Normal distribution in SciPy

Introduction

The normal distribution helps us understand how data spreads around an average value. It is useful because many natural things follow this pattern.

To model heights of people in a population.

To analyze test scores where most students score near the average.

To estimate measurement errors in experiments.

To predict daily temperatures that vary around a mean.

To check if data fits a common pattern for further analysis.

Syntax

SciPy

from scipy.stats import norm

# Create a normal distribution object
norm(loc=mean, scale=std_dev)

# Calculate probability density function (PDF) at x
pdf_value = norm.pdf(x, loc=mean, scale=std_dev)

# Calculate cumulative distribution function (CDF) at x
cdf_value = norm.cdf(x, loc=mean, scale=std_dev)

# Generate random samples
samples = norm.rvs(loc=mean, scale=std_dev, size=n)

loc is the mean (center) of the distribution.

scale is the standard deviation (spread) of the distribution.

Examples

This example uses the standard normal distribution to find the PDF at 0 and CDF at 1.

SciPy

from scipy.stats import norm

# Standard normal distribution (mean=0, std=1)
pdf_at_0 = norm.pdf(0)
cdf_at_1 = norm.cdf(1)

This example calculates PDF and CDF for a normal distribution centered at 10 with spread 2.

SciPy

from scipy.stats import norm

# Normal distribution with mean=10 and std=2
pdf_at_12 = norm.pdf(12, loc=10, scale=2)
cdf_at_8 = norm.cdf(8, loc=10, scale=2)

This example shows how to create random data points from a normal distribution.

SciPy

from scipy.stats import norm

# Generate 5 random samples from N(5, 1.5)
samples = norm.rvs(loc=5, scale=1.5, size=5)
print(samples)

Sample Program

This program calculates the probability density at 55 and cumulative probability up to 45 for a normal distribution with mean 50 and standard deviation 5. It also generates 10 random samples from this distribution.

SciPy

from scipy.stats import norm

# Define mean and standard deviation
mean = 50
std_dev = 5

# Calculate PDF at 55
pdf_55 = norm.pdf(55, loc=mean, scale=std_dev)

# Calculate CDF at 45
cdf_45 = norm.cdf(45, loc=mean, scale=std_dev)

# Generate 10 random samples
samples = norm.rvs(loc=mean, scale=std_dev, size=10)

print(f"PDF at 55: {pdf_55:.4f}")
print(f"CDF at 45: {cdf_45:.4f}")
print("Random samples:")
print(samples)

Important Notes

The PDF value shows how likely a specific value is, but it is not a probability itself.

The CDF value shows the probability that a value is less than or equal to a point.

Random samples help simulate real data following the normal pattern.

Summary

The normal distribution models data around an average with a certain spread.

Use norm.pdf to find how dense the data is at a point.

Use norm.cdf to find the probability of data being below a point.

Practice

1. What does the normal distribution model in data science?

easy

A. Data with no variation

B. Data with only positive values

C. Data sorted in ascending order

D. Data spread around an average value

Solution

1. Step 1: Understand the normal distribution concept

   The normal distribution describes how data points are spread around a central average (mean).

2. Step 2: Identify the correct description

   Among the options, only data spread around an average matches the normal distribution's purpose.

3. Final Answer:

   Data spread around an average value -> Option D

4. Quick Check:

   Normal distribution = data around average [OK]

Hint: Normal distribution centers data around mean [OK]

Common Mistakes:

• Thinking normal distribution only has positive values
• Confusing normal distribution with sorted data
• Assuming no variation in normal distribution

2. Which of the following is the correct way to import the normal distribution functions from scipy.stats?

easy

A. import scipy.norm

B. from scipy.stats import norm

C. from scipy import normal

D. import stats.norm

Solution

1. Step 1: Recall scipy.stats import syntax

   The normal distribution functions are in scipy.stats under the name 'norm'.

2. Step 2: Match correct import statement

   Only 'from scipy.stats import norm' correctly imports the normal distribution functions.

3. Final Answer:

   from scipy.stats import norm -> Option B

4. Quick Check:

   Correct import = from scipy.stats import norm [OK]

Hint: Use 'from scipy.stats import norm' to access normal distribution [OK]

Common Mistakes:

• Using 'import scipy.norm' which does not exist
• Trying 'from scipy import normal' which is incorrect
• Importing 'stats.norm' without scipy prefix

3. What is the output of this code?

from scipy.stats import norm
result = norm.cdf(0, loc=0, scale=1)
print(round(result, 2))

medium

A. 1.00

B. 0.00

C. 0.50

D. 0.84

Solution

1. Step 1: Understand norm.cdf at mean

   norm.cdf(x, loc, scale) gives the probability data is less than x. At x=0 with mean=0, scale=1, it is the 50th percentile.

2. Step 2: Calculate and round the result

   The cumulative probability at mean is 0.5, rounding to 2 decimals gives 0.50.

3. Final Answer:

   0.50 -> Option C

4. Quick Check:

   norm.cdf(mean) = 0.5 [OK]

Hint: CDF at mean of normal = 0.5 [OK]

Common Mistakes:

• Confusing PDF with CDF values
• Expecting 0 or 1 instead of 0.5 at mean
• Rounding errors ignoring decimals

4. Find the error in this code snippet:

from scipy.stats import norm
prob = norm.pdf(1, mean=0, scale=1)
print(prob)

medium

A. Parameter 'mean' should be 'loc'

B. Missing import statement

C. norm.pdf does not exist

D. scale must be an integer

Solution

1. Step 1: Check parameter names for norm.pdf

   norm.pdf uses 'loc' for mean, not 'mean'. Using 'mean' causes an error.

2. Step 2: Identify correct parameter

   Replace 'mean=0' with 'loc=0' to fix the code.

3. Final Answer:

   Parameter 'mean' should be 'loc' -> Option A

4. Quick Check:

   Use loc for mean in norm functions [OK]

Hint: Use 'loc' not 'mean' for normal distribution mean [OK]

Common Mistakes:

• Using 'mean' instead of 'loc' parameter
• Assuming norm.pdf is missing
• Thinking scale must be integer

5. You want to find the probability that a value from a normal distribution with mean 10 and standard deviation 2 is less than 13. Which code correctly calculates this using scipy?

hard

A. norm.cdf(13, loc=10, scale=2)

B. norm.pdf(13, loc=10, scale=2)

C. norm.cdf(10, loc=13, scale=2)

D. norm.sf(13, loc=10, scale=2)

Solution

1. Step 1: Identify the function for cumulative probability

   To find probability less than a value, use norm.cdf with the value, mean (loc), and std dev (scale).

2. Step 2: Match parameters to the problem

   Value is 13, mean is 10, std dev is 2, so norm.cdf(13, loc=10, scale=2) is correct.

3. Final Answer:

   norm.cdf(13, loc=10, scale=2) -> Option A

4. Quick Check:

   Use norm.cdf(value, loc=mean, scale=std) [OK]

Hint: Use norm.cdf for probability less than a value [OK]

Common Mistakes:

• Using norm.pdf instead of norm.cdf
• Swapping loc and value parameters
• Using norm.sf which gives probability greater than value