Normal Distribution vs. Standard Normal Distribution

Attribute	Normal Distribution	Standard Normal Distribution
Mean	μ	0
Standard Deviation	σ	1
Probability Density Function	1/(σ√(2π)) * e^(-(x-μ)^2/(2σ^2))	1/√(2π) * e^(-x^2/2)
Cumulative Distribution Function	Φ((x-μ)/σ)	Φ(x)
Standardized	No	Yes

Introduction

Normal distribution and standard normal distribution are two important concepts in statistics that are often used in various fields such as finance, science, and engineering. While they are related, they have distinct attributes that set them apart. In this article, we will explore the differences between normal distribution and standard normal distribution.

Definition

Normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric around its mean, with the majority of the data points falling close to the mean and fewer data points further away. It is characterized by its bell-shaped curve. Standard normal distribution, on the other hand, is a specific type of normal distribution with a mean of 0 and a standard deviation of 1. It is a standardized version of the normal distribution.

Parameters

Normal distribution can be defined by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the center of the distribution, while the standard deviation measures the spread of the data points around the mean. In standard normal distribution, the mean is always 0 and the standard deviation is always 1. This standardization allows for easier comparison and analysis of data.

Standardization

One of the key differences between normal distribution and standard normal distribution is the process of standardization. To convert a normal distribution to a standard normal distribution, you need to subtract the mean from each data point and then divide by the standard deviation. This process transforms the data into z-scores, which represent the number of standard deviations a data point is from the mean. Standardization is useful for comparing data sets with different units or scales.

Use Cases

Normal distribution is commonly used to model real-world phenomena such as heights, weights, test scores, and IQ scores. It is also used in hypothesis testing, regression analysis, and quality control. Standard normal distribution, on the other hand, is often used in statistical tables and calculations. It simplifies calculations and makes it easier to interpret results. Z-scores from a standard normal distribution are used to calculate probabilities and make statistical inferences.

Probability

Normal distribution and standard normal distribution are both used to calculate probabilities. In a normal distribution, the area under the curve represents the probability of a data point falling within a certain range. The total area under the curve is equal to 1. In a standard normal distribution, the z-score is used to calculate probabilities. For example, a z-score of 1 represents the area under the curve to the left of that point, which is approximately 0.8413.

Graphical Representation

When graphed, normal distribution and standard normal distribution have different shapes. Normal distribution has a bell-shaped curve that is symmetrical around the mean. The curve extends infinitely in both directions. Standard normal distribution, on the other hand, has a symmetrical bell-shaped curve centered at 0 with a standard deviation of 1. The curve is standardized and does not vary in shape or location.

Conclusion

In conclusion, normal distribution and standard normal distribution are important concepts in statistics that are used to model and analyze data. While normal distribution is characterized by its mean and standard deviation, standard normal distribution is a standardized version with a mean of 0 and a standard deviation of 1. Both distributions have their own unique attributes and applications, making them essential tools for statisticians and researchers.