Category: Education

Introduction to Central Limit Theorem and Steps to Check Whether an Equation Is Linear


Central Limit Theorem

Statement of Central Limit Theorem: Suppose large random samples are considered from the population with replacement consisting of mean μ and standard deviation σ, then the sample means’ distribution is normally distributed. The theorem is valid irrespective of whether the population is skewed or normal. The only criteria is that the sample should be ≥ 30. Central Limit Theorem is accurate provided that the population approaches a binomial distribution if the minimum [f (np, nq) > 5], where, n denotes the sample size, p is the probability of success of the population and 1 – p = q since p + q = 1.

The computation of sample means and sample standard deviation that are obtained from the population are as follows:

sample mean and standard deviation formula

Few illustrations of Central Limit Theorem are discussed below

1] Suppose a coin is flipped many times, the probability of getting a sure number of heads will approach a normal distribution, provided the mean = half of the total number of flips of the coin. It will equate to a normal distribution at the limit of an infinite number of flips.

2] The total distance covered in a morning or an evening walk will approach towards a normal distribution.

3] A single quantity can be weighted as the average of several small effects in the case of electronic noise or exam grades.

The process involved in obtaining the solution of the problem based on the Central Limit Theorem that consists of “>” “<” or “=” are as follows.

1] From the given problem, the known quantities such as mean, population and sample size, standard deviation and the values associated with the respective sign is to be identified.

2] A graph with a mean as its centre is to be drawn.

3] Calculate the z score using the formula:

z-score formula

4] Obtain the z-value from the z-table.

5] The various cases associated with different signs are

a] Central Limit Theorem with “>”: 0.5 – [z-score]

b] Central Limit Theorem with “<”: 0.5 + [z-score]

c] Central limit theorem that involves “between”.

Step 3 to be repeated.

6] The values of x{bar} and z-value has to be found and is expressed in percentage by converting it from decimal to percentage in the end.

Uses of Central Limit Theorem

1] To find a range of values that includes the population mean, sample mean is used.

2] Since there is an increase in the samples that are taken from the population, it leads to a decrease in the sample mean deviation which in turn helps in calculating the population mean more accurately.

3] Central Limit Theorem is implemented in political election polls to find the number of voters in percentage who are in favour of a particular candidate as confidence intervals which are shown by the news channels.

Check more JEE questions on topics like sets, relations and functions, permutation and combination, mathematical equations and inequalities, three dimensional geometry and many more at BYJU’S. This page answers the question – is an equation linear. For more information on linear equations, equations of a line, different forms of linear equations, solutions to linear equations, please refer to the above page.

Deciding Where To Move After College Graduation? 8 Factors To Consider

College Graduation

If you’re in the final stretch of your college career — first of all, congratulations! You’re one step closer to earning your degree and embarking on an exciting career path.

But if we’re being completely honest, throwing your grad cap in the air is also a little bittersweet. Graduating from college means packing your bags and leaving behind the place you’ve called home for several years. But don’t worry! Your next hometown can be just as amazing — if you know what to look for, of course. To help you determine the right fit, here are eight factors to consider when moving to a new city after college.

Job Opportunities

Landing your first job out of college can be tough, especially if you live in a city with a tough job market. Spend some time researching the economies and job markets in cities that you might want to live. What is the unemployment rate? How strong and diverse is the economy? What are the major industries in the city?

Additionally, be sure to look for potential opportunities within your industry. For instance, if you’re hoping to score a research position in a lab, you may want to search for a city with a slew of universities and medical institutions.

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Size of City or Town

Do you prefer to live in a big city or a small town? (Spoiler alert: There is no wrong answer to this one.) While many young professionals enjoy living in a large metro with an active social scene, others prefer the slower pace and tightknit community of a small town. Perhaps you fall somewhere in between, in which case, a mid-sized city like Madison, Wisconsin or Asheville, North Carolina may be more to your liking. Another important thing to keep in mind is that smaller towns typically don’t have the same access to internships, co-ops, and other career opportunities compared to larger cities.

Cost of Living

The cost of living is another big factor that new college grads need to carefully consider. Large metros like New York City, San Francisco, and Washington, D.C. are notoriously expensive places to live. However, these cities may be worth it if you can manage to snag a job with a high salary.

When researching cities to move to after college, use PayScale’s Cost of Living Calculator to figure out how much you’d need to make to maintain your current standard of living. Just plug in your current location, job title, and salary and the calculator will compare the cost of living between two cities.


Many college graduates often overlook demographics when choosing a city, which could be a big mistake. If you’re single and want to meet other college graduates, be sure to look at age demographics. How many other young professionals are living in the city?

BIPOC (black, indigenous, and people of color) can also attest to the fact that demographics are important to consider in a new city. It can be incredibly awkward — not to mention, racially fatiguing — to live someplace where you are the only BIPOC person in the office.

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Commute Time

Even the highest-paying jobs may not be worth the insane amount of traffic you might experience in some cities (looking at you, Boston). If you easily succumb to road rage, you should probably narrow your search to cities that have decent infrastructure.

Additionally, think about how you will commute to work. Do you prefer to ride your bike? Check the bike-ability score for each city or community on your shortlist. Do you plan to rely on public transportation? Look at average commute times for public transportation and the size of the transportation system. Metros with large transportation systems indicate that many commuters consider it a viable option to get around their city.


When it comes to weighing job offers and internship opportunities, don’t forget to consider the climate! If you’re about to wrap up four years of college at Arizona State University, moving to a city like Minneapolis or Detroit is probably going to be a huge adjustment for you, climate-wise.

Moving to a city that gets brutal winters can also be dangerous if you aren’t used to them. If you decide to go for it anyway, make sure that you research how to prepare your automobile for the cold, how to winterize your home, etc. Moving to a sunny locale? Pile on the sunscreen every day and be aware of heat exhaustion.


