Normal distribution in SciPy - Step-by-Step Execution

Concept Flow - Normal distribution

Start: Define mean (mu) and std dev (sigma)

↓

Use scipy.stats.norm to create distribution

↓

Calculate PDF or CDF for values

↓

Output probabilities or plot

↓

End

We start by setting the mean and standard deviation, then use scipy to get probabilities or plots for values in the normal distribution.

Execution Sample

SciPy

from scipy.stats import norm
mu, sigma = 0, 1
x = 0
pdf_val = norm.pdf(x, mu, sigma)
print(pdf_val)

This code calculates the probability density at x=0 for a normal distribution with mean 0 and std dev 1.

Execution Table

Step	Variable	Value	Action	Result
1	mu	0	Set mean	mu = 0
2	sigma	1	Set std dev	sigma = 1
3	x	0	Set point to evaluate	x = 0
4	pdf_val	0.3989422804014327	Calculate PDF at x	pdf_val = norm.pdf(0, 0, 1)
5	-	-	Print pdf_val	Output: 0.3989422804014327

💡 All steps done, PDF value computed and printed.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	Final
mu	undefined	0	0	0	0	0
sigma	undefined	undefined	1	1	1	1
x	undefined	undefined	undefined	0	0	0
pdf_val	undefined	undefined	undefined	undefined	0.3989422804014327	0.3989422804014327

Key Moments - 2 Insights

Why do we need to specify mean and standard deviation before calculating PDF?

Because the normal distribution shape depends on mean and std dev. The execution_table rows 1 and 2 show setting these before calculating PDF at step 4.

What does the PDF value represent at x=0?

It shows the height of the normal curve at x=0, not a probability but a density. See execution_table step 4 where pdf_val is calculated.

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of pdf_val at step 4?

A 0.3989

B 1.0

C 0.0

D 0.5

💡 Hint

Check the 'Value' column at step 4 in execution_table.

At which step is the standard deviation set?

A Step 3

B Step 1

C Step 2

D Step 4

💡 Hint

Look at the 'Variable' column for 'sigma' in execution_table.

If we change mu to 1, how does it affect pdf_val at x=0?

A pdf_val increases

B pdf_val decreases

C pdf_val stays the same

D pdf_val becomes zero

💡 Hint

Changing mean shifts the curve; check variable_tracker for mu and think how PDF at x=0 changes.

Concept Snapshot

Normal distribution uses mean (mu) and std dev (sigma).
Use scipy.stats.norm.pdf(x, mu, sigma) to get density at x.
PDF shows curve height, not direct probability.
Changing mu shifts curve; sigma changes spread.
Common for modeling natural data variations.

Full Transcript

This visual execution shows how to use scipy's normal distribution. First, we set mean and standard deviation. Then, we pick a point x to evaluate. Using norm.pdf, we get the density value at x. The tables track each variable's value step-by-step. Key moments clarify why mean and std dev matter and what PDF means. The quiz tests understanding of these steps and effects of changing parameters.

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