Normal and Standard Normal Distribution
Hello all, in this post I will discuss:
1: Normal/Gaussian Distribution
2: Standard Normal Distribution
Before learning about Normal Distribution, you should know about some basic terms.
1: — Mean
Mean is the average of all the numbers present in a dataset.
For e.g. I have a dataset D which contains information about weights in kg of students in class such that
D = [54,65,58,75,62,82]
Here I have 6observation so n = 6
Mean = (54+65+58+75+62+82)/6 = 66 kg
2: — Standard Deviation
Standard Deviation measures the spread of data points relative to its mean.
Normal Distribution:
Normal Distribution also known as Gaussian distribution is one of the most important distributions in probability and statistics. Normal distribution tells:
1: In a dataset how observation is distributed around the mean.
2: Characteristics or features of that distribution.
Normal/Gaussian Distribution has two parameters i.e. Mean and Standard Deviation.
Let us have a random variable X which belongs to normal or gaussian distribution with mean = µ and standard deviation = σ
i.e. X ~ N (µ, σ)
As X follows normal distribution it has the following properties known as an empirical formula:
1. P (µ — σ ≤ X ≤ µ + σ) = 68.27%
which is a random variable X follows normal distribution then 68.27% of the points belonging to that random variable X fall within the range of 1 standard deviation.
2. P (µ — 2σ ≤ X ≤ µ + 2σ) = 95.45%
95.45% of data points belonging to random variable X fall within the range of 2 standard deviations.
3. P (µ — 3σ ≤ X ≤ µ + 3σ) = 99.73%
99.73% of data points belonging to random variable X fall within the range of 3 standard deviations.
Examples of Normal Distribution:
1. Heights of people.
2. Normal Distribution is used in the economy for income distribution.
3. Rolling a dice etc.
Properties of Normal/Gaussian Distribution:
1. The Curve is bell-shaped.
2. The curve is bilaterally symmetrical.
3. Area under the curve is 100%
4. Mean = Median = Mode
Standard Normal Distribution:
The standard normal distribution is a normal/gaussian distribution with mean, µ = 0, and standard deviation, σ = 1.
Standard Normal Variate:
Standard Normal Variate (Z) is a random variable following a normal/gaussian distribution with mean, µ = 0 and standard deviation,
σ = 1.
Z ~ N (0,1)
Standardization
It is a transformation technique used to convert/transform any random variable, X following a normal/gaussian distribution to a standard normal variate, Z.
Let us have a random variable X follows a normal distribution with mean, µ, and standard deviation, σ.
So, we can convert any random variable following a normal distribution having to mean, µ, and standard deviation, σ. i.e. X ~ N (µ, σ) to Standard normal variate Z ~ N (0,1) by:
