We come across statements like "the average monthly income of a family is ₹15,000 or the average monthly rainfall of a place is 1000 mm" quite often. Average is typically referred to as Arithmetic Mean but the fact is one of the types of average. In statistics, the Arithmetic Mean (AP) is nothing but the ratio of all observations to the total number of observations in a data set. Some of the examples include the average rainfall of a place, the average income of employees in an organization.
We will be focussing here only on Arithmetic Mean. Let’s first understand the meaning of the term "Mean", followed by arithmetic with few solved examples in the end.
What Is Arithmetic Mean?
Arithmetic mean is often referred to as the mean or arithmetic average. Arithmetic mean is calculated by adding all the numbers in a given data set and then dividing it by the total number of items within that set. The arithmetic mean for evenly distributed numbers is equal to the middle most number. Further the arithmetic mean is calculated using numerous methods, which is based on the amount of the data, and the distribution of the data.
Let's discuss an example where we find the use of arithmetic mean. The mean of the numbers 6, 8, 10 is 8 since 6 + 8 + 10 = 24 and 24 divided by 3 [there are three numbers] is 8. The arithmetic mean maintains its place in calculating a stock’s average closing price during a particular month. Let's assume there are 24 trading days in a month. How can we calculate the mean? All you need to do is take all the prices, add them up, and divide by 24 to get the arithmetic mean. You can learn more about the Difference Between Average and Mean here.
Arithmetic Mean Formula
The general formula to find the arithmetic mean of a given data is:
Mean (x̄) = Sum of all observations/ Number of observations
It is denoted by x̄, (read as x bar). Data can be presented in different forms. For example, when we have raw data like the marks of a student in five subjects, we add the marks obtained in the five subjects and divide the sum by 5, since there are 5 subjects in total.
Now consider a case where we have huge data like the heights of 40 students in a class or the number of people visiting an amusement park across each of the seven days of a week.
Will it be convenient to find the arithmetic mean with the above method? The answer is a big NO! So, how can we find the mean? We arrange the data in a form that is meaningful and easy to comprehend. Let's understand how to compute the arithmetic average in such cases. We will study more in detail about finding the arithmetic mean for ungrouped and grouped data.
Arithmetic Mean for Ungrouped Data
Here the arithmetic mean is calculated using the formula:
Mean x̄ = Sum of all observations / Number of observations
Example: Compute the mean of the first 6 odd, natural numbers.
Solution: The first 6 odd, natural numbers: 1, 3, 5, 7, 9, 11
x̄ = (1+3+5+7+9+11) / 6 = 36/6 = 6.
Thus, the arithmetic mean is 6
Arithmetic Mean for Grouped Data
There are three methods (Direct method, Short-cut method, and Step-deviation method) to find the arithmetic mean for grouped data. The choice of the method to be used depends on the numerical value of xi and fi. xi is the sum of all data inputs and fi is the sum of their frequencies. ∑ (sigma) symbol represents summation. If xi and fi are sufficiently small, the direct method will work. But, if they are numerically large, we use the assumed arithmetic mean method or step-deviation method. In this section, we will be studying all three methods along with examples.
1. Direct Method for Finding the Arithmetic Mean
Let x1, x2, x3 ……xn be the observations with the frequency f1, f2, f3 ……fn.
Then, mean is calculated using the formula:
x̄ = (x1f1+x2f2+......+xnfn) / ∑fi
Here, f1+ f2 + ....fn = ∑fi indicates the sum of all frequencies.
Example I (discrete grouped data): Find the mean of the following distribution:
|10||7||10×7 = 70|
|30||8||30×8 = 240|
|50||10||50×10 = 500|
|70||15||70×15 = 1050|
|89||10||89×10 = 890|
Add up all the (xifi) values to obtain ∑xifi. Add up all the fi values to get ∑fi
Now, use the mean formula.
x̄ = ∑xifi / ∑fi = 2750/50 = 55
Mean = 55. The above problem is an example of discrete grouped data.
Let's now consider an example where the data is present in the form of continuous class intervals.
Example II (continuous class intervals): Let's try finding the mean of the following distribution:
When the data is presented in the form of class intervals, the mid-point of each class (also called class mark) is considered for calculating the mean.
The formula for mean remains the same as discussed above.
