The arithmetic mean in statistics, 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 often 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. Show
We will be focusing here only on Arithmetic Mean. Let’s first understand the meaning of the term "Mean", followed by arithmetic with a few solved examples in the end.
What Is Arithmetic Mean?Arithmetic mean is often referred to as the mean or arithmetic average. It 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 (AM) for evenly distributed numbers is equal to the middlemost number. Further, the AM 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 AM. You can learn more about the difference between average and mean here. Arithmetic Mean FormulaThe 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. The below-given image presents the general formula to find the arithmetic mean: Properties of Arithmetic MeanLet 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. Calculating Arithmetic Mean for Ungrouped DataHere the arithmetic mean is calculated using the formula: Mean x̄ = Sum of all observations / Number of observations Example: Compute the arithmetic 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. Calculating Arithmetic Mean for Grouped DataThere are three methods (Direct method, Short-cut method, and Step-deviation method) to calculate 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) the 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. Direct Method for Finding the Arithmetic MeanLet 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 Example I (discrete grouped data): Find the mean of the following distribution:
Solution:
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:
Solution: 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. Note: Class Mark = (Upper limit + Lower limit) / 2
x̄ = ∑xifi/ ∑fi = 1390/35 = 39.71. We have, ∑fi = 35 and ∑xifi = 35 Mean = 39.71 Short-cut Method for Finding the Arithmetic MeanThe 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.
Now we use the formula, x̄ = A + (∑fidi/∑fi) = 62.5 + (−25/100) = 62.5 − 0.25 = 62.25 ∴ Mean = 62.25 Step Deviation Method for Finding the Arithmetic MeanThis is also called the change of origin or scale method. The following steps describe this method: Step 1: Calculate the class marks of each class (xi). Step 2: Let A denote the assumed mean of the data. Step 3: Find ui = (xi−A)/h, where h is the class size. Step 4: Use the formula: x̄ = A + h × (∑fiui/∑fi) Example: Consider the following example to understand this method. Find the arithmetic 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
Using mean formula: x̄ = A + h × (∑fiui/∑fi) =35 + (16/50) ×10 = 35 + 3.2 = 38.2 Mean = 38. Advantages of Arithmetic MeanThe 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.
After having discussed some of the major advantages of arithmetic mean, let's understand its limitations. Disadvantages of Arithmetic MeanLet us now look at some of the disadvantages/demerits of using the arithmetic mean.
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.
☛Tips and tricks on arithmetic mean:
☛Related Topics: Given below is the list of topics that are closely connected to the arithmetic mean. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
FAQs on Arithmetic MeanWhat Is Arithmetic Mean Definition?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 mean as 212/4 = 53 How to Calculate the Arithmetic Mean?In statistics, arithmetic mean (AM) 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 AM 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 (which is nothing but AM or mean) is calculated as follows: AM = (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 (AM) are as follows:
What Is the Sum of Deviations from Arithmetic Mean?The sum of deviations from the arithmetic mean is equal to zero. What is the Arithmetic Mean Formula Used for Grouped Data?Then, arithmetic mean for ungrouped data is calculated using the formula: x̄ = (x₁f₁+x₂f₂+......+xₙfₙ) / ∑fi What is the Arithmetic Mean Formula Used for Ungrouped Data?Then, arithmetic mean for grouped data is calculated using the formula: Mean x̄ = Sum of all observations / Number of observations. Is the total of the sum of all values in a collection of numbers divided by the number of items in a collection?MEAN-the sum of a set of numbers divided by the number of items in the set, also referred to as the average. MEDIAN- with a list of numbers that are arranged in numerical order, the median is the middle number. If there are two middle numbers, the median is the average of the two middle numbers.
What is calculated by summing the values of the observations in the sample and then dividing the sum by the number of observations in the sample?Arithmetic mean refers to the average amount in a given group of data. It is defined as the summation of all the observation is the data which is divided by the number of observations in the data.
What is the value calculated by adding all the values in a data set and dividing the sum by the number of values?To calculate mean, add together all of the numbers in a set and then divide the sum by the total count of numbers.
What is obtained by dividing the total values of the various items by their number?The mean of a data set is the sum of the values divided by the number of values. The median of a data set is the middle value when the values are written in numerical order. If a data set has an even number of values, the median is the mean of the two middle values.
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