What Is Normal Distribution and its definitions

What Is Normal Distribution

In mathematical terms, a normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In the A-level math syllabus, the normal distribution intends to highlight a few key points about the normal distribution, represented by N ( μ , σ2 ).   

refers to the mean, and σ refers to the variance.

Primarily, for both H1 math and H2 math students should be aware that:

  • The location and shape of a normal curve are determined by the values of μ and σ. 
  • The normal curve is symmetrical about μ
  • The area under a normal curve between x = a and x = b is the probability P (a ≤ x ≤ b)

To clarify on how a normal distribution will shift due to its mean and variance: 

If σ remains unchanged and μ changes from μ1 to μ2, where (μ1 < μ2 ), then the normal curve will shift to the right. (Location)

 

If μ remains unchanged and σ changes from σ1 to σ2 where (σ1<σ2), then the normal curve will become flatter and sharper. (Shape

In mathematical terms, a normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In the A-level math syllabus, the normal distribution intends to highlight a few key points about the normal distribution, represented by N ( μ , σ2 ).   

refers to the mean, and σ refers to the variance.

Primarily, for both H1 math and H2 math students should be aware that:

  • The location and shape of a normal curve are determined by the values of μ and σ. 
  • The normal curve is symmetrical about μ
  • The area under a normal curve between x = a and x = b is the probability P (a ≤ x ≤ b)

To clarify on how a normal distribution will shift due to its mean and variance: 

If σ remains unchanged and μ changes from μ1 to μ2, where (μ1 < μ2 ), then the normal curve will shift to the right. (Location)

 

If μ remains unchanged and σ changes from σ1 to σ2 where (σ1<σ2), then the normal curve will become flatter and sharper. (Shape

What is CDF and PDF in Normal Distribution?

When calculating a probabilities of normal distribution, it is common to come across the terms “CDF of normal distribution” or a “normal distribution PDF”. 

The term CDF, cumulative distribution function, refers to the probability that a random variable will take a value less than or equal to the random variable. 


Step by step to get the CDF using a graphic calculator: 

  • Press 2nd, vars for the distribution menu. 
  • Press 2 to select normalcdf(. 
  • Key in the numbers for the lower limit, upper limit, mean and standard deviation.
  • Paste and press enter. 
  • The CDF should be shown. 

The term PDF, or probability density function, refers to the probability that a random variable will take a value exactly equal to the random variable. 

The Standard Normal Distribution 

It is important to note that the standard normal distribution is a special group among the normal types that are presented. 

It has mean 0 and a standard deviation of 1. 

The random variable for the standard normal distribution is denoted by the letter Z. 

Normal Distribution as an Approximation to Binomial Distribution

In the normal distribution formula where N ( μ , σ2 ), it is shown that as n gets larger, the binomial distribution gets closer and closer in shape to the normal distribution curve. 

To understand and graph the shape of any intended binomial distributions with n number of trials, with different probabilities of success, p, the graphing calculator can be used. 

Step by step to get the histogram using a graphic calculator: 

(Generate a string of integers in sequence from 0 to 12.) 

  • Press stat, 1:edit. 
  • Highlight L1 and press 2nd, stat, >, 5:seq
  • Key in the fields respectively 

(Generate the respective probabilities in List 2) 

  •  Highlight L2, press 2nd, vars. 
  • Choose A:binompdf 
  • Key in the fields respectively 
  • Go to Paste and enter. 

(Draw the Graph) 

  • Press 2nd, y= and enter 
  • Select and key in the fields 
  • Press graph to see the histogram. 

Math Question

The weights of monkeys in a particular jungle follow an unknown distribution with mean 15kg and standard deviation 6kg. Find the probability that a randomly selected group of 60 monkeys has a total weight exceeding 1000 kg. 

Hint: This will require the usage of the central limit theorem.  Answer:0.0157(3 s.f)

Psychology

Do I Need Math to Study Psychology?

Unlike computer science, it is not compulsory to score well for math to study psychology. But there are aspects of math that relate to psychology, such as in object-oriented programming. It may come as a surprise, but psychologists often use mathematical methods for measuring and quantifying various aspects of human behavior and cognition. They use statistical techniques to analyze data from experiments, surveys, and observations, enabling them to draw conclusions and make inferences about psychological phenomena.

Moreover, the process of studying maths itself may also be incorporated into psychology in the form of cognitive psychology and mathematical cognition. Check it out in the example below! 

Have you ever wondered why you struggle in maths? Studies published by the British Psychological Society currently suggest two broad theories. The first, that an early deficit in foundational numerical skill causes mathematical difficulties. The second, that difficulties in learning are caused by external factors such as working memory, inhibitory functions or attentional functions. 

What is the Purpose of Learning Normal Distributions?

As we learn about standard deviations, CDF, PDF and binomial distribution in the h2 mathematics syllabus or math classes, it is reasonable to ask why we learn these terms and how it is used. Take a breather and understand more about why this chapter of statistics is taught!

Statistical Analysis: The normal distribution, also known as the Gaussian distribution or bell curve, is one of the most important probability distributions in statistics. It is used to describe many real-world phenomena, such as heights, weights, IQ scores, and measurement errors. By understanding the properties of the normal distribution, you can perform statistical analysis, make predictions, and draw conclusions about various data sets.

Hypothesis Testing: In hypothesis testing, the normal distribution is frequently used to assess the probability of obtaining certain sample statistics. By knowing the properties of the normal distribution, you can calculate probabilities, perform hypothesis tests, and make inferences about population parameters based on sample data.

*In the A level math syllabus hypothesis testing is taught, this will also be taught later in the lessons!

In real life situations, understanding the normal distribution helps you make more informed decisions in various fields. For example, in quality control, you can use the normal distribution to set tolerance limits and identify potential defects in a manufacturing process. In finance, the normal distribution is often used to model returns on investments, which aids in risk assessment and portfolio management.

Overall, learning about the normal distribution equips you with a foundational understanding of probability and statistics, enabling you to analyze data, make predictions, and draw conclusions in a wide range of fields. It provides a valuable framework for quantitative reasoning and decision making.

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