Where, \(a\) is the assumed mean, \(\) denoted the frequency of \(\) class which is having the deviation of \(\) from the assumed mean. 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. Follow this page for any further details related to NCERT examinations. 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.

Properties of arithmetic mean

The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value , or the most likely value . For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

If you have any queries on this page, post your comments in the comment box below and we will get back to you as soon as possible. A. All the approaches related to finding arithmetic mean is important. Students need to practice to be able to identify the correct approach considering the data type. We can calculate the arithmetic mean in three different types of series as listed below. Now, if \(n\) arithmetic numbers are to be inserted between \(a\) and\(b,\) then we first find the common difference \(d\) which will make the sequence as arithmetic progression. Here we will learn about all the properties and proof the arithmetic mean showing the step-by-step explanation.

It distributes the expression as it simplifies the calculation. This property is valid for addition and multiplication not for subtraction and division. The division is the operation that computes the quotient of two numbers. The two values involved in it are known as dividends by the divisor and if the quotient is more than 1 if the dividend is greater than the divisor the result would be a positive number.

In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values. Equality holds if all the elements of the given sample are equal. We hope that the above article on Arithmetic Mean is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams. Now that you are provided with all the necessary information on the arithmetic mean and its formulas, we hope this article is helpful to you.


Where , \(n\) is the number of observations, \( \ldots .\) are the different values of frequency and \( \ldots .\) are the different values of midpoints for ranges. To get more ideas students can follow the below links to understand how to solve various types of problems using the properties of arithmetic mean. To solve different types of problems on average we need to follow the properties of arithmetic mean.

Arithmetic probably has the longest history during the time. It is a method of calculation that is been in use from ancient times for normal calculations like measurements, labeling, and all sorts of day-to-day calculations to obtain definite values. The term got originated from the Greek word “arithmos” which simply means numbers. The Fréchet mean gives a manner for determining the “center” of a mass distribution on a surface or, more generally, Riemannian manifold. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. Angles, times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers.

The power mean is an abstraction of quadratic, arithmetic, geometric, and harmonic means, while the f-mean generalizes the concept further. There are different types of the mean such as arithmetic means, geometric mean and harmonic mean . This type of mean calculation is not fit for data having extremely asymmetrical distributions. Mean is dependent on the change of origin but it is independent of the change of scale. Q.5. In an examination, the mean of marks scored by a class of \(40\) students was calculated as \(72.5\).

Arithmetic Mean of Grouped & Ungrouped Data with Formula & Examples

Or we can say that the placement of adding numbers can be changed but it will give the same results. It states that the operation of addition on the number does not matter what is the order, it will give us the same result even after swapping or reversing their position. 0 is said to be the identity element of addition as while adding 0 to any value it gives the same result. For example, if we add 0 to 11 the result would be the same that is 11.


Assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. Geometric visualization of the mode, median and mean of an arbitrary probability density function. Comparison of the arithmetic mean, median, and mode of two skewed (log-normal) distributions. It’s also a useful means of central tendency, as it tends to give useful results, even with huge groupings of numbers. In the AM calculation, each value of the data set is considered.

Has an advantage in that it is a calculated quantity that is not depending on the order of terms in a series. 5) The presence of extreme observations has the least impact on it. Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average. The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group.

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If all the observations assumed by a variable are constants, say “k”, then arithmetic mean is also “k”. 7) The Sum of the squared deviations of the items from A.M. Is minimum, which is less than the sum of the squared deviations of the items from any other values. Or we can say for a real number, it reverses its sign from positive number to negative and negative number to positive.

The collection is often a set of results from an experiment, an observational study, or a survey. The term “arithmetic mean” is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic. 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. The result is then added to the assumed mean value of the final answer. In the assumed mean method, students need to first assume a certain number within the data as the mean.

Arithmetic Mean: Assumed Mean Method

The mean of a probability distribution is the long-run average value of a random variable with that distribution. The arithmetic mean as the name suggest is the ratio of summation of all observation to the total number of observation present. The arithmetical average of a group of two or more quantities is known as the mean. 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.

As the formula to find the arithmetic mean is rigid, the result does not change. Students need to practice a significant number of sums to be able to prepare themselves for the final paper. In this article, we will cover the arithmetic mean, its properties and most importantly, its use in real life. It is for this reason that it is the most widely used central tendency measure. The average value can vary considerably from most values in the sample and can be larger or smaller than most.

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The properties of arithmetic mean of the squares of the deviations of a set of data is lowest when carried about the mean. Arithmetic mean in simple words 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. There are different approaches that can be used to calculate arithmetic mean and students need to gain the knowledge of when to use which approach.

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The two values involved in the operation of multiplication are known as multiplicand and multiplier. It combines two values that is multiplicand and multiplier to give a single product. While, if the subtrahend is greater than minuend the difference between them will be negative. In geometry, there are thousands of different definitions for the center of a triangle that can all be interpretted as the mean of a triangular set of points in the plane.

For instance, if there are a set of “n” numbers, add the numbers commonly for example a + b + c + d and so on. Similarly, if you multiplied or divided every value of the data set by a specified weight, then the mean is also multiplied/divided by the exact digit. For example the height of 60 students in a class or the number of individuals attending a park over each of the seven days of a week. To estimate the arithmetic average in such cases we need to study the arithmetic mean for ungrouped and grouped data. The arithmetic mean is affected by extreme values in the data set. It’s also a useful measure of central tendency, as it tends to provide useful results, even with a large group of numbers.

The arithmetic mean can be conceived of as a gravitational centre in a physical sense. The average distance the data points are from the mean of a data set is referred to as standard deviation. In the physical paradigm, the square of standard deviation (i.e. variance) is comparable to the moment of inertia. Arr-ith-MET -ik), arithmetic average, or just the mean or average , is the sum of a collection of numbers divided by the count of numbers in the collection.

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