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    What Is The Scope Of Statistics In Our Daily Lives?

    By Sumit Saini
    28 Nov'22  6 min read
    What Is The Scope Of Statistics In Our Daily Lives?
    Synopsis

    Mathematics plays an important role in our lives, especially statistics. Statistics is a branch of mathematics that deals with collection, interpretation, analysis, and representation of data. This article will help you understand what the applications of statistics are, as well as solutions to some previous year questions of JEE mains on this topic.

    What Is The Scope Of Statistics In Our Daily Lives?
    Synopsis

    Mathematics plays an important role in our lives, especially statistics. Statistics is a branch of mathematics that deals with collection, interpretation, analysis, and representation of data. This article will help you understand what the applications of statistics are, as well as solutions to some previous year questions of JEE mains on this topic.

    Statistics play a crucial role in scientific research, discoveries, public welfare activities, economic decisions, forecast, and stock analysis. We live in the data centric and information age and most of the information is determined using statistics. In addition, statistical data analysis makes decision-making more objective. In schools and colleges, it is used to calculate the average percentage of the students. Similarly, it is used in households to determine daily expenses.

    As a branch of mathematics, statistics deals with the collection, interpretation, analysis, and representation of data. It is the science behind summarising, identifying and organising data. The data could be both, either quantitative or qualitative. These mathematical concepts of statistics are used in decision-making. Basically, it is a combination of methods which allow us to make optimal reasonable decisions in cases of any uncertainty in our data. It compares data through mode, median and mean. With the help of statistics, we can predict what will happen in the future and what occurred in the past. Mainly there are two types of statistics: Descriptive and Inferential.

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    Descriptive Statistics

    The goal of descriptive statistics is to describe a data set in a very specific manner. It utilises the graphical (bar graphs or pie charts) and numerical (mean, median, mode) methods to look for patterns in a data set, to present the information and to summarise the information in a suitable manner so that individuals can use it to arrive at decisions.

    Inferential Statistics

    The goal of inferential statistics is to derive a conclusion about a population based on the sample evaluation from that population. It utilises sample data to make predictions, estimates, decisions, or other generalisations about a larger set of data. Hypothesis testing is the most commonly used inferential technique which is an educated guess about the relationship between two or more variables.

    Why Statistics Is Used In Various Fields?

    Statistics helps us in decision-making and in research in different fields due to its various statistical properties. Some of them are given below:

    Unbiasedness: We collect data randomly from the data set without any adjustment. It is purely based on mechanism and parameter therefore, it is reliable.

    Uniformity: Most of the data is similar, which we used in evaluations. To maintain consistency; data is collected according to a set of parameters.

    Sufficiency With Efficiency: The sample data analysis includes all the information about the whole population, which makes this method most useful for vast data sets. If the obtained information deviates from sample data, it can still be interpreted using specific methods and formulas.

    Quantitative Analysis: Numerical formulas and methods of statistical data makes it simple and easier to understand results and draw conclusions.

    Applications of Statistics

    Statistics is used in every aspect of our daily life and it is needed for almost every research project. Proper analysis of data could be used to collect important demographic information, to prevent disease, and test potential pharmaceutical products. It also helps us to increase and maintain the profitability and efficiency of an organisation. It is used to solve different problems in various fields. Some important examples are given below:

    In Research

    Statistics is beneficial for examining the correctness of choices and decision making. It is essential for all the branches of science. It plays an essential role in various fields in the work of researchers. It is used extensively to present results in research papers.

    In Statistical Modeling

    In statistical modelling scientists and engineers build predictive models based on knowledge and pattern recognition. It is used for various purposes such as predicting election results, geographical and environmental studies, survival analysis of populations, and more. Meteorologists also use statistical models and tools to predict the weather.

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    In Robotics

    Robot is a machine made by humans using engineering, science and technology. Statistics is also used in Robotics. Robots are required to perform certain tasks. Data analysis is essential for the robots to control their speed, use of force, to evaluate their performance. By estimating the probability density function value, the robot always senses the present state. Robots continuously update themselves with the help of new inputs and accordingly give priority to the current actions. Efficiency of a robot can be increased significantly with the application of statistics in machine learning and computer science, algorithms’

    In Data Science

    Statistics helps data scientists to obtain the relevant outputs from the sample of the population. A data scientist uses different techniques to study the data set, such as regression, classification, time series analysis, hypothesis testing and much more.

    In Clinical Trials

    Designing clinical trials is one of the most important uses of statistical analysis. When a new treatment or drug is discovered, it is essential to test it on a group of people to understand its safety and efficacy. A clinical trial includes selecting a sample size, defining the time range to monitor the treatment. Statistics help doctors and scientists to design the phases, and select parameters so that they can decide how effective the treatment is.

