H2 Mathematics - Normal Distribution (Basics Operations)

Title Junior College ‘A’ Levels H1/H2 Mathematics – Normal Distribution Author AprilDolphin Date 10/4/2025 In many situations out there, many types of continuous random variables, such as test scores, height and weight of students of a certain age would show the following characteristics once data got collected and plotted as it would be on a probability distribution. • You would realize that majority of the data are centred in the middle, with extreme small and big values tailing off symmetrically on the left and right of the average measurement respectively. When this happens, the random variable 𝑋 is said to follow a Normal Distribution with parameters 𝜇 and 𝜎2, where 𝑋~𝑁(𝜇, 𝜎2), with 𝜇 referring to the mean and 𝜎2 referring to the variance of the Normal Distribution. (*Be careful when dealing with notations, certain books, software and calculators deal with Normal Distribution using the Standard Deviation 𝜎 parameter rather than variance which is 𝜎2. In such cases, the obvious first step you should take is to square-root the variance to get the value of standard deviation 𝜎.) A basic visual look at the Normal Distribution and some properties to note. • The probability value is the area between the curve and the x-axis. (Which is also the definite integral of the Normal Distribution in question.) • The probability of getting a very specific value in a Normal Distribution is basically zero since area under curve cannot be created on a continuous random variable just with specific values. Instead of defining specific value on a Normal Distribution, we usually define a range of values to calculate probability in a Normal Distribution. [Therefore 𝑃(𝑋 = 𝑥) = 0]

• Any Normal Distribution is symmetrical at the mean 𝜇. (This property is important as certain questions you will encounter requires you to understand this symmetrical property of any Normal Distribution.) Understanding the concept of a Standard Normal Distribution • Any Normal Distribution can technically be transformed to a Standard Normal Distribution. • A Standard Normal Distribution has the property of which the area under curve from negative infinity to positive infinity is exactly 1. • A Standard Normal Distribution also has property of mean 𝜇 = 0 and 𝜎2 = 1, for this reason, a Standard Normal Distribution will also have standard deviation 𝜎 = 1 as well. (Note: In order to input “negative infinity” in Graphing Calculator, press -E99. In order to input a value of “positive infinity” in Graphing Calculator, press E99.) A visual look at the standard normal distribution. Understanding the concept of 𝑍-score in Standard Normal Distribution • 𝑍-score refers to the number of standard deviations from the mean in a standardized normal distribution where the 𝑍 −score can take on any finite values. (−∞ < 𝑍 < ∞) • 𝑍-score of any normal distribution can be computed using the below formula. 𝑍 = 𝑋 − 𝜇 𝜎 𝑍 refers to the 𝑍