Class Mark = (Upper limit + Lower limit) / 2
|Class- Interval||Class Mark (xi)||Frequency (fi)||xifi|
x̄ = ∑xifi/ ∑fi = 1390/35 = 39.71We have, ∑fi = 35 and ∑xifi = 35
Mean = 39.71
The below-given image presents the general formula to find the arithmetic mean:
2. Short-cut Method for Finding the Arithmetic Mean
The short-cut method is called as assumed mean method or change of origin method. The following steps describe this method.
Step1: Calculate the class marks (mid-point) of each class (xi).
Step2: Let A denote the assumed mean of the data.
Step3: Find deviation (di) = xi – A
Step4: Use the formula:
x̄ = A + (∑fidi/∑fi)
Example: Let's understand this with the help of the following example. Calculate the mean of the following using the short-cut method.
Solution: Let us make the calculation table. Let the assumed mean be A = 62.5
Note: A is chosen from the xi values. Usually, the value which is around the middle is taken.
Classmark/ Mid-points (xi)
|fi||di = (xi - A)||fidi|
Now we use the formula,
x̄ = A + (∑fidi/∑fi) = 62.5 + (−25/100) = 62.5 − 0.25 = 62.25
∴ Mean = 62.25
3. Step Deviation Method for Finding the Arithmetic Mean
This is also called the change of origin or scale method. The following steps describe this method:
Step1: Calculate the class marks of each class (xi).
Step2: Let A denote the assumed mean of the data.
Step3: Find ui=(xi−A)/h, where h is the class size.
Step4: Use the formula:
x̄ = A + h × (∑fiui/∑fi)
Example: Consider the following example to understand this method. Find the mean of the following using the step-deviation method.
Solution: To find the mean, we first have to find the class marks and decide A (assumed mean). Let A = 35 Here h (class width) = 10
|0-10||5||4||-3||4 x (-3)=-12|
|10-20||15||4||-2||4 x (-2)=-8|
|20-30||25||7||-1||7 x (-1)=-7|
|30-40||35||10||0||10 x 0= 0|
|40-50||45||12||1||12 x 1=12|
|50-60||55||8||2||8 x 2=16|
|60-70||65||5||3||5 x 3=15|
Using mean formula:
x̄ = A + h × (∑fiui/∑fi) =35 + (16/50) ×10 = 35 + 3.2 = 38.2
Mean = 38.
Properties of Arithmetic Mean
Let us have a look at some of the important properties of the arithmetic mean. Suppose we have n observations denoted by x1, x2, x3, ….,xn and x̄ is their arithmetic mean, then:
1. If all the observations in the given data set have a value say ‘m’, then their arithmetic mean 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 deviations of a set of observations 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 increases or decreases by a fixed value, then the mean also increases/decreases by the same number. 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.
4. If each value in the data gets multiplied or divided by a fixed value, then the mean also gets multiplied or divided by the same number. Let the mean of x1, x2, x3 ……xn be X̄, then the mean of kx1, kx2, kx3 ……xn+k will be kX̄. Similarly, the mean of x1/k, x2/k, x3/k ……xn/k will be X̄k.
Note: While dividing each value by k, it must be a non-zero number as division by 0 is not defined.
Advantages of Arithmetic Mean
The uses of arithmetic mean are not just limited to statistics and mathematics, but it is also used in experimental science, economics, sociology, and other diverse academic disciplines. Listed below are some of the major advantages of the arithmetic mean.
1. As the formula to find the arithmetic mean is rigid, the result doesn’t change. Unlike the median, it doesn’t get affected by the position of the value in the data set.
2. It takes into consideration each value of the data set.
3. Finding arithmetic mean is quite simple; even a common man having very little finance and math skills can calculate it.
4. It’s also a useful measure of central tendency, as it tends to provide useful results, even with large groupings of numbers.
5. It can be further subjected to many algebraic treatments, unlike mode and median. For example, the mean of two or more series can be obtained from the mean of the individual series.
6. The arithmetic mean is widely used in geometry as well. For example, the coordinates of the “centroid” of a triangle (or any other figure bounded by line segments) are the arithmetic mean of the coordinates of the vertices.
After having discussed some of the major advantages of arithmetic mean, let's understand its limitations.
Disadvantages of Arithmetic Mean
Let us now look at some of the disadvantages/demerits of using the arithmetic mean.
