Mean Absolute Deviation

Introduction

Various statistical terms and methods represent, analyse, and compare the data in mathematics. A data set contains many numbers or values, which is quite difficult to analyse. In this direction, various mean and deviation terms are used to predict the behaviour of the process. In this tutorial, we will learn about central tendencies, variance, deviation, mean, absolute, and standard deviation with some examples.

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. However, some of the important terms have been described below.

• Mean − Mean or arithmetic mean is the ratio of adding all values of samples to the total number of samples.

• Median − The middlemost value of the dataset (however dataset is arranged in ascending order).

• Mode − It represents the most frequent value of the dataset.

• Geometric mean − The geometric mean is defined as n^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.

• Arithmetic Mean − The arithmetic mean includes the average of all data in the dataset.

• Harmonic mean − The harmonic mean is the inverse of the arithmetic mean.

• Weighted mean − This type of mean is used to summarize the dataset so that maximum importance is given to the specific values in the dataset. Midrange − It is defined as the arithmetic value of the maximum and minimum values of the dataset.

Variance and Deviation

The variance and deviation are two statistical parameters mostly used in finance. Mathematically, both parameters are correlated to each other. Let’s discuss each term in detail.

Variance

The variance is defined as the arithmetic mean of the squared differences from the mean (arithmetic). It tells us how far is the individual data from the mean of the dataset. In other words, it is used to find the expected deviation difference from the actual value. The variance of the dataset is mathematically correlated with the deviation as per the following expression.

$$\mathrm{Variance = (standard\: deviation)^2=\sigma ^2}$$

The general formula to evaluate the variance in a dataset is given below.

$\mathrm{ Variance =\sigma ^2=\frac{∑(X-μ)^2}{P}}$(For grouped data)

$\mathrm{Variance =\sigma^2=\frac{\sum(X-( \underline{\bar{x}}) )^2}{P-1}}$ (For ungrouped data)

where X, μ, , and P are the value of individual data, the mean value of the population, mean of the data, and the total number of data, respectively.

Deviation

The deviation measures the difference between the variance and the individual data. It tells us about the degree of dispersion of data. A low deviation value represents that the observed data are closer to the mean value. In the case of a higher deviation value, there is a significant difference between the observed and the mean values.

Types of Deviation

Several types of deviations exist in statistics, which are enlisted below.

• Mean absolute deviation

• Standard deviation

• Average absolute deviation

• Maximum absolute deviation

Mean Absolute Deviation

It is defined as the mean distance of each observed data from the mean of the whole dataset. It is abbreviated as “MAD”. The following formula can be used to determine the mean absolute deviation.

$$\mathrm{MAD =\sum \frac{Absolute\: value\: deviation\: from\: central\: measure}{Total\: number\: of\: samples}= \sum \frac{|X-\underline{\bar{x}}|}{P}}$$

The following steps need to be followed to evaluate the mean absolute deviation.

• Evaluate the mean of the whole dataset.

• Find the differences between each observation and the mean value.

• Find the arithmetic mean of the obtained difference values. The resulted value will give the required mean absolute deviation.

Standard Deviation

The standard deviation is the square root of the variance, and it tells us about the degree of dispersion of data. It is denoted by a symbol “σ” and abbreviated as SD. A low SD value represents that the observed data are closer to the mean value. In the case of a higher SD value, there is a significant difference between the observed and the mean values. The general formula to evaluate the SD in a dataset is given below.

$\mathrm{ SD =\sigma =\sqrt{\frac{\sum (X-μ)^2}{P}}}$ (For grouped data)

$\mathrm{ SD =\sigma =\sqrt{\frac{\sum (X-\underline{\bar{x}})^2}{P-1}}}$(For ungrouped data)

where $\mathrm{X, \mu, \underline{\bar{x}}, and\: P }$ are the value of individual data, the mean value of the population, mean of the data, and the total number of data, respectively.

Similarities and Differences between Standard and Mean Absolute Deviation

The similarities between the standard and mean absolute deviation are described below.

• Both deviations are used to quantify the deviation in a dataset.

• The arithmetic mean is used to calculate both deviations.

However, several differences exist between these two deviations, which are summarized below.

S.N.	Standard deviation	Mean absolute deviation
1	It is often used in statistics.	It is an alternate way to find the deviation.
2	It can be calculated using the formula,$\mathrm{\sigma =\sqrt{\frac{\sum (X-\underline{\bar{x}})^2}{P-1}}}$	It can be calculated using the formula, $\mathrm{MAD=\sum \frac{ |X-\underline{\bar{x}}|}{P}}$
3	The value of SD is always smaller than MAD.	The value of MAD is always greater than SD.

Solved Examples

Example 1:

Evaluate the mean absolute deviation of the following dataset: 10, 5, 19, 30, 45, 60, 55, and 72.

Solution:

The mean of the data = $\mathrm{\underline{\bar{x}}=\frac{10+5+19+30+45+60+55+72}{8}=37}$

The mean absolute deviation can be obtained using the formula,

$$\mathrm{MAD=\sum \frac{|X-\underline{\bar{x}|}}{P}}$$

$$\mathrm{\Rightarrow MAD=\frac{|10-37|+|5-37|+|19-37|+|30-37|+|45-37|+|60-37|+|55-37|+|72-37|}{8}}$$

$$\mathrm{\Rightarrow MAD=21}$$

∴ The mean absolute deviation of the given dataset is 21.

Example 2:

Evaluate the variance and standard deviation of the grouped data.

Class interval	0-4	4-8	8-12	12-16	16-20	20-24
Frequency	2	3	1	5	6	8

Solution:

Class interval	Frequency (f)	Class Mark (xi)	fxi	fxi2
0-4	2	2	4	8
4-8	3	6	18	108
8-12	1	10	10	100
12-16	5	14	70	980
16-20	6	18	108	1944
20-24	8	22	176	3872
	$$\mathrm{\sum \mathit{f}=25}$$		$$\mathrm{\sum \mathit{f}x_i=386}$$	$$\mathrm{\sum \mathit{f}x_i^2=7012}$$

$$\mathrm{Mean\: value = \bar{x}=\frac{\sum fx_i}{\sum f}=\frac{386}{25}=15.44}$$

$$\mathrm{Variance =\frac{1}{\sum f-1}[\sum fx_i^2-\frac{1}{\sum f}(\sum fx_i )^2]}$$

$$\mathrm{\Rightarrow Variance =\frac{1}{25-1}[7012-\frac{1}{25}(386)^2]}$$

$$\mathrm{\Rightarrow Variance =43.84}$$

Now, the standard deviation $\mathrm{= \sqrt{variance}=\sqrt{43.84}=6.62}$

∴ The variance and the standard deviation of the given dataset are 43.84 and 6.62, respectively.

Conclusion

The present tutorial gives a brief introduction to about the central tendencies and their various mean values. The basic definition of mean absolute and standard deviation have been stated in this tutorial. In addition, the similarities and the difference between these two deviations have been illustrated. 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 mean absolute deviation.

FAQs

1. What implies that if the standard deviation is zero?

The value of standard deviation is zero means all the observed data in the dataset are equal.

2. What are the limitations of standard deviation?

The difficulty in calculating deviation is a major disadvantage of standard deviation. In addition, it cannot be calculated in open intervals.

3. What should be the good standard deviation value?

Most mathematicians prefer to get the standard deviation value within the range of ±2.

4. What implies that if the variance is zero?

The zero variance of a dataset denotes that the data values are constant.

5. What are the advantages of the mean absolute deviation?

The major advantages of the mean absolute deviation include simplicity and effectiveness compared to the standard deviation.