Normal Probability Calculator for Sampling Distributions (2024)

This normal probability calculator for sampling distributions finds the probability that your sample mean lies within a specific range.

It calculates the normal distribution probability with the sample size (n), a mean values range (defined by X₁ and X₂), the population mean (μ), and the standard deviation (σ).

Keep reading to learn more about:

  • What is the sampling distribution of the mean?
  • How to find the standard deviation of the sampling distribution.
  • How to calculate probabilities for sampling distributions.
  • How to use our normal probability calculator for sampling distributions.

🔎 If you need a calculator that makes the same, but for sample proportions, check our sampling distribution of the sample proportion calculator. If you're interested in the opposite problem: finding a range of possible population values given a probability level, look at our sampling error calculator.

Calculation of normal probability for sampling distributions

Many real-life phenomena follow a normal distribution. For example, the American men's height follows that distribution with a mean of approximately 176.3 cm and a standard deviation of about 7.6 cm. In the following plot, you can see the distribution graph of those heights.

Normal Probability Calculator for Sampling Distributions (1)

Usually, we use samples to estimate population parameters like a population mean height. The most common example is using the sample mean to estimate the population mean.

If you take different samples from a population, you'll probably get different mean values each time. Therefore, the sample mean is also a random variable we can describe with some distribution. This distribution is known as the sampling distribution of the sample mean, which we will name the sampling distribution for simplicity.

If the original population follows a normal distribution, the sampling distribution will do the same, and if not, the sampling distribution will approximate a normal distribution. The central limit theorem describes the degree to which it occurs.

A common task is to find the probability that the mean of a sample falls within a specific range. We can do it using the same tools for calculating normal distributions (using the z-score). The only difference is that the standard deviation of the sampling distribution (σXˉσ_{\bar X}σXˉ) is equal to the population standard deviation divided by the square root of the sample size:

σXˉ=σn\footnotesize σ_{\bar X}=\frac{σ}{\sqrt{n}}σXˉ=nσ

Then, the formula to calculate the z-score is:

z=Xμσ/n\footnotesize z= \frac{X-μ}{σ/ \sqrt{n}}z=σ/nXμ

With the z-score value, you can calculate the probability using available tables or, even better and faster, using our p-value calculator. Read on to look at an example of how to do it.

🙋 If you're interested in the σXˉσ_{\bar X}σXˉ term, you can learn more about it in our standard deviation of the sample mean calculator.

How this sampling distribution calculator works: an example

The average height of the American women (including all race and Hispanic-origin groups) aged 20 and over is approximately 161.3 cm, with a standard deviation of about 7.1 cm. Let's suppose you randomly sample 7 American women. What is the probability that the average height falls below 160 cm?

To know the answer, follow these steps:

  1. Input the population parameters in the sampling distribution calculator (μ = 161.3, σ = 7.1)
  2. Select left-tailed, in this case.
  3. Input the sample data (n = 7, X = 160).
  4. Your result is ready. It should be 0.314039. Therefore, the probability that the average height of those women falls below 160 cm is about 31.4%.

Alternatively, we can calculate this probability using the z-score formula:

zscore=Xμσ/n=160161.37.1/7=0.484433\footnotesize\begin{align*}z_{score}&=\frac{X-μ}{σ/\sqrt{n}}\\\\&= \frac{160-161.3}{ 7.1/ \sqrt{7}}=-0.484433\end{align*}zscore=σ/nXμ=7.1/7160161.3=0.484433

P(Xˉ<170)=P(zscore<0.484433)=0.314039\footnotesize\begin{align*}P(\bar X<170)&=P(z_{score}<−0.484433)\\&=0.314039\end{align*}P(Xˉ<170)=P(zscore<0.484433)=0.314039

How to find the mean of the sampling distribution?

If you know the population mean, you know the mean of the sampling distribution, as they're both the same. If you don't, you can assume your sample mean as the mean of the sampling distribution.

FAQ

What is the sampling distribution of the mean?

The sampling distribution of the mean describes the distribution of possible means you could obtain from infinitely sampling from a given population.

How to calculate probability in sampling distribution?

  1. Define your population mean (μ), standard deviation (σ), sample size, and range of possible sample means.
  2. Input those values in the z-score formula zscore = (X̄ - μ)/(σ/√n).
  3. Considering if your probability is left, right, or two-tailed, use the z-score value to find your probability.
  4. Alternatively, you can use our normal probability calculator for sampling distributions.

