## Arithmetic Mean Calculator

#### Result Summary:

 Type Result Given set of numbers 0 Total numbers provided 0 Sum of Numbers 0 Arithmetic Mean Value 0

### What is Arithmetic mean?

Arithmetic mean is commonly used in statistics. For a given data set having n values, the arithmetic mean is the ratio of the sum of all the values of the data set to the number of values in the data set. It is a measure of central tendency. In other words, the arithmetic mean would help to determine a number which lies in the centre of the data set.

The arithmetic mean is calculated by adding up all the values of the data set, and then dividing by the total number of values. The arithmetic mean may or may not be equal to a value in the data set. For example, if we were to find the arithmetic mean of the numbers 10, 11 and 12, we would first find the sum = 10+11+12 = 33. Then, we divide the sum by the number of values, which is 3. Hence, the arithmetic mean =33/3 =11. Hence, 11 is the arithmetic mean of these three numbers and it lies in the centre of the data set. The arithmetic mean may not always belong to the data set, but it is the central value.

The arithmetic mean is also known as the “average” in mathematics. It can be used to model situations apart from statistical ones. If we arrange the data set in ascending/descending order, the arithmetic mean would help to find the difference of each value in the set from the mean, which is measure of the standard deviation. Such measures find applications in physics.

### Properties of Arithmetic mean

Given below are some important properties of the arithmetic mean.

Data set – The values for which the arithmetic mean is to be calculated.

Sum – The total obtained after adding up all the values in the data set.

Arithmetic mean – The value obtained by dividing the sum by the total number of values in the set.

Weighted arithmetic mean – When the values in the data set make unequal contributions to the data set depending on some properties, we compute the weighted arithmetic mean. Here, each value is assigned a fixed weight, depending on how much they influence the data. To find the weighted arithmetic mean, all the values are multiplied with their respective weights and then added up. Finally, this sum is divided by the sum of all the weights. In the case of a non-weighted arithmetic mean, each value in the data set has the same weight.

Standard deviation – This is the difference of each value in the data set from the arithmetic mean.

Range – The difference between the smallest and largest points of the data set.

### Arithmetic mean Formula

The formula for arithmetic mean is: $$arithmetic \,mean \,= \,{sum \,of \,all \,the \,values \over number \,of \,values}$$

This can also be written as $$\bar x = {1\over n}\sum_{i=1}^n x_i \\ \\ \bar x = {{x_1 + x_2 + x_3 + ... + x_n} \over n}$$

where $$W = \sum_{i=1}^n w_i = w_1 + w_2 + ⋯ + w_n,$$ and each wi denotes the weight corresponding to xi.

$$\bar x_w = { { {x_1w_1}+{x_2w_2}+{x_3w_3}+...+{x_nw_n} } \over {w_1+w_2+w_3+...+w_n} }$$

The arithmetic mean can be positive, negative as well as a real number.

### Characteristics of Arithmetic mean

• In a data set, then the sum of the deviations of each value from the mean is zero
• If the data set is an arithmetic progression, then the arithmetic mean is, arithmetic mean=(first value+last value)/2
• For example, consider the data set 4, 8, 12, 16, 20.

The arithmetic mean = (4 + 8 + 12 + 16 + 20)/5 = 60/5 =12

Since the data set is an arithmetic progression, the arithmetic mean can be computed faster using the above formula. Arithmetic mean = (first value + last value)/2 = (4+20)/2 = 24/2 = 12.

• If each value of the data set is multiplied (or divided) by a number x, then the arithmetic mean also gets multiplied (or divided) by x.
• If a number x is added to (or subtracted from) each value of the data set, then the arithmetic mean increases (or decreases) by x.
• If the data set is asymmetrical, then the arithmetic mean is not very helpful. This is one of the key limitations of the arithmetic mean.

• ### Areas of application

The arithmetic mean has quite a few real-world applications.

• In data science, the arithmetic mean is used extensively to analyse data trends. It is used in advanced modelling situations. It is also used in regression algorithms to find and classify errors.
• In physics, the arithmetic mean can be compared to the centre of gravity of a body, where the different values of the data set indicate the points on the body.
• Industries and corporates use arithmetic mean to monitor their sales over some period of time.
• We commonly use arithmetic mean in daily life to calculate the average expenses per day, or per month.
• For tracking the weather at a place, mathematicians use the arithmetic mean as a representative for a large set of data, so the weather data can be condensed for a day or for a month.
• Given below is an example making use of the arithmetic mean formula.

Question: Find the arithmetic mean of the first six prime numbers.

Answer: The first six prime numbers are 2, 3, 5, 7, 11 and 13.

To find the mean, we use the formula: $$\bar x = {1\over n}\sum_{i=1}^n x_i \\ \\ \bar x = {{x_1 + x_2 + x_3 + ... + x_n} \over n}$$

Therefore, $$\bar x = {{2 + 3 + 5 + 7 + 11 + 13} \over 6} = {41 \over 6} = 6.833$$

Hence, the mean of the first six prime numbers is 6.833.