In statistics, the Arithmetic Mean (AM) or called average is the ratio of the sum of all observations to the total number of observations. The arithmetic mean can also inform or model concepts outside of statistics. In a physical sense, the arithmetic mean can be thought of as a centre of gravity. From the mean of a data set, we can think of the average distance the data points are from the mean as standard deviation. The square of standard deviation (i.e. variance) is analogous to the moment of inertia in the physical model.

One of the characteristics of any given frequency distribution is central tendency. The characteristic by virtue of which the values of a variable tend to cluster around at the central part of the frequency distribution is called central tendency. In statistics, arithmetic mean (AM) is defined as the ratio of the sum of all the properties of arithmetic mean 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. 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.

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. In real life, the importance of displaying a single value for a huge amount of data makes it simple to examine and analyse a set of data and deduce necessary information from it. Sometimes a measure of central tendency is called a measure of location because it locates the position of the frequency distribution on the axis of the variable. Arithmetic Mean is a fundamental concept in mathematics, statistics, and various other fields.

- Also, the arithmetic mean fails to give a satisfactory average of the grouped data.
- The arithmetic mean is defined as the average value of all the data set, it is calculated by dividing the sum of all the data set by the number of the data sets.
- For evenly distributed terms arranged in ascending or descending order arithmetic mean is the middle term of the sequence.
- If all the observations assumed by a variable are constants, say “k”, then arithmetic mean is also “k”.
- The arithmetic mean is commonly referred to as the average, because it is a common measure of central tendency among a data set.

## Arithmetic Mean – Definition, Formula, and Examples

We see the use of representative value quite regularly in our daily life. When you ask about the mileage of the car, you are asking for the representative value of the amount of distance travelled to the amount of fuel consumed. This doesn’t mean that the temperature in Shimla in constantly the representative value but that overall, it amounts to the average value. Average here represents a number that expresses a central or typical value in a set of data, calculated by the sum of values divided by the number of values.

## What is the difference between the arithmetic mean, median, and mode about outliers?

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. You can use arithmetic mean calculator to find the mean of grouped and ungrouped data. For ungrouped data, the arithmetic mean is relatively easy to find.

## Short-cut Method for Finding the 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. 5) It is least affected by the presence of extreme observations. For example, if the height of every student in a group of 10 students is 170 cm, the mean height is, of course 170 cm. Here we will learn about all the properties andproof the arithmetic mean showing the step-by-step explanation.

## Arithmetic Mean Formula

Let’s learn to find the arithmetic mean for grouped and ungrouped data. Where,n is number of itemsA.M is arithmetic meanai are set values. An examination was held to decide about the award of a scholarship in an institution.

## What is the Arithmetic Mean Formula for Ungrouped Data?

Sometimes it doesn’t represent the situation accurately enough. Say there are 10 students in the class and they recently gave a test out of 100 marks. The average marks obtained by a class of 70 students was found to be 65. Later on it was detected that the marks of one student was wrongly recorded as 85 instead of 58. The arithmetic mean is a good parameter when the values of the data set are minorly different. But if there are very high or low values present, the arithmetic mean will not be a good option.

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. Find the arithmetic mean for a class of eight students, who scored the following marks for a maths test out of 20. This value is called weighted Arithmetic mean or simple weighted mean (W.P), and it is donated by XÌ„w. Its formula is derived from the arithmetic mean and that is why, both A.P and W.M are learned together. The arithmetic mean can be visualized as a balancing point on a scale.

Range, as the word suggests, represents the difference between the largest and the smallest value of data. This helps us determine the range over which the data is spread—taking the previous example into consideration once again. There are 10 students in the class, and they recently gave a test out of 100 marks.

The marks obtained by 3 candidates (A, B, and C) out of 100 are given below. If the candidate getting the average score is to be awarded the scholarship, who should get it. Also, the arithmetic mean fails to give a satisfactory average of the grouped data. Arithmetic mean and Average are different names for the same thing.

This is because it is highly skewed by the outliers, values relatively very high or lower than the rest of the data. But in day-to-day life, people often skip the word arithmetic or simply use the layman’s term “average”. In some document formats (such as PDF), the symbol may be replaced by a “¢” (cent) symbol when copied to a text processor such as Microsoft Word.

The 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. It allows us to know the center of the frequency distribution by considering all of the observations. 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.