# Normal Distribution Calculator

Enter the raw score, mean, and standard deviation to find the probability of getting a number above or below that score using the normal distribution calculator below.

## Probabilities:

Learn how we calculated this below

## How to Find a Normal Distribution

A normal distribution, also referred to as a Gaussian distribution, is a continuous probability distribution that is symmetric about the mean.[1] Most of the observations in the sample cluster around a central peak and then tapers off away from the mean equally on both sides.

An ideal normal distribution would be perfectly symmetrical and shaped like a bell, which is why its graph is often referred to as a bell curve. However, real-life data rarely follows an ideal normal distribution.

In some cases, a normal distribution is not possible due to sample size limitations or skewness in the data set. Data that does not follow a normal distribution are called non-normal data.[2]

While a normal distribution is one of the most common probability distributions in statistics, there are also other non-normal probability distributions, such as the binomial distribution or Poisson distribution.

While all normal distributions are symmetrical, the exact shape of the bell curve is defined by two parameters of the data: the mean and standard deviation.

### Standard Normal Distributions

A standard normal distribution, sometimes called a z-distribution, is a special normal distribution that has been standardized by converting its values to z-scores. It is characterized by having a mean equal to zero and a standard deviation equal to 1.

The total area under the curve of a standard normal distribution is exactly equal to 1.

Converting a normal distribution to a standard normal distribution allows comparing scores on distributions that have different means and standard deviations and allows you to normalize scores for decision-making purposes.

Finding the area under a normal distribution bell curve is important as it allows us to calculate the probability of observing a score within a range of the distribution. The proportion of the area under the curve between two points indicates the probability that a score will fall within that range.

It is not trivial to calculate the area under a normal distribution curve, but many formulas and tools have been created to simplify this task.

### Normal Distribution Formula

You can define the curve of a normal distribution using the probability density function using the population mean and standard deviation. The formula states that for any value of x, you can solve the probability density with the function:[3]

f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}

Where:
μ = mean
σ = standard deviation
x = random variable

### Cumulative Distribution Function Formula

In every probability distribution, including normal distributions, the area under the curve is defined by a cumulative distribution function, or CDF. For a normal distribution, the CDF formula[4] denoted by the Greek letter phi (Φ) is:

\Phi(x) = \int_{-\infty }^{x} \frac{e^{-x^{2}/2}}{\sqrt{2\pi}}

Using this formula, you can find the probability of any value being less than a score x. More commonly, other techniques such as using the z-score are used to calculate this probability.

### How to Find the Z-Score

To use the z-score (sometimes called a standard score) to calculate this probability, you first must find the mean and standard deviation that defines the distribution.

Then, calculate the z-score for a given raw score. You can do this using a z-score calculator or using a simple formula: z = x – μ / σ.

z = \frac{x-\mu}{\sigma}

The z-score for a score x is equal to x minus the population mean μ, divided by the population standard deviation σ.

### How to Find the Cumulative Probability

After you calculate the z-score for a given value, you can calculate the cumulative probability of a value being below the observation using spreadsheet software, a z-table, or a graphing calculator.[5]

The probability of a score is greater than or equal to the raw score is known as a p-value. Learn more about calculating p-values with our p-value calculator.

## Key Properties of a Normal Distribution

A normal distribution has several key properties:

• The mean, median, and mode are all equal.
• The distribution is symmetric about the mean, so 50% of values are below it, and 50% of values are above it.
• The value of the distribution is non-zero over the entire real line but is very close to zero for observations that are more than a few standard deviations away from the mean.

## Central Limit Theorem

The central limit theorem states that as you take larger samples, as you calculate their means, they form a normal distribution. In fact, this holds true even if the original population is not normally distributed, provided the sample size is large enough (generally >30 data points).[6]

This is known as the sampling distribution of the mean, which should have the same mean and variance as the original distribution divided by the sample size. This behavior is useful because it allows for much easier statistical analysis.

## The Empirical Rule

The empirical rule states that 99.7% of data that is normally distributed will fall within three standard deviations of the mean, 95% will fall within two standard deviations, and 68% of the data will fall within one standard deviation.

You can use our empirical rule calculator to calculate the amount of data that will fall within these intervals.

## References

1. Chen, J, Normal Distribution, Investopedia, https://www.investopedia.com/terms/n/normaldistribution.asp
2. Krithikadatta J., Normal distribution, J Conserv Dent, 2014, 17(1), 96-97. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3915399/
3. Menke, W., Menke, J., Probability Density Function, Science Direct, https://www.sciencedirect.com/topics/earth-and-planetary-sciences/probability-density-function
4. National Institute of Standards and Technology, 1.3.6.6.1.Normal Distribution, NIST/SEMATECH e-Handbook of Statistical Method, April, 2012, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
5. Kozak, K., Finding Probabilities for the Normal Distribution, LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Statistics_Using_Technology_(Kozak)/06%3A_Continuous_Probability_Distributions/6.03%3A_Finding_Probabilities_for_the_Normal_Distribution
6. LaMorte, W., Central Limit Theorem, Boston University School of Public Health, https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/BS704_Probability12.html