Normal Distribution

Properties of the Normal Distribution

The normal distribution is a symmetrical, bell-shaped probability distribution. It is characterized by its mean (μ) and standard deviation (σ). These parameters fully define the distribution.

• Symmetrical: The left and right sides of the distribution are mirror images of each other.
• Bell-shaped: The distribution is highest at the mean and gradually decreases towards both ends.
• Mean (μ): The average value of the distribution. The peak of the bell is at the mean.
• Standard Deviation (σ): A measure of the spread or dispersion of the data. A smaller standard deviation indicates that the data points are clustered closely around the mean, while a larger standard deviation indicates that the data points are more spread out.
• Empirical Rule (68-95-99.7 Rule):
• Approximately 68% of the data falls within one standard deviation of the mean (μ ± σ).
• Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).
• Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).

Standard Normal Distribution

A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is often denoted by Z.

We can transform any normal distribution with mean μ and standard deviation σ into a standard normal distribution using the Z-score formula:

$$Z = \frac{X - \mu}{\sigma}$$

Where:

• X is a data point from the original distribution.
• μ is the mean of the original distribution.
• σ is the standard deviation of the original distribution.

Applications of the Normal Distribution

The normal distribution is widely used in statistics because many natural phenomena tend to follow it. Some common applications include:

• Central Limit Theorem: This theorem states that the distribution of the sum or average of a large number of independent random variables approaches a normal distribution, regardless of the original distribution of the variables.
• Sampling Distributions: The distribution of a sample statistic (e.g., sample mean) is approximately normal, especially when the sample size is large.
• Quality Control: Used to monitor and control the quality of products by identifying deviations from expected values.
• Biological Measurements: Many biological measurements, such as height and weight, tend to follow a normal distribution.

Approximations using the Normal Distribution

When the population standard deviation (σ) is known, we can use the standard normal distribution table (Z-table) to find probabilities associated with the normal distribution.

For cases where the population standard deviation is unknown and we use the sample standard deviation (s), we use a t-distribution instead of the normal distribution. However, for large sample sizes (n ≥ 30), the t-distribution approximates the standard normal distribution.

Z-score	Probability (P(Z < Z-score))
-3	0.00135
-2	0.0228
-1	0.1587
0	0.5
1	0.1587
2	0.0228
3	0.00135

Calculating Probabilities using the Z-table

To find the probability that a value X from a normal distribution with mean μ and standard deviation σ falls within a certain range, we first convert the X values to Z-scores using the formula above. Then, we use the Z-table to find the corresponding probabilities.

For example, to find the probability that a value X is less than 70, given that μ = 75 and σ = 5:

1. Calculate the Z-score: $Z = \frac{70 - 75}{5} = -1$
2. Find the probability P(Z < -1) from the Z-table. This probability is approximately 0.1587.