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The probability that a normally distributed variable X {\textstyle X} with known μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^ {2}} is in a particular set, can be calculated by using the fact that the fraction Z = (X − μ) / σ {\textstyle Z= (X-\mu )/\sigma } has a standard normal distribution.
Apr 23, 2022 · The mean, median, and mode of a normal distribution are equal. The area under the normal curve is equal to 1.0 1.0. Normal distributions are denser in the center and less dense in the tails. Normal distributions are defined by two parameters, the mean (μ μ) and the standard deviation (σ σ).
A normal distribution is completely determined by its mean μ μ and its standard deviation σ σ, which means there are an infinite number of normal distributions. The mean μ μ determines the center of the distribution—a change in the value of μ μ causes the graph to shift to the left or right.
- Why Is It So Important?
- Main Characteristics
- The Standard Normal Distribution
- The Normal Distribution in General
- Density Plots
- How to Cite
The normal distribution is extremely important because: 1. many real-world phenomena involve random quantities that are approximately normal (e.g., errors in scientific measurement); 2. it plays a crucial role in the Central Limit Theorem, one of the fundamental results in statistics; 3. its great analytical tractability make it very popular in sta...
Sometimes it is also referred to as "bell-shaped distribution" because the graph of its probability density functionresembles the shape of a bell. As you can see from the above plot, the density of a normal distribution has two main characteristics: 1. it is symmetric around the mean(indicated by the vertical line); as a consequence, deviations fro...
The adjective "standard" indicates the special case in which the mean is equal to zero and the variance is equal to one.
While in the previous section we restricted our attention to the special case of zero mean and unit variance, we now deal with the general case.
This section shows the plots of the densities of some normal random variables. These plots help us to understand how the shape of the distribution changes by changing its parameters.
Please cite as: Taboga, Marco (2021). "Normal distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/normal-distribution.
Apr 23, 2022 · A normal distribution with a mean of \(0\) and a standard deviation of \(1\) is called a standard normal distribution. Areas of the normal distribution are often represented by tables of the standard normal distribution.
Feb 22, 2024 · A normal distribution, often referred to as a bell curve, is symmetrical on the left and right, with the mean, median, and mode being the value in the center. There are lots of data values near the center, then fewer and fewer as the values get further from the center.
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The location of the mean simply indicates the location of the line of symmetry, in a normal distribution. This means there are an infinite number of normal probability distributions. One distribution of special interest is called the standard normal distribution.
