Averages

As per its definition, an average is a single number representing a group of numbers. It is often the sum of numbers divided by the total number of digits in the group (the arithmetic mean). For example, the average of the integers 2, 3, 4, 7, and 9 (adding up to 25) is 5. Based on the circumstances, an average could be another statistic, including the median or mode. Because the mean would be misleadingly high if it included the personal incomes of a few billionaires, the average individual income is generally stated as the median—the value below which 50% of personal revenues fall and above which 50% of personal revenues fall.

Why Calculate the Average?

The average is a numeric statistic representing a massive quantity of data in a single number. The average mark in a classroom is calculated by averaging the results of the entire class on a given topic. It is more important to know the all-around performance of the class than each student’s performance. The average is helpful in this case. 

The total of a set of values divided by individual values equals the mean of the group of values. In addition, the average is used in situations where the numbers are changing. For example:

• The average temperature of a location for a season is used to determine the temperature of a place.

• The earnings of various employees in a corporation are averaged to determine the total income of all employees.

Making decisions based on a single piece of data or a vast group of facts can be tricky at times. As a result, the average value represents all of the values in a single number.

Some General Properties Regarding Average

1. If all of the numbers in a list are the same, their average is likewise the same. Each of the numerous varieties of average has this attribute.

2. Another universal condition is monotonicity which means if two sets of numbers A and B have the same no. of elements and each element of list A is as great as the corresponding entry on list B. The average of list A is at least as large as the average of list B. Furthermore, all standards fulfil linear homogeneity: if all of the numbers in a list are multiplied by the same positive value, the average changes by the same factor.

3. Before calculating the average, several versions of the average assign varying weights to the elements in the list. The weighted arithmetic average or mean, the weighted geometric average or mean, and the median are among them. 

4. Additionally, the value of an item in some forms of moving averages is determined by its place in the list. 

5. The majority of averages, on the other side, are permutation-insensitive. All elements count equally in establishing their average value, and their order in the list is immaterial. For instance, the average of (1, 2, 3, 4, 6) is the same as the average of (3, 2, 6, 4, 1).

How to Estimate an Average

We know that average is the sum of numbers divided by the total count of the numbers.

In mathematical terms, it can be written as:

Average= (Sum of quantities)/(Number of quantities)

Or

Average=a+b+c…..n terms/n

How to Calculate Average

The steps to calculate the average of a given data set are as follows.

1. Find the sum of the total elements of elements present in the data set.

2. Count the total number of elements of the data set.

3. Divide the obtained sum of elements with the total count of the elements to get the average.

For instance, you must find the average of 4,5,7 and 11.

For the first step, we estimate the sum of all the elements of the given data set.

Sum=4+5+7+11=27

Now, we count the total number of elements in a data set.

4 elements are present in the set.

Lastly, divide the total sum of the set by the number of elements to get the average.

Average=114=2.75

Types of Mean and Their Estimation Formulas

The mean or average is divided into three categories or kinds of broader categorisation.

1. Arithmetic Mean: 

The Arithmetic Mean (AM), also referred to as the average, is the ratio of the entire set of measurements to the sum of all observations. Outside of statistics, the arithmetic mean can explain or describe topics. The arithmetic mean can be seen as a gravitational centre in a physical sense.

AM=1/n∑i=1nai=1/n(a1+a2+a3+a4……..)

2. Geometric Mean: 

The geometric average or mean is a way to determine a set of numbers’ central tendency by taking the nth root of the products of n integers. It is not the same as the arithmetic mean, which involves adding the observations and then dividing the result by the number of observations. 

In geometric mean, on the other side, we find products of all the observations and then the nth root of the product, assuming n is the number of observations.

3. Harmonic Mean: 

The reciprocal of the average of the reciprocals of the provided data values is the harmonic mean.

Conclusion

You may want to calculate the average for a variety of reasons. With the average formulae given above, you will be able to make sense of a vast set of numbers.

Instead of trying to figure out the implications of the complete data collection, you only need to select one number to sum up the entire set of numbers. When you locate the averages of several data sets, it is much easier to compare them and extract useful information for your use.