2025-10-20
The Normal Distribution is one of the most important concepts in statistics and is commonly represented by a bell-shaped curve that is symmetric around the mean,showing concentration of that data.
This presentation explores:
The Normal Distribution is a continuous probability distribution that models how values of a random variable are dispersed around the mean.
It has the following characteristics:
The Normal Distribution is defined by its probability density function:
\[ f(x) = \frac{1} {\sigma \sqrt{2\pi}} e^{ -\frac{(x - \mu)^2}{2\sigma^2} } \]
where: - \(\mu\): the mean — determines where the center is defined
\(\sigma\): the standard deviation that measures deviation from mean
\(e\): Euler’s number
\(\pi\): the constant (\(\approx 3.14\))
# Generate 1000 random values from a normal distribution set.seed(111) x <- rnorm(1000, mean = 50, sd = 10) # Display the summary statistics summary(x)
## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 16.77 43.46 50.20 50.11 56.76 79.26
sd(x)
## [1] 9.879296
Summary includes the mean and we also have the standard deviation.
The blue curve is bell curve with mean = 0 and sd = 2. The red curve has lower sd, making curve narrower and green curve shows that increasing the mean shifts the distribution to the right.
The z-score measures the standard deviations between a point \(x\) and mean \(\mu\) in normal distribution. Formula: \[ z = \frac{x - \mu}{\sigma} \]
Principle: - A z-score of 0 means the value is same as mean.
- Positive z-scores shows it’s above mean.
- Negative z-scores shows it’s below mean.
- The z-score allows comparison of values from different normal distributions through standardization.
Example:
If \(X \sim N(20, 2^2)\) and we observe \(x = 30\):
\[ z = \frac{30 - 20}{2} = 5 \]