The **normal** **distribution** in **R** is categorized as **rnorm**, **dnorm**, and **pnorm**. A **normal distribution**, also known as a Gaussian distribution, is a symmetrical distribution of continuous data with a bell-shaped curve. The normal distribution is widely used in statistical analysis to model the distribution of various data types.

The **r****norm**, **dnorm**, and **pnorm** functions in R help us calculate the normal density, distribution, and quantile function for a set of values.

**pnorm in R**

The **pnorm() in R** is a built-in mathematical function that returns the** Cumulative Density Function (CDF) value **of the **normal distribution** given a specific **random** **variable** **q,** the **population mean** **μ**, and the **population standard deviation** **σ**.

The **pnorm() function** takes several parameters as input, including the normal distribution’s mean, standard deviation, and quantile. It returns the cumulative probability of the normal distribution for the given parameters.

The **pnorm() function** calculates the c. d. f. where X is normal. Optional arguments described in the online documentation specify the parameters of the particular normal distribution.

**Syntax**

`pnorm(q, mean, sd, lower.tail = TRUE, log.p = FALSE) `

**Parameters**

**q:** It is a vector of quantiles.

**mean:** It is a vector of means.

**sd:** It is a vector of standard deviations.

**lower.tail:** It is logical; if TRUE (default), probabilities are otherwise.

**log, log.p:** It is a logical argument.

If **mean** or **sd** are not specified, they assume the default values of 0 and 1, respectively.

**Example**

Find the percentage of males taller than 78 inches in a population with mean = 74 and sd = 2.

`pnorm(78, mean = 74, sd = 2, lower.tail = FALSE)`

**Output**

`[1] 0.02275013`

Find the percentage of otters that weight less than 33 lbs in a population with mean = 40 and sd = 8. Let’s see the following code example.

`pnorm(33, mean=40, sd = 8)`

**Output**

`[1] 0.190787`

The pnorm() function returns the integral from **−∞** to q of the normal distribution pdf, where q is a Z-score. Try to guess the value of pnorm(0).

`pnorm(0)`

**Output**

`[1] 0.5`

The pnorm() function also takes the argument **lower.tail**. If the **lower.tail** is set equal to **FALSE,** then pnorm returns the integral from q to ∞ the pdf of the normal distribution.

Use the **pnorm() function** to calculate the probability density of a normal distribution by setting the **lower.tail** argument to **FALSE**.

Note that pnorm(q) is the same as 1-pnorm(q, lower.tail = **FALSE**).

`pnorm(2)`

**Output**

`[1] 0.9772499`

**How to calculate cumulative distribution of normal distribution in R**

To calculate the **cumulative** **distribution** of a **normal** **distribution **in** R, ** use the **pnorm() function** and pass the **lower.tail** argument to **TRUE**.

`pnorm(1, mean = 0, sd = 1, lower.tail = TRUE)`

In the above code, it returns the cumulative distribution of the normal distribution with mean 0 and standard deviation 1 at the value 1.

The **pnorm()** function has an optional argument called **log.p**, which is set to **FALSE** by **default**. If set to **TRUE**, the **pnorm()** function returns the logarithm of the **probability** **density** or **cumulative** **distribution**.

**More examples**

```
# Import the stats package
library(stats)
# Calculate the cumulative probability of a normal distribution with a mean of 0,
# a standard deviation of 1, and a quantile of 0.5
prob <- pnorm(q = 0.5, mean = 0, sd = 1)
# Print the cumulative probability
print(prob)
```

**Output**

`[1] 0.6914625`

We first imported the stats package in the above code.

In the next step, we use the **pnorm() function** to calculate the cumulative probability of a normal distribution with a mean of 0, a standard deviation of 1, and a quantile of 0.5.

The **pnorm() function** returns the cumulative probability of the normal distribution as a numeric value, which we store in the prob variable.

At last, we used the print() function to print the prob variable, which displays the cumulative probability of the normal distribution.

**Applications of the pnorm() function in R**

- The
**pnorm() function**is mostly used in**statistical****analysis**, particularly when you need to model the distribution of continuous data. - The
**pnorm() function**allows you to calculate the probability density or cumulative distribution of normal distribution at specific values, which can help you make predictions about the likelihood of certain events occurring. - Calculating the probability of obtaining the
**specific****test****statistic**or**sample****mean**under the premise that the**NULL**hypothesis is**TRUE**. The**pnorm() function**helps to determine if the observed result is statistically significant.

**Conclusion**

Using the pnorm() function, we can calculate the cumulative probability of a normal distribution in R.

That’s it.

Krunal Lathiya is a Software Engineer with over eight years of experience. He has developed a strong foundation in computer science principles and a passion for problem-solving. In addition, Krunal has excellent knowledge of Data Science and Machine Learning, and he is an expert in R Language.