The norm.ppf function is useful for working with normal distributions in Python. It is part of the **scipy.stats module, which provides a variety of statistical functions and distributions. **Normal distributions are one of the most common and important statistical probability distributions. They are often used to model natural phenomena, such as heights, weights, IQ scores, test scores, errors, etc. **Normal distributions have a bell-shaped curve, which is symmetric and centered around the mean. A standard deviation typically measures how spread out the values are from the mean.**

The norm.ppf function allows us to find the x-value corresponding to a given probability on the normal distribution. For example, if we want to know what value of x has a 90% chance of being less than or equal to it, we can use norm.ppf(0.9) to find out. The norm.ppf function can also take optional parameters for the mean and standard deviation of the normal distribution, which can be different from the standard values of 0 and 1. This way, we can work with any normal distribution we want.

The norm.ppf function has many statistical applications, such as finding z-scores, confidence intervals, and p-values. These concepts are helpful for performing hypothesis tests, estimating population parameters, and making inferences from data. In this article, we will explore what the mentioned function does, how to use it, and some examples of its applications.

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## What is norm.ppf?

The norm.ppf function stands for **the percent point function**. It is also known as the **inverse cumulative distribution **or **quantile function**. It takes a probability value (between 0 and 1) and returns the corresponding value on the x-axis of the normal distribution. In other words, it tells us what value of x has a certain probability of being less than or equal to it.

For example, the norm.ppf(0.5) returns 0 because 50% of the area under the normal curve is to the left of 0. Similarly, the norm.ppf(0.95) returns 1.6449 because 95% of the area under the normal curve is to the left of 1.6449.

This function can also take optional parameters for the mean and standard deviation of the normal distribution. By default, these are set to 0 and 1, respectively, corresponding to the standard normal distribution. However, we can change them to any values we want to work with different normal distributions.

For example, norm.ppf(0.95, loc=10, scale=2) returns 13.2898, because 95% of the area under the normal curve with a standard deviation of 2 and a mean of 10 is to the left of 13.2898.

## How to use norm.ppf?

To use this function, we need to import the scipy.stats module first:

`import scipy.stats as stats`

Then, we can call this function with the probability value we want, and optionally, the mean and standard deviation of the normal distribution. For example:

```
# Find the 90th percentile of the standard normal distribution
stats.norm.ppf(0.9)
```

**Output: 1.2815515655446004**

```
# Find the 75th percentile of the normal distribution alongwith a mean value of 5 and a standard deviation of 3
stats.norm.ppf(0.75, loc=5, scale=3)
```

**Output: 7.023469250588246**

We can also pass an array of probability values to this function, and it will return an array of corresponding x-values. For example:

```
# Find the percentiles of the standard normal distribution
stats.norm.ppf([0.25, 0.5, 0.75]) 25th, 50th, and 75th respectively.
```

**Output: array([-0.67448975, 0. , 0.67448975])**

## Examples of norm.ppf applications

*This function can be used for various purposes, such as finding the z-score of a given probability, the confidence interval of a sample mean, and the p-value of a hypothesis test.*

### Finding the **z-score** of a given probability.

** The z-score is the number of standard deviations away from the normal distribution mean**. For example, the norm.ppf(0.95) returns 1.6449, which means that 95% of the values in the standard normal distribution are within 1.6449 standard deviations from the mean.

### Finding the confidence interval of a sample mean.

The confidence interval is the range of values containing the true population mean with a certain confidence level. For example, if we have a sample of size 100 with a mean of 50 and standard deviation of 10, and we want to find the 95% confidence interval of the population mean, we can use this function to find the margin of error:

```
# Find the 95% confidence interval of the population mean
sample_size = 100
sample_mean = 50
sample_std = 10
confidence_level = 0.95
# Find the z-score for the confidence level
z = stats.norm.ppf((1 + confidence_level) / 2)
# Find the margin of error
margin_of_error = z * sample_std / sample_size**0.5
# Find the lower and upper bounds of the confidence interval
lower_bound = sample_mean - margin_of_error
upper_bound = sample_mean + margin_of_error
# Print the confidence interval
print(f"The 95% confidence interval of the population mean is ({lower_bound:.2f}, {upper_bound:.2f})")
```

** Output: The 95% confidence interval of the population mean is (48.21, 51.79)**

### Finding the p-value of a hypothesis test.

The p-value is the probability of finding a result at least as extreme as the observed one, assuming the null hypothesis is true. For example, if we want to test whether the mean height of a population is 170 cm, and we have a sample of size 50 with a mean of 172 cm and a standard deviation of 5 cm, we can use this function to find the p-value:

```
# Find the p-value of the hypothesis test
population_mean = 170
sample_size = 50
sample_mean = 172
sample_std = 5
# Find the z-score of the sample mean
z = (sample_mean - population_mean) / (sample_std / sample_size**0.5)
# Find the p-value by using the survival function (1 - cdf)
p = stats.norm.sf(z)
# Print the p-value
print(f"The p-value of the hypothesis test is {p:.4f}")
```

**Output: The p-value of the hypothesis test is 0.0023**

## FAQs

### What is the difference between norm.ppf and norm.cdf?

The norm.cdf function is the **cumulative distribution function**, which takes a value on the x-axis of the normal distribution and returns the probability of being less than or equal to it. The norm.ppf function is the **percent point function**, which does the opposite: it takes a probability and returns the corresponding value on the x-axis. They are inverse functions of each other, so norm.cdf(norm.ppf(p)) = p and norm.ppf(norm.cdf(x)) = x for any p and x.

### How do we find the norm.ppf value for a two-tailed test?

A two-tailed test is when we are interested in the values on both sides of the mean of the normal distribution. For example, if we want to find the 95% confidence interval of the population mean, we need to find the values that exclude 2.5% of the area on each tail. To do this, we can use norm.ppf with the probability of 0.975, the sum of 0.5 (the area to the left of the mean) and 0.475 (the area to the right of the mean up to the 95th percentile). Alternatively, we can use norm.isf, the **inverse survival function**, with the probability of 0.025, the area to the right of the 97.5th percentile.

### How do we plot the norm.ppf function in Python?

To plot the norm.ppf function in Python, we can use matplotlib.pyplot module, which provides various functions for creating and customizing graphs. For example, we can plot the norm.ppf function for the standard normal distribution using the code: plt.plot(np.linspace(0.01, 0.99, 100), stats.norm.ppf(np.linspace(0.01, 0.99, 100)))

## Conclusion

The norm.ppf function is a powerful and versatile tool for working with normal distributions in Python. It allows us to find the x-value corresponding to a given probability or vice versa. It can also be used for various applications, such as finding z-scores, confidence intervals, and p-values. The norm.ppf function is part of the scipy.stats module, which offers many other statistical functions and distributions. We hope this article has helped you understand the norm.ppf function and how to use it in your Python projects.

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