Understanding the Main Types of Statistics in Maths

Any raw Data, when collected and organized in the form of numerical or tables, is known as Statistics. Statistics is also the mathematical study of the probability of events occurring based on known quantitative Data or a Collection of Data.

Statistics attempts to infer the properties of a large Collection of Data from inspection of a sample of the Collection thereby allowing educated guesses to be made with a minimum of expense. There are generally 3 kinds of averages commonly used in Statistics. They are: (i) Mean, (ii) Median, and (iii) Mode. 

Statistics is the study of Data Collection, Analysis, Interpretation, Presentation, and organizing in a specific way. Mathematical methods used for different analytics include mathematical Analysis, linear algebra, stochastic Analysis, the theory of measure-theoretical probability, and differential equations. Collecting, classifying, organizing, and displaying numerical Data is associated with Statistics. This helps one to grasp different outcomes from it and foresee several possibilities of various events. Statistics discuss information, observations, and Data in the form of numerical Data. We are able to find different indicators of central tendencies and the divergence of various values from the center with the help of Statistics. 

The ability to analyze and interpret statistical Data is a vital skill for researchers and professionals from a wide variety of disciplines. You may need to make decisions on the basis of statistical Data, interpret statistical Data in research papers, do your own research, and interpret the Data.

Types of Statistics

There are two kinds of Statistics, which are descriptive Statistics and inferential Statistics. In descriptive Statistics, the Data or Collection Data are described in a summarized way, whereas in inferential Statistics, we make use of it in order to explain the descriptive kind. Both of them are used on a large scale. Also, there is another kind of Statistics where descriptive transitions into inferential Statistics.

Statistics is mainly divided into the following two categories. 

1. Descriptive Statistics

2. Inferential Statistics

Descriptive Statistics

In the descriptive Statistics, the Data is described in a summarized way. The summarization is done from the sample of the population using different parameters like Mean or standard deviation. Descriptive Statistics are a way of using charts, graphs, and summary measures to organize, represent, and explain a set of Data. 

• Data is typically arranged and displayed in tables or graphs summarizing details such as histograms, pie charts, bars or scatter plots.

• Descriptive Statistics are just descriptive and thus do not require normalization beyond the Data collected.

Inferential Statistics

In the Inferential Statistics, we try to interpret the Meaning of descriptive Statistics. After the Data has been collected, analyzed, and summarised we use Inferential Statistics to describe the Meaning of the collected Data. 

• Inferential Statistics use the probability principle to assess whether trends contained in the research sample can be generalized to the larger population from which the sample originally comes.

• Inferential Statistics are intended to test hypotheses and investigate relationships between variables and can be used to make population predictions.

• Inferential Statistics are used to draw conclusions and inferences, i.e., to make valid generalizations from samples.

Example

In a class, the Data is the set of marks obtained by 50 students. Now when we take out the Data average, the result is the average of 50 students’ marks. If the average marks obtained by 50 students are 88 out of 100, on the basis of the outcome, we will draw a conclusion. 

Mean, Median and Mode in Statistics

Mean: Mean is considered the arithmetic average of a Data set that is found by adding the numbers in a set and dividing by the number of observations in the Data set. 

Median: The middle number in the Data set while listed in either ascending or descending order is the Median. 

Mode: The number that occurs the most in a Data set and ranges between the highest and lowest value is the Mode. 

For n number of observations, we have

Mean = \[\overline {x} = \frac{\sum x}{n} = \sum {x}{n} \]

Median = \[ \frac{\left[ \frac {n}{2} + 1 \right ]^{th} term}{2}\] if n is odd.

Median = \[\frac {\left[ \frac {n}{2} \right ]^{th} term +  \left[ \frac {n}{2} + 1 \right ]^{th} term }{2}\] if n is even.

Mode = The value which occurs most frequently

Measures of Dispersion in Statistics

The measures of central tendency do not suffice to describe the complete information about a given Data. Therefore, the variability is described by a value called the measure of dispersion. 

The different measures of dispersion include:

1. The range in Statistics is calculated as the difference between the maximum value and the minimum value of the Data points.

2. The quartile deviation that measures the absolute measure of dispersion. The Data points are divided into 3 quarters. Find the Median of the Data points. The Median of the Data points to the left of this Median is said to be the upper quartile and the Median of the Data points to the right of this Median is said to be the lower quartile. Upper quartile - lower quartile is the interquartile range. Half of this is the quartile deviation.

3. The Mean deviation is the statistical measure to determine the average of the absolute difference between the items in a distribution and the Mean or Median of that series.

4. The standard deviation is the measure of the amount of variation of a set of values.

Solved Example

1. What is the probability of getting two tails and one head, when 3 coins are tossed at a time?

(A) 15

(B) 3/8

(C) 14

(D) 17

Solution:

Step 1: Number of possible outcomes when one coin is tossed = 2. Outcomes are HHH and TTT. 

Step 2: The possible outcomes, when 3 coins are tossed are {TTT, THT, TTH, THH, HHT, HTH, HTT, HHH}. 

Step 3: Number of favorable outcomes = 3. Favorable outcomes are THT, TTH, HTT. 

Step 4: Substitute.

Step 5: So, the probability of getting two tails and one head is \[\frac{3}{8}\].

Correct answer: (B) \[\frac{3}{8}\].

Stages of Statistics

1. Collection of Data:

This is the first step of statistical Analysis where we collect the Data using different methods depending upon the case.

2. Organizing the Collected Data: 

In the next step, we organize the collected Data in a Meaningful manner. All the Data is made easier to understand.  

3. Presentation of Data: 

In the third step we simplify the Data. These Data are presented in the form of tables, graphs, and diagrams.

4. Analysis of the Data: 

Analysis is required to get the right results. It is often carried out using measures of central tendencies, measures of dispersion, correlation, regression, and interpolation.

5. Interpretation of Data: 

In this last stage, conclusions are enacted. Use of comparisons is made. On this basis, forecasting is made.

Uses of Statistics

• Statistics helps to obtain appropriate quantitative Data. 

• Statistics helps to present complex Data for the simple and consistent Interpretation of the Data in a suitable tabular, diagrammatic, and graphic form.

• Statistics help to explain the nature and pattern of variability through quantitative observations of a phenomenon.

• Statistics help to depict the Data in tabular form, or in a graphical form in order to understand it properly. 

Applications of Statistics

• Statistics is used in Machine Learning and Data Mining.

• Statistics is used in Mathematics.

• Statistics is used in Economics.