properties of arithmetic mean: What are the properties of mean?

data set

For the full article, see arithmetic. Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average. There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income. The arithmetic mean of any amount of equal-sized number groups together is the arithmetic mean of the arithmetic means of each group. By the extreme values in the set of the data, the AM gets affected.


While the basics of arithmetic can seem difficult, practice makes perfect. By knowing how to add, subtract, multiply, and divide numbers, you’re preparing for many other levels of mathematics and more advanced problem-solving. The commutative, associative and identity properties apply to functions in arithmetic and dictate how numbers relate to one another consistently. There are several arithmetic properties.

Arithmetic Mean: Assumed Mean Method

It is responsive to mathematical treatment or properties. 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.

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Is minimum, which is less than the sum of the squared deviations of the items from any other values. 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.

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. Get answers to the most common queries related to the JEE Examination Preparation. Unlike other measures like as mode and median, it can be subjected to algebraic treatment.

The arithmetic mean or average is calculated by dividing the sum of all the individual values of data series by the total number of items. Arithmetic Mean is the most important concept in statistics. It is one of the measures of central tendency that can be directly described as the sum of all quantities to be divided by the number of quantities. Every time we can’t apply the formula of AM to solve the problems on average or mean or arithmetic mean.

Arithmetic Mean Formula

To properties of arithmetic mean the harmonic mean, all elements of the series must be known. In case of unknown elements, we cannot determine the harmonic mean. Given below are other demerits of harmonic mean. If any of the values of a given series is 0 then its harmonic mean cannot be determined as the reciprocal of 0 doesn’t exist.

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5) The presence of extreme observations has the least impact on it. 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.

3 Properties of Arithmetic mean

When the data is presented in the form of class intervals, the mid-point of each class is considered for calculating the mean. Let’s now consider an example where the data is present in the form of continuous class intervals. 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.

  • If the point is near 2, the figure will tip over to the right.
  • For ungrouped data, the arithmetic mean is relatively easy to find.
  • In the AM calculation, each value of the data set is considered.
  • In a data set, if some observations have more importance as compared to the other observations then taking a simple average Is misleading.

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. Here we will learn about all the properties and proof the arithmetic mean showing the step-by-step explanation. As a counterexample, we pick two numbers 12 and 15. The extreme values in a series greatly affect the harmonic mean.


Its formula is derived from the arithmetic mean and that is why, both A.P and W.M are learned together. In this case, different weights are assigned to different observations according to their relative importance And then the average is calculated by considering weights as well. The sum of the squares of the deviations of a set of data is lowest when carried about the mean.

For instance, if there are a set of “n” numbers, add the numbers commonly for example a + b + c + d and so on. One of the major drawbacks of arithmetic mean is that it is changed by extreme values in the data set. If all the observations in the given data set have a value say ‘y′, then their arithmetic mean is also ‘y′. 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. 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.

It is a foundational branch of mathematics. When arithmetic problems are solved, the order of operations should be used. There are several properties of arithmetic that hold true of addition and multiplication. These properties do not hold for subtraction or division.


It is somewhere between the smallest and largest values in the collection. The definition and the example above point to some properties of the mean. The reciprocal for a number “a”, denoted by 1/a, is a number which when multiplied by “a” yields the multiplicative identity 1. To calculate result you have to disable your ad blocker first. Arithmetic mean is not always practical to use. It is because it is highly skewed by the outliers, values relatively very high or lower than the rest of the data.

Weighted average

To find the middle value, the median is calculated. The difference is on the basis of the importance of outliers. For a data set that is positively skewed, the large value drives A.P up the graph. The sum of the squared deviations from the observations is minimum. This value is called weighted Arithmetic mean or simple weighted mean (W.P), and it is donated by X̄w.

There is not enough to give every child a fourth piece. So, that means that 13 divided by 4 is 3, with a remainder of 1 left over. In other words, the number 4 goes into the number 13 evenly 3 times, with a remainder of 1. As you can imagine, addition is a pretty big part of arithmetic.

The first is the property. Basically, this tells us that 2 + 1 is the same as 1 + 2. The order you follow when adding numbers doesn’t matter. If we are given a data series or a set of observations then the harmonic mean can be defined as the reciprocal of the average of the reciprocal terms. This implies that the harmonic mean of a particular set of observations is the reciprocal of the arithmetic mean of the reciprocals.

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Ans.1 Arithmetic Mean is the ratio of all observations or data to the cumulative number of observations in a data set. 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. If the number of classes is less and the data has values with a smaller measurement, then the direct method is preferred over the three methods to get the arithmetic mean.

In the physical paradigm, the square of standard deviation (i.e. variance) is comparable to the moment of inertia. Arithmetic mean is one of the most important chapters of Maths. It is introduced in lower grades and is referred to as average however, in 10th boards, students are taught different approaches 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 AM can be calculated by adding all the 5 given observations divided by 5. 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.

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