Everyone wants to feel safe where they live. While no large metro is without crime, some are better than others. Be sure to look up crime and safety statistics for any of the cities on your shortlist. Start with statistics from the most recent FBI crime reports. To get a more complete picture, you can hop on online forums like Reddit and Quora and ask long-time residents how safe they feel in their hometown.

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Food and Entertainment

Although everyone’s social lives have taken a major hit since the pandemic, that won’t be the case forever. Once things pick back up again, you’ll want to hang out with other young people and have fun in your city.

Spend some time researching what residents your age do for fun by checking out local Meetup groups. What activities and hobbies are they into there? If you’re spiritual, research local churches, temples, and mosques in the city. And don’t forget to scope out the cultural scene. Look up sporting events, festivals, concerts, theater, and recreational activities that will help you strike the perfect balance between work and play.

How To Prepare For Cbse Class 11th Commerce?

How To Prepare For Cbse Class 11th Commerce?

If you have chosen the Commerce stream after your 10th boards, you might want to know several things? Many questions would have popped up in your head. How to prepare? What books to study? Which subjects to give more emphasis? And many more. Well, you have landed on the right page then. Here we will let you know how you can score good in your 11th Class and ace the examination. Read On!

How To Prepare For Cbse Class 11 Commerce

Here we would list some of the strategies and tips for you to start with.

Plan A Schedule And Follow It

Firstly, make a routine of sleeping, eating, and studying. All of these are equally important. Sleep early and wake up early in the morning. Do some exercise for a calm mind and a healthy body. Do have breakfast and sit back to study. Pick any of the decided subjects and thoroughly read it. Do take little breaks in between. But make sure you don’t waste much time on it.  Ensure checking your plan of study by ticking the To-Do list activities.

Be Consistent And Remain Focused

Preparing a plan of action is easy but executing it daily is a bit difficult. Make sure you follow every activity as planned.  Yes, sometimes circumstances can play foul play but make sure most of the time you remain focused. By maintaining consistency, you would be more confident, organized, and complete your syllabus on time.

Multiple Ways Of Studying

Gone are the days when only books used to be the only mode of study. Well, you would have your Class 11th commerce books with you. Take the help of guidebooks and reference books available easily. Books are provided for a thorough understanding of the chapters and not only remember it like a parrot. Also, you can take CBSE 11th commerce classes online. There are expert teachers online who can guide you and help to solve your doubts.

Extract Time For Playful Activities

Do you like to listen to music? Or have a passion for drawing? Well, it is good to take up your interests in routine life. They are good for building self-confidence and self-motivation. At least give 30 minutes of your days to your favorite activity. As said, All work and no play made Jack a dull boy.

Solving Previous Years Sample Papers

When your course is about to end in each subject, try practicing with solving more and more last years’ question papers—practice, practice, and practice. Practice makes a man perfect. Ensure you solve sample papers available in the school library or online. CBSE 11th Commerce classes online tutorials and mock tests are another way to evaluate how much you have understood.

Every Commerce subjects must be given equal importance. There are five subjects, so allocate fixed hours for each of them. Do not clutter with every subject daily but do manage to finish each of them on time. You cannot ace the exam by studying Class 11th Commerce books of your favorite subject only. So do follow the above-mentioned strategies and come out with flying colors!

Gre Study Plan & Preparation

Gre Study Plan & Preparation

Top universities and colleges from the US and other countries offer several graduate, post-graduate, and Ph.D. courses to young aspirants who wish to start their careers from the best institutes in the world. Thousands of students every year try their luck to crack the entrance exam of GRE to get a seat in top B schools.

GRE is also known as the Graduate Record Test is an important step for young minds towards the best graduate and professional schools. Students start preparing well for this exam in advance and usually, the test preparation requires 6 to 7 months.

The best strategy to crack this exam involves a detailed weekly study plan covering all the concepts and topics. It is advised to start with the basics of Math and English and proceeding towards Intermediate and High Levels. GRE test covers three sections:

  • Verbal Reasoning – This is the most important section of all as it requires a strong vocabulary, understanding the comprehensions, figure out different levels of meaning, and concluding statements.
  • Quantitative Reasoning – This section involves solving the problems using mathematical tools and models, knowledge of algebra, geometry, and arithmetic.
  • Analytical Reasoning – It determines the analytical skills of the candidate and how a complex problem can be solved with relevant solutions. This section of the test involves a broad range of topics.

The format of this exam is either paper-delivered or computer-delivered. This test is conducted across more than 500+ test centres in different countries and can be taken several times a year. Several coaching institutes have GRE prep online provision with a perfect study plan and test facilities. Cracking this exam is not a cakewalk, so one should utilize the time carefully to figure out what needs to be done daily. A daily plan would help to retain the focus, slowly and gradually one can reach towards the target.

The best tip is to stay focused and that can be achieved if a strategic plan is devised. Understanding the concepts in the beginning and solving numerous practice problems of all the sections is the best strategy for exam preparation. Sample GRE tests are available at the official and other websites that can be recommended at the preparation stage. Along with the tests, study materials for all the topics can be found at these websites. Before the exam, one must look up to the GRE expert guide.

Initially, when a candidate decides to give GRE, the probability of losing the focus is more, and understanding the basics can take time. However, with time when a study plan is prepared, a routine is set up, and one can practice time-management as well. The GRE prep online has a complete set of all the sections of study material and tests.

Week by week the time duration or the study hours can be increased. English and Maths being the important part, it’s advised to focus more on these topics and study them thoroughly inside-out. The outcome of all the inputs would be bagging a seat in a top-rated college/university.