Normal Distribution calculator calculates the area under a bell curve and gives the probability which is higher or lower than any arbitrary $X$. It's an online
statistics and probability tool requires range of variable, $X_1$ to $X_2$, mean of normal distribution and standard deviation to find the corresponding probabilities. More precisely, this Guassian distribution calculator help us to find: probability less than a $X_1$, probability greater than a $X_1$, probability less than a $X_2$, probability greater than a $X_2$ and probability between
$X_1$ and $X_2$. Show
Input: Four real numbers. The first two are representing the mean and standard deviation. Normal Distribution Formula: The probabilities are determined by the formulas $$\begin{align} P(X \leq X_1)&=\frac1{\sqrt{2\pi \sigma^2}}\int_{-\infty}^{X_1} e^{-\frac{(x-\mu)^2}{2\sigma^2 }}dx\\ P(X \geq X_1)&=\frac1{\sqrt{2\pi \sigma^2}}\int^{\infty}_{X_1} e^{-\frac{(x-\mu)^2}{2\sigma^2 }}dx\end{align}$$ where $\mu$ is the population mean and $\sigma^2$ is the population variance and $e\approx 2.7183$ and $\pi\approx 3.1416$. What is Normal Distribution?The normal distribution plays an important role in probability theory. In 1809, C.F. Gauss gave the first application of the normal distribution. He modeled observational errors in
astronomy. , Bell curve or Gaussian function is used to represent the distribution of probability density function. The mean determines the location of the center of the Bell curve, and the standard deviation determines the height and width of the Bell curve. Note that the total area under the curve is 1. A random variable with a normal distribution is called normally distributed. For a normally distributed random variable $X$ with the population mean $\mu$ and the population variance $\sigma^2$, it holds that The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. It is denoted as $N(0,1)$. In this case, the probability density function is $$f(x)=\frac1{\sqrt{2\pi}} e^{-\frac{x^2}{2 }}$$ and the graph of this function is: Note that this concept applies mainly to populations rather than samples. The continuous normal distribution cannot be obtained from a sample. How to Calculate Normal Distribution?To find the probability $P(X_1 < X < X_2)$, we need to find the area under the normal curve between the points $X_1$ and $X_2.$ Because the standard normal distribution is used very often,
there exist tables to help us calculate probabilities (Standard Normal Table).
The normal distribution calculator to finding the probability less than $1.5$, probability greater than $1.5$, probability less than $1$, probability greater than $1$ and probability between $1$ and $1.5$ with a mean of $0.5$ and standard deviation of $2$. For any other values, just supply the four real numbers and click on the "CALCULATE" button. The grade school students may use this Guassian distribution calculator to verify the results derived by hand or do their homework problems efficiently. How do you find probability with mean and standard deviation?In a normally distributed data set, you can find the probability of a particular event as long as you have the mean and standard deviation. With these, you can calculate the z-score using the formula z = (x – μ (mean)) / σ (standard deviation).
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