Normal distribution, also known as the Gaussian distribution or bell curve, is a statistical distribution that is symmetric and bell-shaped. It is widely used in various fields to model continuous data that tends to cluster around a mean value. The distribution is characterized by its mean (μ) and standard deviation (σ), which determine the shape and spread of the curve.
The density of the normal distribution describes the probability of observing a particular value within the distribution. The density function, also known as the probability density function (PDF), represents the relative likelihood of different values occurring within the distribution. The total area under the curve of the density function is equal to 1, indicating the complete probability distribution.
Example: Let's consider an example of a normal distribution representing the heights of adult males. Suppose the mean height is 175 cm with a standard deviation of 5 cm.
1. Normal Distribution: The normal distribution curve for this example would be symmetric and bell-shaped. The peak of the curve represents the mean height of 175 cm. As we move away from the mean, the likelihood of observing heights decreases, following the shape of the bell curve. The standard deviation of 5 cm indicates the spread or dispersion of heights around the mean.
2. Density of Normal Distribution: The density of the normal distribution describes the probability of observing a particular height value. For instance, let's say we want to calculate the probability of randomly selecting an adult male with a height between 170 cm and 180 cm. We can use the density function to determine this probability. By integrating the density function between the heights of 170 cm and 180 cm, we can find the probability of obtaining a height within this range.The density function of the normal distribution provides a continuous range of probabilities for different height values. It allows us to determine the likelihood of observing specific heights or ranges of heights within the distribution.
The normal distribution and its density function are widely used in statistical analysis, hypothesis testing, and modeling various natural phenomena. They provide a useful framework for understanding the behavior and characteristics of data that follow a bell-shaped pattern.
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