How to find the standard deviation of the sampling distribution?

Depending on the information you possess, there are two ways:

  • If you know the population standard deviation (σ), divide it by the square root of the sample size: σ = σ/√n.
  • If you don't have σ, estimate it with the sample standard deviation (s): σ = s/√n.

What is the probability of getting a sample mean greater than the population mean?

The probability of getting a sample mean greater than μ (population mean) is 50%, as long as your sampling distribution follows a normal distribution (this occurs if the population distribution is normal or the sample size is large).

Normal Probability Calculator for Sampling Distributions (2024)

FAQs

How to calculate probability for sampling distribution? ›

Define your population mean (μ), standard deviation (σ), sample size, and range of possible sample means. Input those values in the z-score formula zscore = (X̄ - μ)/(σ/√n). Considering if your probability is left, right, or two-tailed, use the z-score value to find your probability.

How to calculate normal probability distribution? ›

The probability of P(a < Z < b) is calculated as follows. Then express these as their respective probabilities under the standard normal distribution curve: P(Z < b) – P(Z < a) = Φ(b) – Φ(a). Therefore, P(a < Z < b) = Φ(b) – Φ(a), where a and b are positive.

How to find the probability of a sample mean from a normal distribution? ›

Once we have determined that the sample mean is Normally distributed, we can compute probabilities with ˉx. We first compute z=ˉx−μσ/√n and then use the Normal Probability Applet to compute probabilities about z.

What is the formula for the sampling normal distribution? ›

For samples of any size drawn from a normally distributed population, the sample mean is normally distributed, with mean μˉX=μ and standard deviation σˉX=σ/√n, where n is the sample size. The effect of increasing the sample size is shown in Figure 6.4 "Distribution of Sample Means for a Normal Population".

What is the formula for the probability of a sample? ›

To calculate probability, you'll use simple multiplication and division. Probability equals the number of favorable outcomes divided by the total number of outcomes.

What is the probability of normal distribution exactly? ›

The normal distribution is a continuous distribution and hence the probability of getting exactly 1.23 is zero. Further, the probability of getting either 1.23 or -1.23 is also zeo. Said differently, the set {−1.23;1.23} has measure zero (the probability that any of its values are produced by a normal is zero).

What is the formula for the normal distribution with the mean? ›

What is the normal distribution formula? For a random variable x, with mean “μ” and standard deviation “σ”, the normal distribution formula is given by: f(x) = (1/√(2πσ2)) (e[-(x-μ)^2]/^2).

How to find the z-score of a sampling distribution? ›

Z Score = (x − x̅ )/σ
  1. x = Standardized random variable.
  2. x̅ = Mean.
  3. σ = Standard deviation.

What is the formula that generates the normal probability distribution? ›

A continuous random variable X is normally distributed or follows a normal probability distribution if its probability distribution is given by the following function: f x = 1 σ 2 π e − x − μ 2 2 σ 2 , − ∞ < x < ∞ , − ∞ < μ < ∞ , 0 < σ 2 < ∞ .

How do you determine normal distribution? ›

You can test the hypothesis that your data were sampled from a Normal (Gaussian) distribution visually (with QQ-plots and histograms) or statistically (with tests such as D'Agostino-Pearson and Kolmogorov-Smirnov).

How to get a random sample from normal distribution? ›

Equivalently, we can think of the sample as being obtained by considering the x--y plane and choosing n points randomly from the region under the curve: {(x,y):0<y<fX(x)}, where fX(x) is the pdf of X.

How do you find the probability of a distribution? ›

Probability Distribution Function

It can be written as F(x) = P (X ≤ x). Furthermore, if there is a semi-closed interval given by (a, b] then the probability distribution function is given by the formula P(a < X ≤ b) = F(b) - F(a). The probability distribution function of a random variable always lies between 0 and 1.

How to find probability of sample proportion? ›

How to find the sample proportion?
  1. Determine the number of successes in your sample.
  2. Determine your sample size.
  3. Divide the number of successes by the sample size. This result represents the fraction or percentage of successes in your sample. That's how you find the sample proportion.
Jun 27, 2024

What is the probability of sampling method? ›

Probability sampling refers to the selection of a sample from a population, when this selection is based on the principle of randomization, that is, random selection or chance. Probability sampling is more complex, more time-consuming and usually more costly than non-probability sampling.

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