This free online calculator calculates the probability density function for normal distribution using the means, standard deviation, and the specific point where function x is to be evaluated.

## What is the Probability Density Function (PDF)?

In statistics, a probability function that describes the density of any continuous random variable within a specific range of values is called a probability density function or probability distribution function.

The PDF must always be non-negative and the integral of the probability distribution function over the entire range of possible values for the random variable should be equal to one.

**Note **

The PDF itself does not give the probability of a specific value but rather the probability density at that point.

## Formula to Calculate Probability Density Function for Normal Distribution

The probability density function (PDF) for a normal distribution (also known as a Gaussian distribution) can be calculated by the following formula:

**f(x) **is the probability density function.**x **denotes the quantity for which probability is to be calculated.**σ **is the standard deviation.**π (pi)** is a mathematical constant (**π ≅ 3.14159**)**e **is a Euler constant (**e ≅ 2.718**)**μ **is the mean of the distribution.

## How to Find PDF for Data Set (or Ungrouped Data)?

Let’s learn how to find the probability density function if the means and standard deviation are not given directly.

- First, find the mean by adding up all the values in the dataset and dividing it by the number of data sets.

i.e. **means = (x**_{1, }x_{2}, x_{3}, …, x_{n}) / n

- Use the above formula to find the standard deviation
**σ**.

**σ = √ ((Σ(x**_{i }- x̄)^{2}) / n)

- Consider the number of data sets as
**n**. - Now, put the values of mean, standard deviation, and x into the formula of the probability density function and simplify it.

## Examples of Probability Density Function (PDF)

Use the above Probability Density Function Calculator to calculate (PDF) to save time and get error-free results. Alternatively, you can manually compute the PDF using the examples given below.

**Example 1: (When Raw Score, Means, and Standard deviation are given)**

Find the Probability Density Function (**PDF**) for a raw score of **80 **with a mean of **75 **and a standard deviation of **5**.

**Solution:**

**Step 1:** Identify the parameters required for the formula.

In this example,

Raw score (x) = 80

Means (μ) = 75

Standard deviation (σ) = 5

Also, we know that

**π ≅ 3.14159**

**e ≅ 2.718**

**Step 2: **Substitute the values of x, μ, σ, π, and e into the PDF formula.

**Step 3: **Simplify the obtained result from step 2.

f(80) = 0.08 × 0.607

**f(80) = 0.04856**

**Example 2: (For Dataset or Ungrouped Data)**

Find the Probability Density Function (PDF) of Normal distribution for the following data set:

15, 27, 9, 26, 34, 11

**Solution:**

**Step 1:** Calculate the Mean (μ) and Standard Deviation (σ) for the given Dataset.

Means (μ) = (15 + 27 + 9 + 26 + 34 + 11) / 6

**Means (μ) = 20.3**

To calculate Standard deviation.

**Data Set (x**_{i}) | **x**_{i} - x̄ | **(x**_{i} - x̄)^{2} |

15 | 15 - 20.3 = 5.3 | 28.09 |

27 | 27 - 20.3 = 6.7 | 44.89 |

9 | 9 - 20.3 = - 11.3 | 127.69 |

26 | 26 - 20.3 = 5.7 | 32.49 |

34 | 34 - 20.3 = 13.7 | 187.69 |

11 | 11- 20.3 = - 9.3 | 86.49 |

----- | ----- | Σ(x_{i} - x̄)^{2} = 507.34 |

∴ σ = √ ((Σ(x_{i }- x̄)^{2}) / n)

σ = √ (507.34 / 6)

σ = √ (84.72578)

**Standard deviation = σ = 9.2**

**Step 2:** Put the following values into the PDF formula.

Raw score (x) = Total number of data sets = 6

Means (μ) = 20.3

Standard deviation = σ = 9.2

π ≅ 3.14159

e ≅ 2.718

**Step 3: **Simplify the obtained result from step 2.

f(6) = 0.043 × 0.299

**f(6) = 0.01286**