# Normal Probability Calculator for Sampling Distributions

Instructions: This Normal Probability Calculator for Sampling Distributions will compute normal distribution probabilities for sample means $$\bar X$$, using the form below. Please type the population mean ($$\mu$$), population standard deviation ($$\sigma$$), and sample size ($$n$$), and provide details about the event you want to compute the probability for (for the standard normal distribution, the mean is 0 and the standard deviation is 1): Population Mean ($$\mu$$) Population St. Dev. ($$\sigma$$) Sample Size ($$n$$)
Two-Tailed:
≤ X ≤
Left-Tailed:
X ≤
Right-Tailed:
X ≥

When a sequence of normally distributed variables $$X_1, X_2, ...., X_n$$ is averaged, we get the sample mean

$\bar X = \frac{1}{n}\sum_{i=1}^n X_i$

Since any linear combination of normal variables is also normal, the sample mean $$\bar X$$ is also normally distributed (assuming that each $$X_i$$ is normally distributed). The distribution of $$\bar X$$ is commonly referred as to the sampling distribution of sample means.

Assuming that $$X_i \sim N(\mu, \sigma^2)$$, for all $$i = 1, 2, 3, ...n$$, then $$\bar X$$ is normally distributed with the same common mean $$\mu$$, but with a variance of $$\displaystyle\frac{\sigma^2}{n}$$. This tells us that $$\bar X$$ is also centered at $$\mu$$ but its dispersion is less than that for each individual $$X_i$$. Indeed, the larger the sample size, the smaller the dispersion of $$\bar X$$.

If you want to compute normal probabilities for one single observation $$X$$, you can use this calculator with $$n=1$$, or you can use our regular normal distribution calculator.

In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.