Mean in Maths

The term ‘Mean’ is used constantly in the field of Statistics and is one of the basic methods used to obtain a result. It is also known as arithmetic mean or the average of a given set of data. It also measures the central tendency of data. The definition of a mean for a given set of data is the average calculated for a given set of numbers or data. This is referred to as the total of all the values of data provided divided by the number of data values in total for any given set of data.

The mathematical symbol or notation for the mean is ‘x-bar’. This symbol appears on scientific calculators and in mathematical and statistical notations.

The ‘mean’ or ‘arithmetic mean is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data set). At least two numbers are needed in order to calculate the mean.

The formula denoting the mean of a given set of data is as follows:

Mean = Sum of Observations/Total number of observations

The other two statistical methods used are median and mode to obtain a result for a given set of data. The median is defined as the value present in the middle of a given set of data and the mode is the frequency with which a particular number occurs in a given set of data.

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In order to find the mean of 4, 5, 6, 3, and 7, first, we have to add the numbers and then divide the sum by the number of items. 

4 + 5 + 6 + 3 + 7 = 25 i.e. the sum of the numbers is 25.

Mean =   =   =   = 5

So, the mean of the data set 4, 5, 6, 3, and 7 is 5.

How to Find Mean?

The mean value for a given set of data is calculated in a two-step process:

1. The values given in the data set are added up together.

2. The total of the values obtained is then divided by the number of values given.

Mean Formula

The measure of central tendencies is used to describe data clusters around a central value. The mean definition indicates a varied formula used to calculate the mean depending on the data provided. The general formula to calculate the mean is as follows:

\[Mean = \frac{\text{Sum of Given Data}}{\text{Total Number of Data}}\]

When using the Sigma (∑) notation, the mean formula is:

\[\frac{\sum_{i=1}^{n} X_{i}}{N}\]

Here,

N = it is the Total number provided in a given data set.

∑ Xi = Total sum of all the data values.

Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers till now. But what if the numbers in the observation list include negative numbers. Let us understand with an instance,

Example: Find the mean of 9, 6, -3, 2, -7, 1.

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

Mean Formula with Example

Find the mean for the given set of random data,

3, 5, 9, 17, 19

• The given set of data contains the numbers 3, 5, 9, 17, 19

• The total number of numerals given is 5

• Sum of the given numbers in the data set = 3 + 5 + 9 + 17 + 19 = 53

Therefore, Mean = Sum of given data/Total number of data

=\[ \frac{53}{5}\] =10.6

Hence, the mean for the given data is 10.6.

Different Types of Mean

A. Arithmetic Mean

The arithmetic mean is one of the foremost methods used to obtain the central tendency of a set of data. It encompasses all the values provided by the data set. It is referred to as the ratio of the total sum of given observations to the total number of observations. The arithmetic mean can be positive, negative, or zero. There are two types of Arithmetic Mean,

• Simple Arithmetic Mean.

• Weighted Arithmetic Mean.

The formula to calculate Arithmetic mean is as follows:

\[X = \frac{\sum_{i=1}^{n} x_{i}p_{i}}{N}\]

The arithmetic mean is easy to calculate and is rigidly defined.

B. Geometric Mean

The second type of Mean is the Geometric Mean (GM). It is defined as the average value signifying the set of numbers of central tendencies by calculating the product of their values. Multiplication of the numbers provided and take out the nth root of the multiplied numbers.

Here, n is the total number of values.

Taking an example of two numbers in a given set of data as 4 and 2, the geometric mean is equal to. \[\sqrt{(4+2)} = \sqrt{6} = 2.5\]

The difference between the arithmetic mean and the geometric mean is the method. In the arithmetic mean, we add the numbers whereas in the geometric mean we calculate the product of the numbers.

\[\text{Geometric Mean = } \sqrt[n]{\prod_{i=1}^{n} x_{i}}\]

C. Harmonic Mean

This is one of the methods of central tendency used in Statistics. It is the reciprocal of the arithmetic mean for a given set of data. The Harmonic Mean is based on all values from the data set and it is defined rigidly. It also provides the weightage of the mean in terms of large or small values depending on the data set. This is applied in time and average analysis.

To calculate the harmonic mean for a given set of data, where  x1, x2, x3,…, xn are the individual items up to n terms, then,

\[\text{Harmonic Mean = } \frac{n}{[(\frac{1}{x_{1}}) + (\frac{1}{x_{2}}) + (\frac{1}{x_{3}}) + . . . + (\frac{1}{x_{n}})]}\]

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Properties of Mean

• The sum of the deviations taken from the arithmetic mean is zero.
  If the mean of n observations x1, x2, x3….,xn is x then (x1-x)+(x2-x)+(x3-x)…+(xn-x)=0. In short, ∑ (x-x)=0

• If each observation is increased by p, the mean of new observations is also increased by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of (x1+p), (x2+p), (x3+p),….,(xn+p) is (x+p).

• If each observation is decreased by p, the mean of new observations is also decreased by p.
  If the mean of n observations x1, x2, x3….,xnis x then the mean of (x1-p), (x2-p), (x3-p),….,(xn-p) is (x-p).

• If each observation is multiplied by p (where p≠0), the mean of new observations is also multiplied by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of px1, px2,px3,pxn is px.

• If each observation is divided by p (where p≠0), the mean of new observations is also divided by p.
  If the mean of n observations x1, x2, x3….,xn is x then the mean of

\[ \frac{X_{1}}{p} ,\frac{X_{2}}{p} ,\frac{X_{3}}{p} ,...\frac{X_{n}}{p} is \frac{\bar{X}}{p}\]

Why is the Average Called the Mean?

To find the mean, add all the data points and divide it by the total number of data points. In the case of Mathematics, we have been always taught that the average is the middle point of all the given numbers. The central value which is called the average in mathematics is called the mean in statistics.

Important points:

• The mean is the mathematical average of a set of two or more numbers.

• The arithmetic mean and the geometric mean are two types of mean that can be calculated.

• Summing the numbers in a set and dividing by the total number gives you the arithmetic mean.

• The geometric mean is more complicated and involves the multiplication of the numbers taking the nth root.

• The mean helps to assess the performance of an investment or company over a period of time, and many other uses.