Descriptive Statistics – Moment

By / August 15, 2021

In the earlier notes of descriptive statistics, we have seen two different aspects to measure or quantify the data: the first one was the central tendency of data, and the second one was variation in data. Even we have seen how to compare two different variables’ variability using the Coefficient of Variance (CV), this makes the variability of variables comparable irrespective of the scale of the variables.

In this note, we will see a new concept called the moment. Moreover, the concept of moment explains the frequency curve’s symmetry or how the values are concentrated in the entire frequency distribution or the hump of the frequency curve.

Table of Contents hide
1 Moment
1.1 Measures of Central Tendency using Arithmetic Mean
1.2 Measures of Variability or Dispersion of Data using Variance
1.3 Measures of Variability or Dispersion of Data using Absolute Mean Deviation
2 Significance of Moments
3 References
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Moment

Moment is a particular type of mathematical function, and using this; we can quantify different types of information contained inside the data. It describes different characteristics and features of a frequency distribution or data, viz., central tendency, dispersion, symmetry, and peakedness (hump) of frequency curse.

To understand the moment, we will first reiterate the various concepts we studied in the earlier notes of descriptive statistics. Then we will show the relationship with the moment.

Measures of Central Tendency using Arithmetic Mean

To measure the central tendency of data, one of the way is using arithmetic mean. It tells where the average value of a variable lies. It adds all the data points from a variable and divides them by the total number of data points count. In other words, it is a summation of n data points divided by n. It is represented as follows:

\bar x = \frac {1}{n}\sum_{i=1}^n {x_i} = \frac {x_{1}+x_{2}+\cdots +x_{n}}{n}

where x is a variable name and x_{1}, x_{2}, \cdots ,x_{n} are n observations or data points.

Measures of Variability or Dispersion of Data using Variance

To measure the variability of data, one of the way is using variance. In this we take the mean squared distance of each data points from the arithmetic mean and it is represented as follows:

s^2 = \frac {1}{n} \sum_{i=1}^n { {(x_i - \bar{x})}^2 }

where \bar{x} = \sum_{i=1}^n {x_i} and n is the total number of data points or observation values.

Even there is another way to measure the variability or dispersion of data using Absolute Mean Deviation. It is beneficial when the data contains one or more outliers or extreme values, which may corrupt the arithmetic mean, and as a consequence, the actual outcome will be erroneous.

Measures of Variability or Dispersion of Data using Absolute Mean Deviation

To measure the variability of data using absolute mean deviation when the data contains lots of outlier. In this case, instead of measuring deviation from mean, we consider median of a variable. It is calculated as the average absolute deviation of each data points from the median. It is represented as follows:

\frac {1}{n} \sum_{i=1}^n { | x_i - \bar{x}_{med} |}

where \bar{x}_{med} is the median of a variable x and n is the total number of data points in a variable.

Significance of Moments

Suppose we have n observations from a variable x and let us consider the variables, A and r, and these take any arbitrary value. Then the general equation of moment looks as follows:

\frac {1}{n} \sum_{i=1}^n { {(x_i - A)}^r }

  • Case 1: Suppose A = 0 and r = 1 then general equation of moment represents arithmetic mean and it gives the central tendency of data.
  • Case 2: Suppose A = \bar{x} = arithmetic mean, then in this case, it represents something else, i.e., it represents variability of data.

So we have seen just changing the values of A and r, we get different characteristics of data. It motivates us to learn different types of the moment and what it represents when changing the values of r and A.

References

  1. Descriptive Statistic, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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