Statistics
Statistics is a branch of mathematics that deals with the study of collecting, analyzing, interpreting, presenting, and organizing data in a particular manner. It is referred to as arriving at conclusions of data with the use of data. Statistics is the process of collecting data, evaluating data, and summarizing it into a mathematical form. By definition, statistics is defined as the classified facts representing the conditions of a people in a state – especially the facts that can be stated in numbers or any other tabular or classified arrangement.
Mathematical Statistics
Statistics is used mainly to gain an understanding of the data and focus on various applications. Initially, statistics were related to the science of the state where it was used in the collection and analysis of facts and data about a country such as its economy, population, etc. Mathematical statistics refers to when math is applied to statistics. The techniques used for different analysis consists of mathematical analysis, linear algebra, stochastic analysis, differential equation, and measuretheoretic probability theory. Different sectors such as psychology, sociology, geology, probability, and so on also use statistics to function.
There are two methods of analyzing data in mathematical statistics that are used on a large scale:
Descriptive Statistics  This method of statistics is used to summarize the data and its properties.
Inferential Statistics  This method of statistics is used to get a conclusion from the data.
Different Models of Statistics
Statistics being a broad term used in various forms, different models of statistics are used in different forms. Listed below are a few models:
Skewness  In statistics, the word skewness refers to a measure of the asymmetry in a probability distribution where it measures the deviation of the normal distribution curve for data. The value of skewed distribution could be positive or negative or zero. The curve is said to be skewed when it shifts from left to right. If the curve mores towards the right it is called a positive skewed and if the curve moves towards the left, it is called leftskewed.
ANOVA Statistics  The word ANOVA means Analysis of Variance. The measure used in calculating the mean difference for the given set of data is called the ANOVA statistics. This model of statistics is used to compare the performance of stocks over a period of time.
Degrees of freedom  This model of statistics is used when the values are changed. Data that can be moved while estimating a parameter is the degree of freedom.
Regression Analysis  In this model, the statistical process determines the relationship between the variables. The process signifies how a dependent variable changes when an independent variable changed.
Basics of Statistics
The measure of central tendency and the measure of dispersion are considered as the basics of statistics. Variance and standard deviation come under dispersion whereas under the central tendencies are mean, median, and mode used in mathematics and its formulas. Let us look at the formulas for each.
Mean, Median and Mode
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. The middle number in the data set while listed in either ascending or descending order is the median. Lastly, the number that occurs the most in a data set and ranges between the highest and lowest value is the mode.
\(\begin{array}{cc}
\hline \text { Mean } (\bar{x}) & \bar{x}=\frac{\sum x}{n} \\
\hline \text { Median (M)} & \begin{array}{c}
\text { If } \mathrm{n} \text { is odd, then } \\
\mathrm{M}=\left(\frac{n+1}{2}\right)^{t h} \text { term } \\
\text { If } \mathrm{n} \text { is even, then } \\
\mathrm{M}=\frac{\left(\frac{n}{2}\right)^{t h} \text { term }+\left(\frac{n}{2}+1\right)^{t h} \text { term }}{2}
\end{array} \\
\hline \text { Mode } & \text { The value which occurs most frequently } \\
\hline
\end{array}\)
where,
x = Observations given
\(\bar{x}\) = Mean
n = Total number of observations
Standard Deviation and Variance
Standard deviation is a measure of how the numbers are spread and it is expressed with a Greek letter sigma. Whereas a variance is the average of the squared differences of the mean.
\(\begin{array}{cc} \hline \text { Variance }(\sigma^{2}) & \sigma^{2}=\frac{\sum(x\bar{x})^{2}}{n} \\
\hline \text { Standard Deviation } (S)& S=\sigma=\sqrt{\frac{\sum(x\bar{x})^{2}}{n}} \\ \hline
\end{array}\)
where,
x = Observations given
\(\bar{x}\) = Mean
n = Total number of observations
Representation of Data
The collection of observations and facts is known a data. These observations and facts can be in the form of numbers, measurements, or statements. There are two different kinds of data i.e. Qualitative data and quantitative data. Qualitative data is when the data is descriptive and quantitative data is when the data is numerical information. Data can be represented in different forms of graphs such as a bar graph, line graph, and so on. Let us look at a different kind of data representation.
Bar Graph
A group of data represented with rectangular bars with lengths proportional to the values is a bar graph. The bars can either be vertically or horizontally plotted.
Pie Chart
The pie chart is a type of graph in which a circle is divided into Sectors where each sector represents a proportion of the whole.
Line graph
The line graph represents the data in a form of series that is connected with a straight line. These series are called markers.
Pictograph
Data shown in the form of pictures is a pictograph. Pictorial symbols for words, objects, or phrases can be represented with different numbers.
Histogram
The histogram is a type of graph where the diagram consists of rectangles, the area is proportional to the frequency of a variable and the width is equal to the class interval. Here is an example of a histogram.
Frequency Distribution
The frequency distribution table showcases the data in ascending order along with their corresponding frequencies. The frequency of the data is often represented by f. Here is an example of the table.
Solved Examples

Example 1: Find the mean of the list of values 13, 20, 13, 15, 11, 20, 15
Solution: The given list is 13, 20, 13, 15, 11, 20, 15. Using the above formula
Mean = 13 + 20 + 15 + 21 + 20 + 15 / 6 = 17.34
Therefore, the mean is 17.34

Example 2: Find the mode from the given list of values {23, 26, 29, 23, 28, 29, 23, 21, 22}
Solution: The number that is repeated more than once is the mode.
Mode = {23}
FAQs on Statistics
What is Statistics?
Statistics is a branch of mathematics that deals with the study of collecting, analyzing, interpreting, presenting, and organizing data in a particular manner. It is referred to as arriving at conclusions of data with the use of data.
What are the Two Types of Statistics?
The two different types of statistics are:
Descriptive Statistics: It is used to summarize the data and its properties.
Inferential Statistics: It is used to get a conclusion from the data.
How is Statistics Used in Mathematics?
Statistics is a part of applied mathematics that uses probability theory to simplify the sample data we collect. The concept of probability comes under statistics where we can determine if the data is true or false but mostly, the data is true.
What is the Purpose of Statistics?
Statistics helps in better understanding and accurate description. It also helps in proper planning in the statistical study. Finally, statistics uses tables, diagrams, and graphs as representing the information in a certain manner.
What is the Importance of Statistics in Real Life?
Statistics helps to utilize strategies to gather the information, exam them, and successfully present the outcomes. Measurement is a significant cycle behind how we make disclosures in science, settle on choices dependent on information, and make forecasts.