3.1.4. Kurtosis

Kurtosis measures the degree of peakedness or flatness of a curve. The normal curve is called mesokurtic. If the curve is more peaked than normal, it is called leptokurtic. If it is flatter than normal, it is called platykurtic. Figure 2 illustrates the nature of different types of kurtosis.

The moment measure of kurtosis is

$$\beta\_2 = \frac{\mu\_4}{\mu\_2^2}$$

where

$$\mu\_4 = \sum\_{i=1}^{n} \frac{(x\_i - \bar{x})^4}{n}$$

If *β*<sup>2</sup> = 3, the distribution is mesokurtic, if *β*<sup>2</sup> > 3, the distribution is leptokurtic, and if *β*<sup>2</sup> < 3, the distribution is platykurtic.

(**a**) (**b**) (**c**)

**Figure 2.** (**a**) Leptokurtic, (**b**) mesokurtic, and (**c**) platykurtic.

#### *3.2. Inferential Statistics*

In inferential statistics, the concept of probability is important for studying the uncertainties in the environment. For example, whether it will rain or not tomorrow can be best inferred by using probability. Several theoretical probability distributions, such as the Bernoulli distribution, the binomial distribution, the Poisson distribution, etc., are useful for

modeling the probability distribution of real environmental data. For example, decisions, such as coin tossing, rain/no rain, yes/no, etc., are explained by Bernoulli variables since their outcomes are binary. In addition, if we are interested in counting the number of times floods occurred in the Dhemaji district of Assam, India, out of the total number of floods that occurred, because we are counting the number of times a flood (*X*), a Bernoulli event, occurs with a probability of *p* out of a total, i.e., out of n trials, the probability distribution of such variables is given by a binomial distribution. In addition, if we do not know the total number of flood occurrences but know the meaning of the flood occurrences, the distribution is modeled by the Poisson distribution. Statistical tools such as estimation, hypothesis testing, etc., play a vital role in analyzing environmental data. Some of the frequently used statistical tests in atmospheric and environmental science are the "Z-test," "T-test," "F-test," etc. Another statistical approach is time series analysis, which studies environmental quantities with respect to time. For example, the monthly/yearly mean temperature, rainfall, humidity, etc., are best studied by time series (see [8]).

#### **4. Illustrations**

In this section, we are discussing the available statistical techniques that are used in the field of environmental sciences along with some practical examples in the context of data based on environmental sciences. There are examples of how collaboration between environmental scientists and quantitative researchers has aided future learning in both fields, based primarily on two works: [4], which deals with statistical techniques, and [5], which deals with practical examples.

The statistical techniques available are given below, based on [4].
