Mean Definition

Introduction

In an organization, a lot of data has to be collected and handled for analysis and representation purposes. In this direction, statistical parameters help us to analyse the data effectively. There are various mean used in statistics for the above purpose. In this tutorial, we will learn the meaning of mean, its types, their relationship, and a brief discussion about other central tendencies with solved examples.

Data

• The data is a collective word that represents the collection or arrangement of quantities or categories values.

• The purpose of data collection is to analyse the specific event in a process.

• When data are arranged systematically, then it is known as information.

• The data can be categorized based on values or the source of collection.

A brief discussion about the various types of data are summarized below.

S.N.	Types of data	Meaning
1	Quantitative	These data types can be measured or quantified and expressed in terms of arithmetic values. The algebraic operation can be carried out over them.
2	Qualitative	Data cannot be measured or expressed in terms of numerical values. They are either characteristics or attributes.
3	Primary	Data are collected for the first time and no statistical operation has not been done over those data.
4	Secondary	These types of data are collected from published or unpublished sources. Statistical operations have already been done over those data.
5	Discrete	It represents the specific value of the data. Line graph or charts are mainly used to represent the data.
6	Continuous	It represents the value of data within a specific range. Histograms are generally used to represent the data.

Central Tendencies

• Central tendency is an important concept of descriptive statistics.

• In statistics, central tendencies are defined as a unique value that represents the central value of the probability distribution.

• It does not state the individual information of the dataset; however, it refers to an index value, which summarizes the whole dataset.

• Various terms have been used to describe the central tendencies of the dataset. There are various central tendencies used in statistics, such as mean, median, mode, geometric mean, arithmetic mean, harmonic mean, etc.

Mean

• Mean is a statistical term used to describe the central tendency of a set of data.

• It is defined as the ratio of the addition of all data points to the number of data points.

• It is usually denoted by a symbol, i.e., $\mathrm{\underline{M}}$. Mathematically, it can be written as

$$\mathrm{\underline{M}=\frac{Sum\:of\: all\: data\: points}{Total\: number\: of\: data\: points}}$$

$$\mathrm{or\:\underline{M}=\frac{m_1+m_2+m_3+m_4+.......+m_r}{r}}$$

$$\mathrm{or\:\underline{M}=\frac{\sum_{i=1}^r m_i}{r}}$$

where r = Total number of data points

m₁,m₂,m₃,...= The value of the individual data

The symbol "∑ " denotes the addition or summation of the values.

Types of Mean

There are four types of means used in statistics.

• Arithmetic Mean − The meaning of arithmetic mean includes the average of all data present in the dataset. In addition, it is abbreviated as AM. Statistically, it is expressed as Arithmetic mean $\mathrm{(AM) =\underline{M}= \frac{Σ m}{r}}$

  where r = total number of data

  Σ m = addition of all individual data

• Geometric Mean − The geometric mean is defined as r^th root of the product of all data. In other words, it is also defined as the arithmetic mean of the logarithmic value of all data points. In addition, it is abbreviated as GM. Therefore, the geometric mean can be expressed as

  Geometric mean (GM) =$\mathrm{=\sqrt[r]{m_1×m_2×m_3×m_4×.....×m_r}}$

  $$\mathrm{or\: GM= m_1×m_2×m_3×m_4×.....×m_r )^{\frac{1}{r}} }$$

  where m_1,m_2,m_3,...= The value of the individual data

• Harmonic Mean − The harmonic mean is the inverse of the arithmetic mean. It is abbreviated as HM. It is found to be more effective when the group size is different from another group. Therefore, the harmonic mean can be expressed as Harmonic mean $\mathrm{(HM) =\frac{r}{\sum \frac{1}{m}}(where\: r = number\: of\: elements) }$

• Weighted Mean − This type of mean is used to summarize the dataset in such a way that maximum importance is given to the specific values in the dataset. In addition, it is abbreviated as WM. The weighted mean can be expressed as follows Weighted mean $\mathrm{(WM) = \frac{Σw_i x_i}{w_i}}$

  where w_i= weights applied to each value and x_i= individual data to be averaged.

Relationship between AM, GM, and HM

To establish a relationship between AM, GM, and HM, consider two arithmetic values, i.e., p and q.

Now the AM of these two numbers is = $\mathrm{AM =\frac{p+q}{2}}$

The GM of these two numbers is = $\mathrm{GM =\sqrt{pq}}$

The HM of these two numbers is $\mathrm{=\frac{2pq}{p+q}}$

By performing an algebraic operation between AM and GM, we will get

$$\mathrm{\frac{p+q}{2}\times \frac{2pq}{p+q}=(\sqrt{pq})^2}$$

That means, AM×HM=GM²

Therefore,the square of GM is the product of AM and HM.

Other Central Tendencies

Some of the important central tendencies have been enlisted below.

• Generalized Mean

• Truncated mean

• Interquartile mean

• Midrange

• Midhinge

• Quasi-arithmetic mean

• Tri-mean

Mode and Median

Mode − It is defined as the most frequent value occurring in a dataset.

Median − The median is an index or value, which shows the middle position of the dataset when the samples are arranged in ascending or descending order. It splits the dataset into two halves, namely upper-half and lower-half sets. In addition, the median is also known as the positional average.

Solved Examples

Example 1:

Evaluate the AM and GM of the data -6, -9, 10, 8.

Solution:

The arithmetic mean is = $\mathrm{AM=\frac{-6-9+10+8}{4}=0.75}$

The geometric mean can be found using the formula

$$\mathrm{(GM) =\sqrt[4]{(-6)×(-9)×10×8}≃8.11}$$

∴ The AM and GM of the given data is found to be 0.75 and 8.11, respectively.

Example 2:

A set of data is given as: 4, and 16.

Prove the statement AM×HM=GM²using the given dataset.

Solution:

$$\mathrm{AM =\frac{4+16}{2}=10}$$

$$\mathrm{GM = \sqrt{4×16}=8}$$

$$\mathrm{HM =\frac{2}{\frac{1}{4}+\frac{1}{16}}=6.4}$$

From the above values, it can be concluded that AM×HM=GM²

Conclusion

The present tutorial gives a brief introduction mean and its various types. The basic definition of central tendencies and their various parameters have been stated in this tutorial. In addition, a relationship between the AM, GM, and HM have been derived. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of the mean.

FAQs

1.Can the first term of the geometric mean be zero?

No. The first term of the geometric progression or sequence should be a non-zero number.

2.Which one is considered a better option among geometric mean and arithmetic mean?

The arithmetic mean is used to evaluate the average of the data, whereas the geometric mean considers the concept of compounding over a time period. Therefore, it is more suitable to use in finance to determine the returns on fixed investments.

3.In which case will the two means, i.e., AM and GM, be the same?

If the value of all data is equal, then these two means are identical to each other.

4.Is the mean and median of the first five even numbers be same?

The first five even numbers are 2, 4, 6, 8, and 10.

$$\mathrm{Mean =\frac{2 + 4+ 6 + 8 + 10}{4} = 7.5}$$

$$\mathrm{Median =(\frac{n+1}{2})^{th}\:term}$$

$$\mathrm{=(\frac{5+1}{2})^{th}\:term}$$

$$\mathrm{=(3)^{rd}\: term=6}$$

∴ The mean and median of the first five even numbers are not equal.

5.What is the limitation of the geometric mean?

It is difficult to find the geometric mean if a negative integer exists in the data set