arithmetic mean is 6.
Arithmetic Mean for Grouped Data
To determine the arithmetic mean of grouped data, three methods are available (Direct method, Short-cut method, and Step-deviation method). According to the numerical values of xi and fi, the method to be used must be chosen. The sum of all data inputs is xi, and the sum of their frequencies is fi. ∑ represents the summation. Direct methods work when xi and fi are sufficiently small. When the values are large, we use the assumed arithmetic mean method or the step deviation method.
FAQs on Arithmetic Mean Formula
1. What is Meant by the Arithmetic Mean?
The arithmetic mean is the easiest and extensively used measure of a mean, or what we say average. It simply requires taking the sum of a group of numbers, then dividing that sum total by the count of the numbers used in the series.
For example, take the numbers 9, 21, 46, and 68. The sum is 144. The arithmetic mean is 144 divided by 4 (144/4), or 36.
2. How Many Types of Mean are There?
There are several other types of means that are being widely used such as the geometric mean and harmonic mean. These types of means come into play in certain situations like investing, stock trading analysis and finance. Another example includes the trimmed mean, used when computing economic data such as CPI and CPE.
3. How the Arithmetic Mean Works?
The arithmetic mean surprisingly reserves its position in finance as well. For example, mean earnings approximates essentially are arithmetic mean. Say you want to find out the average earnings expectation of the 21 analysts covering a specific stock. Simply add up all the approximates and divide by 21 to obtain the arithmetic mean.
The same holds true when we seek to find out a stock’s average closing price during a specific month. Say there are 22 trading days in the month. Simply consider all the prices, add them up, and divide by 22 to obtain the average or arithmetic mean.
4. What are the Advantages of Arithmetic Meaning?
In addition to statistics and mathematics, the arithmetic mean is used in experimental science, economics, sociology, and numerous other disciplines. The following are some of the main advantages of using an arithmetic mean.
Since the formula for finding the arithmetic mean is rigid, the result remains the same. Unlike the median, it isn't influenced by how a value is positioned in the data set.
It considers every value in the data set.
Mathematical means are quite easy to find; even a person with no financial or math skills can calculate them.
In addition, it provides useful results even with large number of groupings, making it useful for measuring central tendency.
Contrary to mode and median, it can be further treated algebraically. From the mean of an individual series, for example, it is possible to obtain the mean of two or more series.
5. What are the Disadvantages of Arithmetic Mean?
In the next section, we will look at a few disadvantages/demerits of using the arithmetic mean.
Data sets with extreme values have the greatest tendency to affect the arithmetic mean.
Neither visual inspection nor graphic representation can locate the arithmetic mean.
There is no way to use it to display qualitative data such as honesty, milkshake flavor favorites, most popular products, etc.
If one observation is missing or lost, the arithmetic mean cannot be found.
6. What do Arithmetic mean tips and tricks?
Out of the three methods to find the arithmetic mean, the direct method is preferred when there are fewer classes and values of a smaller magnitude.
A step deviation will work best in scenarios in which we have a grouped frequency distribution, where the width is constant for every interval, and we have a large number of intervals in the frequency distribution.
7. How do you calculate the Arithmetic Mean?
According to statistics, the arithmetic mean is the sum of all observations divided by the number of observations. As an example, if the data set consists of five observations, you can calculate the arithmetic mean by adding all the five observations and dividing them by 5.
8. What are the Uses of Arithmetic Mean?
In mathematics, the arithmetic mean represents the central tendency. By considering all the observations, we can determine the center of the frequency distribution.