1.The strongest drawback of arithmetic mean is that it is affected by extreme values in the data set. To understand this, consider the following example. It’s Ryma’s birthday and she is planning to give return gifts to all who attend her party. She wants to consider the mean age to decide what gift she could give everyone. The ages (in years) of the invitees are as follows: 2, 3, 7, 7, 9, 10, 13, 13, 14, 14 Here, n = 10. Sum of the ages = 2+3+7+7+9+10+13+13+14+14 = 92. Thus, mean = 92/10 = 9.2 In this case, we can say that a gift that is desirable to a kid who is 9 years old may not be suitable for a child aged 2 or 14
2. In a distribution containing open-end classes, the value of the mean cannot be computed without making assumptions regarding the size of the class.
|Less than 15||20|
|More than 45||6|
We know that to find the arithmetic mean of grouped data, we need the mid-point of every class. As evident from the table, there are two cases (less than 15 and 45 or more) where it is not possible to find the mid-point and hence, arithmetic mean can’t be calculated for such cases.
3. It's practically impossible to locate the arithmetic mean by inspection or graphically.
4. It cannot be used for qualitative types of data such as honesty, favorite milkshake flavor, most popular product, etc.
5. We can't find the arithmetic mean if a single observation is missing or lost.
Tips and tricks on arithmetic mean:
- If the number of classes is less and the data has values with a smaller magnitude, then the direct method is preferred out of the three methods to find the arithmetic mean.
- Step deviation works best when we have a grouped frequency distribution in which the width remains constant for every class interval and we have a considerably large number of class intervals.
Solved Examples on Arithmetic Mean
Example 1: If the arithmetic mean of 2m+3, m+2, 3m+4, 4m+5 is m+2, find m.
Solution:The data contains 4 observations : 2m+3,m+2,3m+4,4m+52m+3,m+2,3m+4,4m+5
So, n = 4n = 4
Sum of 4 observations = [(2m+3)+(m+2)+(3m+4)+(4m+5)]/4 = (10x+14)/4
Mean = (10m + 14)/4
∴ m + 2 = (10m + 14)/4
4 × (m+2) = (10m + 14)
4m+8 = 10m + 14
−6m = −6
m = −1
Answer: ∴ m = −1
Example 2: The mean monthly salary of 10 workers of a group is ₹1445. One more worker whose monthly salary is ₹1500 has joined the group. Find the arithmetic mean of the monthly salary of 11 workers of the group.
Solution: Here, n = 10, x̄=1445
x̄ = ∑xi/n
∴∑xi = x̄ × n
∑xi = 1445 × 10 = 14450
(Total salary of 10 workers = ₹14450)
Total salary of 11 workers = 14450 + 1500 = ₹15950
Average salary of 11 workers = 15950/11 = 1450
Answer: ∴ Average monthly salary of 11 workers = ₹1450
FAQs on Arithmetic Mean
What Is an Arithmetic Mean?
The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series. For example, take the numbers 34, 44, 56, and 78. The sum is 212. To find the arithmetic mean we will divide the sum 212 by 4(total numbers), this will give us the arithmetic mean as 212/4 = 53
How to Calculate the Arithmetic Mean?
In statistics, arithmetic mean is defined as the ratio of the sum of all the given observations to the total number of observations. For example, if the data set consists of 5 observations, the arithmetic mean can be calculated by adding all the 5 given observations divided by 5.
How to Find the Arithmetic Mean Between 2 Numbers?
Add the two given numbers and then divide the sum by 2. For example, 2 and 6 are the two numbers, the arithmetic mean is calculated as follows: Arithmetic Mean = (2+6)/2 = 8/2 = 4
What Are the Types of Arithmetic Mean?
In mathematics, we deal with different types of means such as arithmetic mean, harmonic mean, and geometric mean.
What Is the Use of Arithmetic Mean?
The arithmetic mean is a measure of central tendency. It allows us to know the center of the frequency distribution by considering all of the observations.
What Are the Characteristics of Arithmetic Mean?
Some important properties of the arithmetic mean are as follows:
- The sum of deviations of the items from their arithmetic mean is always zero, i.e. ∑(x – X) = 0.
- The sum of the squared deviations of the items from Arithmetic Mean (A.M) is minimum, which is less than the sum of the squared deviations of the items from any other values.
- If each item in the arithmetic series is substituted by the mean, then the sum of these replacements will be equal to the sum of the specific items.
- If the individual values are added or subtracted with a constant, then the arithmetic mean can also be added or subtracted by the same constant value.
- If the individual values are multiplied or divided by a constant value, then the arithmetic mean is also multiplied or divided by the same value.
What Is the Sum of Deviations from Arithmetic Mean?
The sum of deviations from the arithmetic mean is equal to zero.