Arithmetic Mean: Definition, Formulas, Examples, and FAQs

When the frequencies divided by N are replaced by probabilities p1, p2, ……,pn we get the formula for the expected value of a discrete random variable. For open end classification, the most appropriate measure of central tendency is “Median. The above properties make “Arithmetic mean” as the best measure of central tendency. To know more about measures of central tendency and arithmetic mean, please download BYJU’S – The Learning App and stay tuned with us.

In the case of larger observations, data can be presented in the form of a frequency table that exhibits the values taken by the variable and the corresponding frequencies. This form of data is called grouped data or discrete frequency distribution. A lot of people seemingly fail to understand the difference between the arithmetic average and the informal use of the word “average” as a synonym to “typical”.

In a physical sense, the arithmetic mean can be thought of as the centre of gravity. For example, you want to visit a place during a particular period of time and want to know the temperature of a place during that period. In such cases, we find the average or mean temperature for days during that period for the past few years. Here we will learn about all the properties andproof the arithmetic mean showing the step-by-step explanation.

  1. In layman’s terms, the mean of data indicates an average of the given collection of data.
  2. The arithmetic mean of the observations is calculated by taking the sum of all the observations and then dividing it by the total number of observations.
  3. However, nowadays we have very powerful and very easy ways to show the whole set of data, the whole distribution, so presenting only the arithmetic mean may be a bad practice.
  4. If any value changes in the data set, this will affect the mean value, but it will not be in the case of median or mode.
  5. Let x₁, x₂, x₃ ……xₙ be the observations with the frequency f₁, f₂, f₃ ……fₙ.

The significance of indicating a single value for a large amount of data in real life makes it easy to study and analyze the collection of data and deduce important information out of it. Let us discuss the arithmetic mean in Statistics and examples in detail. Where ∑ is called sigma which is a Greek letter that represents the summation.

Properties 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. Let’s now consider an example where the data is present in the form of continuous class intervals. Let x₁, x₂, x₃ ……xₙ be the observations with the frequency f₁, f₂, f₃ ……fₙ. 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. Where,n is number of itemsA.M is arithmetic meanai are set values.

Measures of Dispersion – Definition, Formulas & Examples

This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here. 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.

Continuous probability distributions

For instance, the average weight of the 20 students in the class is 50 kg. However, one student weighs 48 kg, another student weighs 53 kg, and so on. This means that 50 kg is the one value that represents the average weight of the class and the value is closer to the majority of observations, which is called mean.

Different items are assigned different weights based on their relative value. In other words, items that are more significant are given greater weights. Arithmetic Mean OR (AM) is calculated by taking the sum of all the given values and then dividing it by the number of values. For evenly distributed terms arranged in ascending or descending order arithmetic mean is the middle term of the sequence.

Thus, the arithmetic mean is used in various scenarios such as in finding the average marks obtained by the student in marks, the average rainfall in any area, etc. Whereas in the second scenario, the range is represented by the difference between the highest value, 75 and the smallest value, 70. The range in the first scenario is represented by the difference between the largest value, 93 and the smallest value, 48. The above formula can also be used to find the weighted arithmetic mean by taking f1, f2,…., fn as the weights of x1, x2,….., xn. 5) It is least affected by the presence of extreme observations.

Arithmetic Mean of Grouped & Ungrouped Data with Formula & Examples

First, add the individual age of all the teachers and then divide the sum by the total number of teachers present in the school. Arithmetic mean in simple words 5 properties of arithmetic mean is often referred to as average and mean. The simplest way to calculate the mean is by adding all the data and dividing it by the total number of data.

The PDF of NCERT books, solution sets and previous year question papers can be found on this page itself. MCQ Test offered by Embibe is curated considering the revised CBSE textbooks, exam patterns and syllabus for the year 2022. The mock tests will hence help the students get access to a range of questions that will contribute towards strengthening their preparations. It is essential for the students to engage in self-analysis and identify their strengths and weaknesses appropriately. The feedback of the mock tests is AI influenced, which improves the accuracy of the analysis.

In layman’s terms, the mean of data indicates an average of the given collection of data. It is equal to the sum of all the values in the group of data divided by the total number of values. In this formula, the deviation of all values from the mean is calculated followed by the summation of the deviation divided by the total number of observations.

We know that to calculate the mean first we need to find the sum of all the observations. To calculate the mean first we need to find the sum of all the observations. E.g. the average between 5° and 355° is 180°, but a more appropriate average might be 0° as it is between the two on a circle. The preferred approach in this case may be to convert the angles to corresponding units on a point circle, e.g. α to (cosα, sinα). After converting from polar to Cartesian coordinates, compute the arithmetic mean for the points, then convert back to polar coordinate.

When looking for a “typical” value, you might want to examine a percentile, say the middle 50% or 60% or 80% and measure their arithmetic average, to get a better idea of what “typical” is. While calculating the simple arithmetic mean, it is assumed that each item in the series has equal importance. There are; however, certain cases in which the values of the series observations are not equally important. A simple arithmetic mean will not accurately represent the provided data if all the items are not equally important. Thus, assigning weights to the different items becomes necessary.

The sum of this product is obtained and finally, by dividing the sum of this product by the sum of frequencies we will obtain the arithmetic mean of the continuous frequency distribution. In the case of open end class intervals, we must assume the intervals’ boundaries, and a small fluctuation in X is possible. This is not the case with median and mode, as the open end intervals are not used in their calculations. 8) If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.) It is amenable to mathematical treatment or properties. The following steps are used to compute the arithmetic mean by the step deviation method. Let n be the number of observations in the operation and n1, n2, n3, n4, …, nn be the given numbers.

With this article you will be able to answer questions like what is the arithmetical mean. The formula for ungrouped and grouped data along with solved examples/ questions. It allows us to know the center of the frequency distribution by considering all of the observations. Arithmetic mean is one of the measures of central tendency which can be defined as the sum of all observations to be divided by the number of observations. In the case of individual observations, i.e. ungrouped data, the following are used to find the median. The mean, also known as the “Arithmetic Mean” of a group of observations is the value that is equally shared out among all the observations.

Let us understand the arithmetic mean of ungrouped data with the help of an example. The deviations of the observations from arithmetic mean (x – x̄) are -20, -10, 0, 10, 20. If all the observations assumed by a variable are constants, say “k”, then arithmetic mean is also “k”. After calculating the class mark, the mean is calculated as discussed earlier. This method of calculating the mean is known as the direct method.

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