    In Social Welfare

    Many administrative decisions and national policies are made on the basis of their data. Statistics provide accurate data, so that the government can make suitable budgets and estimate revenues and expenditures. If the cost of living rises, it is used to revise the pay scale of government employees. Statistics is also used to evaluate the effectiveness of crime and fraud strategies and tactics.

    Finance

    Statistics play an important role in investment and banking. Banks use statistics to analyse the financial market to reduce risk and predict the impact of the crisis.

    Marketing

    In marketing, especially in social advertising and social media, statistical data analysis plays a huge role. Like in marketing, advertising and social media, professionals analyse data to increase the number of followers and to optimise campaign’s performance.

    Apart from the applications in various industries, it is important to go through the previous year questions in order to understand the concepts of statistics and methods to solve problems.

    JEE-Mains Previous Year Questions From Statistics

    Q-1(JEE Main 2022)

    Let 1669610683775 be in geometric progression with 1669610684986 and the common ratio 1669610685446. A new data is constructed replacing each 1669610684101 by 1669610685222. If 1669610684289 is the mean of new data, then the greatest integer less than or equal to 1669610684517 is…….

    Solution:

    \begin{aligned} & \mathrm{\Sigma x_{0}^{\prime}=\frac{3\left(1-\left(\frac{1}{2}\right)^{20}\right)}{1-\frac{1}{2}}=6\left(1-\frac{1}{2^{2}}\right)}\\ &\mathrm{=\sum_{i=1}^{20}\left(x_{i}+i\right)^{2}}\\ &\mathrm{=\sum_{i=1}^{20}\left(x_{i}\right)^{2}+(i)^{2}-2 x_{i} i}\\ &\text { Now } \mathrm{ \sum_{i=1}^{20}\left(x_{i}\right)^{2}=\frac{9\left(1-\left(\frac{1}{4}\right)^{20}\right)}{1-\frac{1}{4}}}\\ &\mathrm{=12\left(1-\frac{1}{2^{40}}\right)}\\ &\mathrm{\sum_{i=1}^{20} i^{2}=\frac{1}{6} \times 20 \times 21 \times 41=287^{\circ}}\\ & \mathrm{\sum_{i=1}^{20} x_{i} i=s=3+2 \cdot 3 \cdot \frac{1}{2}+3 \cdot 3 \cdot \frac{1}{2^{2}}+4 \cdot 3 \cdot \frac{1}{2^{3}}+\cdots}\\ & \mathrm{=6\left(2-\frac{22}{2^{20}}\right)}\\ & \mathrm{\bar{x}=\frac{12-\frac{12}{2^{40}}+2870-12\left(2-\frac{22}{2^{20}}\right)}{20}}\\ & \mathrm{\bar{x}=\frac{2858}{20}+\left(\frac{-12}{240}+\frac{22}{22^{0}}\right) \times \frac{1}{20}}\\ &\mathrm{[\bar{x}]}=\mathrm{142} \end{aligned}


    Q-2 (JEE Main 2022)

    The sum and product of the mean and variance of a binomial distribution are 1669610686771 and 1669610687000 respectively. Then the number of trials in the binomial distribution is____________.

    Solution:

    \begin{aligned} \text { Mean } &\mathrm{=n p} \\ \text { Varianel } &\mathrm{ n p q=n p(1-p) }\\ \text { Sum }=& \mathrm{n p(1+1-p)=82.5 }\\ & \Rightarrow \mathrm{n p(2-p)=82.5 \ldots} \text { (i) } \\ \text { Product }&\mathrm{=n p \cdot n p(1-p)=1350} \\ & \Rightarrow \mathrm{ \frac{(82.5)^{2}}{(2-p)^{2}}(1-p)=1350} \\ & \Rightarrow \mathrm{p=\frac{5}{8}}\\ &\text{Using (i),} \mathrm{n=96} \end{aligned}


    Q-3 (JEE Main 2022)

    Let the mean and the variance of 20 observations 1669610686020 be 1669610687210 and 1669610687691 respectively. For 1669610686560, if the mean of 1669610687485is 178, then the square of the maximum value of 1669610687881 is equal to___________.

    Solution:

    \begin{aligned} & \mathrm{\sum x_{1}=15 \times 20=300 \cdots }\text { (i) } \\ & \begin{aligned}&\mathrm{\frac{\sum x_{1}^{2}}{20}-(15)^{2}=9 \cdots \cdot }\\&\mathrm{\sum x_{1}^{2}=234 \times 20=4680}\end{aligned} \\ &\mathrm{ \frac{\sum\left(x_{1}+\alpha\right)^{2}}{20}=178 \Rightarrow \sum\left(x_{1}+\alpha\right)^{2}=3560} \\ & \mathrm{\Rightarrow \sum x_{1}^{2}+2 \alpha \sum x_{1}+\sum \alpha^{2}=3560} \\ & \mathrm{4680+600 \alpha+20 \alpha^{2}=3660} \\ & \mathrm{\Rightarrow \alpha^{2}+30 \alpha+56=0} \\ & \mathrm{\Rightarrow(\alpha+28)(\alpha+2)=0 }\\ & \mathrm{\alpha=-2,-28} \end{aligned}\\ \text{ square of maximum valu of }\mathrm{\alpha\; is \; 4 }

    Q-4 (JEE Main 2022)

    The mean and variance of 1669610688927 observations were calculated as 1669610688325 and 1669610688553 respectively by a student who took by mistake 1669610689347 instead of 1669610688744 for one observation. Then, the correct standard deviation is______________.

