Arithmetic Mean Definition, Formulas and Examples

The Arithmetic Mean, also known as the average, is a measure of central tendency that provides a simple yet powerful way to summarize a set of numbers. By calculating the sum of all observations and dividing it by the number of observations, one can easily determine the average or mean value. The arithmetic mean is commonly referred to as the average, because it is a common measure of central tendency among a data set. However, there are other ways of measuring an average, including median and mode, so the term should be clarified if there is any uncertainty as to which average a person is using. There are three methods (Direct method, Short-cut method, and Step-deviation method) to calculate the arithmetic mean for grouped data.

1. For instance, the average weight of the 20 students in the class is 50 kg.
2. The Arithmetic Mean provides a single value that represents the central point of the dataset, making it useful for comparing and summarizing data.
3. While calculating the simple arithmetic mean, it is assumed that each item in the series has equal importance.

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.

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.

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. To calculate the central tendency for the given data set, we use different measures like mean, median, mode and so on. Among all these measures, the arithmetic mean or mean is considered to be the best measure, because it includes all the values of the data set. 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.

Representative Values of Data

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.

How to calculate the arithmetic mean?

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.

Now let’s discuss the three methods for finding the arithmetic mean for grouped data in detail. The term weighted mean refers to the average when different items in the series are assigned different weights based on their corresponding importance. If each of the values of a variable occurs equal number of times, then simple A.M. The arithmetic mean is the overall average of the data. 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. Central Tenancies are measures of location that summarise a dataset by giving a “single quantitative value” within the range of the data values.

Properties of Arithmetic Mean

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.

As it provides a single value to represent the central point of the dataset, making it useful for comparing and summarizing data. This formula is widely applicable, whether dealing with ungrouped data or grouped data. Its simplicity and properties of arithmetic mean utility make it indispensable in fields such as economics, finance, and data analysis. Arithmetic Mean, often referred to simply as the mean or average, is a measure of central tendency used to summarize a set of numbers. Arithmetic mean is used in various scenarios such as in finding the average marks obtained by the student , the average rainfall in any area, etc.

So for both the classes, the results mean something different, but the average for both classes are the same. In the first class, the students are performing very varied, some very well and some not so well whereas in the other class the performance is kind of uniform. Therefore we need an extra representative value to help reduce this ambiguity.

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.

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 arithmetic mean or mean is the simplest way to calculate the average for the given set of numbers. It is classified into two different types, namely simple arithmetic mean and weighted arithmetic mean. The average is a pretty neat tool, but it comes with its set of problems.

Symbols and encoding

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