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:
🔎 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.
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.
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 ( σ X ˉ σ_ σ X ˉ ) is equal to the population standard deviation divided by the square root of the sample size:
σ X ˉ = σ n \footnotesize σ_=\frac> σ X ˉ = nThen, the formula to calculate the z-score is:
z = X − μ σ / n \footnotesize z= \fracWith 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 σ X ˉ σ_ σ X ˉ term, you can learn more about it in our standard deviation of the sample mean calculator.
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:
Alternatively, we can calculate this probability using the z-score formula:
z s c o r e = X − μ σ / n = 160 − 161.3 7.1 / 7 = − 0.484433 \footnotesize \begin z_&=\frac>\\\\&= \frac< 7.1/ \sqrt>=-0.484433 \end z score = σ / n
160 − 161.3 = − 0.484433
P ( X ˉ < 170 ) = P ( z s c o r e < − 0.484433 ) = 0.314039 \footnotesize \beginP(\bar X<170)&=P(z_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.
The sampling distribution of the mean describes the distribution of possible means you could obtain from infinitely sampling from a given population.
Depending on the information you possess, there are two ways:
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).