    Solution:

    \begin{aligned} & \mathrm{n=10, \bar{x}=\frac{\sum x_{i}}{10}=15 }\\ &\mathrm {\sigma ^{2}=\frac{\sum x_{1}^{2}}{10}-(\bar{x})^{2}=15 }\\ &\mathrm {\Rightarrow \sum_{i=1}^{10} x_{i}=150} \\ &\mathrm {\Rightarrow \sum_{i=1}^{9} x_{i}+25=150 }\\ &\mathrm {\Rightarrow \sum_{i=1}^{9} x_{i}=125} \\ &\mathrm {\Rightarrow \sum_{j=1}^{9} x_{i}+15=140} \end{aligned}

    \begin{aligned} &\text { Aatual mean }=\mathrm{\frac{140}{10}=14=\bar{x}_{\text {new }}} \\ & \mathrm{\sum_{i=1}^{9} \frac{x_{i}^{2}+25^{2}-15^{2}}{10}=15 }\\ &\mathrm{\Rightarrow \sum_{i=1}^{9} x_{i}^{2}+625=2400} \\ & \mathrm{\sum_{i=1}^{9} x_{i}^{2}=1775} \\ \end{aligned}

    \begin{aligned}&\mathrm{\sum_{i=1}^{9} x_{i}^{2}+15^{2}=2000=\left(\sum x_{i}^{2}\right)_{\text {actual }} }\\ & \mathrm{\sigma ^{2}_{\text{actual}}=\frac{\left ( \sum{x_{i}^{2}} \right )_{\text{actual}}-\left ( \bar{x}_{\text{new}} \right )^{2}}{10}}\\ &=\mathrm{\frac{2000}{10}-14^{2} }\\ &=200-196=4\\ &\mathrm{(\mathrm{S,D})_{\text {actual }}=\sigma =2} \end{aligned}

    Q-5 (JEE Main 2022)

    Let 1669610690239 have a binomial distribution 1669610690441 such that the sum and the product of the mean and variance of 1669610690041 are 24 and 128 respectively. If 1669610690625, then 1669610690819 is equal to :

    Solution:

    \begin{aligned} & \mathrm{n p+n p q=24 \Rightarrow m p(1+q)=24 }\\ &\Rightarrow \mathrm{ n p(2-p)=24 } \\ &\Rightarrow \mathrm{n p=\frac{24}{2-p}}\\ & \text{Also} \mathrm{ n p \cdot n p q=128}\\ & \Rightarrow \mathrm{n^{2} p^{2} q=128 }\\ & \Rightarrow \mathrm{n^{2} p^{2}(1-p)=128} \\ & \Rightarrow \mathrm{ \frac{24^{2}}{(2-p)^{2}}(1-p)=128 }\\ & \Rightarrow \mathrm{\frac{(1-p)}{(2-p)^{2}}=\frac{2}{9}} \\ & \Rightarrow \mathrm{9-9 p=2\left(p^{2}+4-4 p\right)} \\ & \Rightarrow \mathrm{2 p^{2}-8 p+8+9 p-9=0 }\\ & \Rightarrow \mathrm{2 p^{2}+p-1=0} \\ & \Rightarrow \mathrm{p=\frac{1}{2} \quad(\operatorname{as} p>0)} \\ & \Rightarrow \mathrm{n=32} \\ & \therefore \mathrm{P(X>n-3) }\\ & \mathrm{=P(X>29) }\\ & \mathrm{=P(30)+P(31)+P(32)} \\ \end{aligned}

    \begin{aligned} & \mathrm{=^{32}C_{30} \cdot \frac{1}{2^{32}} + ^{32}C_{31} \cdot \frac{1}{2^{32}}+^{32}C_{32} \cdot \frac{1}{2^{32}}}\\ &=\mathrm{\frac{529}{2^{32}} }\\ &\therefore \quad \mathrm{k=529} \\ &\therefore \text { option (B) } \\ \end{aligned}

    In the above questions, concepts of statistics along with algebraic operations are used. Statistics is a great field to explore. Emerging career options like Data science, data analytics, machine learning, etc. heavily relies on the concepts of Statistics.

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