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# Normal Distribution Percentile Calculator + Online Solver With Free Steps

The **Normal Distribution Percentile Calculator** is a simple online tool that determines the value in a normal distribution with a specific proportion of occurrences below it.

When comparing children’s height and weight to one another to determine how they develop compared to their peers, percentiles are frequently employed in standardized examinations like the GRE.

## What Is a Normal Distribution Percentile Calculator?

**A Normal Distribution Percentile Calculator is an online tool that determines the Gaussian distribution of the given data and is among the most commonly used tools in statistical analyses.**

This is typical because many natural processes are dispersed naturally or have a reasonably comparable dispersion. Height, thickness, and measurement error are some instances of regularly distributed data.

**The Normal distribution has a uniform “Bell Curve” pattern, with more data concentrated around the average value at the center and decreasing probability of occurrence as the value moves away from the center.**

The outcome tends toward the normal distribution when independent random factors are included in the** Central Limit Theorem**.

$ Z= \frac{(x-\mu)}{ \sigma } $ distributes as the standard normal distribution when X distributes normally, with $ \mu $ mean and $ \sigma $ standard deviation.

This allows you to construct any normally distributed curve based on the standard normal distribution.

## How To Use a Normal Distribution Percentile Calculator?

You can use the **Normal Distribution Percentile Calculator** to get the desired results by entering the required data and pressing “enter.” Therefore, you can adhere to the guidelines to obtain the desired result.

### Step 1

Enter the **values of Percentile, mean **$ \mu $,** and standard deviation **$ \sigma $** **in the provided entry boxes.

### Step 2

Press the “Submit” button to calculate the** Percentiles of a Normal Distribution** for the given data and to view the detailed, step-by-step solution for the **Normal Distribution Percentile Calculation**.

## How Does a Normal Distribution Percentile Calculator Work?

The **Normal Distribution Percentile Calculator** works by determining the gaussian distribution of the given dataset available.

**The mean value (average) of a normal distribution also serves as the median and the mode (mode is defined as the value that occurs with the greatest frequency).**

50% of values are lower than the mean, and 50% of values are higher than the mean since this distribution is symmetric about the center.

**The standard deviation is a further factor that defines the normal distribution. It describes the extent of the numbers.**

In general, results should be within one standard deviation of the** mean (68%)**,** two standard deviations (95%)**, and** three standard deviations (99.7%)**, respectively.

The z-score is the quantity that represents the number of standard deviations from the mean.

The standard deviation of your distribution could be unknown to you, even though you are aware of the variance. **By simply computing the square root of the variance, the latter is simple to calculate.**

Finding any percentile for a normal distribution X can be done by following the procedures shown below:

- If you need to find x and are given the likelihood (percent) less than x, you translate this as Locate a such that p(X<a) = p. Find the
**pth**percentile for X, in other words. Go to Step 2. - Determine b where p(X > b) = p if you are given the probability (percent) greater than x, and you need to find x.
- This should be rewritten as a percentile (less-than) problem: Locate b in which p(X > b) = 1 – p. This means to determine X’s (1 – p)th percentile.
- Find the probability in the Z-body table that is closest to p (from Step 1a) or 1 – p to determine the matching percentile for Z.
- Discover the row and column in which this probability appears (using the table backward). The intended z-value is this.
- By using, the z-value can be converted back to the original units, the x-value.
- $ x = \mu +z \sigma $
- Finally, you’ve located the required percentile for X. The expression in this step is simply a rewrite of the z-formula.
- $ Z= \frac{(x-\mu)}{ \sigma } $
- Therefore, x has the problem solved.

### Normal Probability Density Function Formula (PDF)

The PDF formula is given as follows:

\[ f(x) = \frac{1}{ \sigma \sqrt{2 \pi}} exp(-\frac{(x – \mu)^2}{2 \sigma^2} ) \]

### Normal Cumulative Distribution Function Formula (CDF)

The CDF formula is given as:

\[ \frac{1}{2} [ 1+ erf( \frac{x – \mu}{ \sigma \sqrt{2}})] \]

Z stands for standard distribution with $ \mu = 0 $ and $ \sigma = 1$.

\[ Z= \frac{(x-\mu)}{ \sigma } \]

## Solved Examples

Following are some examples to help us better understand how the normal distribution percentage calculator works.

### Example 1

What is the 99% percentile ranking given a mean $ \mu $ of 1000 and a standard deviation $ \sigma $ of 50?

### Solution

Below is the normal distribution’s z-score formula.

\[ Z= \frac{(x-\mu)}{ \sigma } \]

Finding x by rearranging this formula gives us:

\[ x = \mu +z \sigma \]

**conf check = 98**

An inverse query for 99% in our normal distribution table yields a z-value of 2.326.

When we enter our numbers, we obtain:

**x = [ 1000 + 2.326*(50) ]**

**x = 1000 + 116.3**

**x = 1116.3**

### Example 2

What is the 88% percentile ranking given a mean $ \mu $ of 265 and a standard deviation $ \sigma $ of 34?

### Solution

Below is the normal distribution’s z-score formula.

\[ Z= \frac{(x-\mu)}{ \sigma } \]

Finding x by rearranging this formula gives us:

\[ x = \mu +z \sigma \]

**conf check = 76**

An inverse query for 88% in our normal distribution table yields a z-value of 1.174986792.

When we enter our numbers, we obtain:

**x = [ 265 + 1.17*(34) ]**

**x = 265 + 39.949**

**x = 304.949**