The normal distribution is the most important and most widely used distribution in statistics. It is sometimes called the "bell curve," Strictly speaking, it is not correct to talk about "the normal distribution" since there are many normal distributions. Normal distributions can differ in their means and in their standard deviations. Normal distributions are symmetric with relatively more values at the center of the distribution and relatively few in the tails.

1. Understanding Statistics
The Normal Probability Distribution
f(x) =
1
ϭ 2π
𝑒−
𝑥−µ 2
2ϭ
2

2. The Normal Probability Distribution
Example: Lets' say you go to scope and buy watermelon; Measure the length many time, random. Each time you get
a result which is the length. The random variable (X) is how long the watermelon is. Soon we will find that most of
the watermelon will be around a value which will be the average (µ). You might still have outliers but there will be
an average.
Properties:
- It is the most common probability distribution, that’s why is called normal
- Follow bell shape
- Continuous
- Normal curve is symmetric on both sides of the mean
- Completely defined by the mean (µ) and Std Dev (ϭ)
- Total area under curve is equal to 1:
- when you add all prob. and cover all outcomes the probability is 1
- Curve does not touch the axes, it goes asymptotically

3. The Normal Probability Distribution
- Formula or Equation for normal distribution:
f(x) =
1
ϭ 2π
𝑒−
𝑥−µ 2
2ϭ2
The graph is the representation of the above function
Usually we use tables most of the time, not the formula.
Graph generated with: https://homepage.stat.uiowa.edu/~mbognar/applets/normal.html

4. Proprieties of the Normal Distribution
- f(x) =
1
ϭ 2π
𝑒−
𝑥−µ 2
2ϭ2
- µ= 60
- Ϭ1 = 5
- Ϭ2= 25
- Note: both have the same mean but the height if the second is lower and is spreader on a larger range (fatter)
- This time consider same std dev, different mean:
- ϭ= 20
- µ1 = 40
- µ2= 50
- Graph are identical in shape only translated on (x) axes according with mean

5. The Area Under The Normal Distribution
- 68% fall under 1 Std Dev
Problem:
µ = 20; ϭ = 5
What is the probability of picking watermelon less than 15 cm?
- The probability will be everything on the left of 1 std (15),
- You cold use the formula but is easier by tables;
(easiest is to use online, freely available tools)
https://homepage.stat.uiowa.edu
- What is the probability of picking watermelon less than 35 cm?
- It cover the whole graph as it represent 3 Std Dev; 99.7%