Normal distribution in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a normal distribution?

A normal distribution is a bell-shaped curve that shows how data points are spread around the mean. Most values are near the mean, and fewer are far away.

beginner

What do the mean and standard deviation represent in a normal distribution?

The mean is the center of the distribution. The standard deviation shows how spread out the data is around the mean.

intermediate

How do you generate random numbers from a normal distribution using scipy?

Use scipy.stats.norm.rvs(loc=mean, scale=std, size=n) to get n random numbers from a normal distribution with given mean and standard deviation.

intermediate

What does the probability density function (PDF) of a normal distribution tell us?

The PDF shows the relative likelihood of different values. Higher PDF means the value is more likely to occur.

intermediate

How can you calculate the cumulative probability up to a value x in a normal distribution?

Use scipy.stats.norm.cdf(x, loc=mean, scale=std) to find the probability that a value is less than or equal to x.

What shape does a normal distribution have?

A Step function

B Uniform line

C Exponential curve

D Bell-shaped curve

A normal distribution is known for its bell-shaped curve centered around the mean.

Which scipy function generates random samples from a normal distribution?

A scipy.stats.norm.rvs()

B scipy.stats.norm.pdf()

C scipy.stats.norm.cdf()

D scipy.stats.norm.mean()

The rvs() function generates random variates (samples) from the normal distribution.

What does the standard deviation control in a normal distribution?

A Spread or width of the distribution

B Height of the peak

C Skewness of the distribution

D Center of the distribution

Standard deviation controls how spread out the data is around the mean.

Which function gives the probability that a value is less than or equal to x in a normal distribution?

A norm.pdf()

B norm.rvs()

C norm.cdf()

D norm.ppf()

The cumulative distribution function (cdf) gives the probability up to x.

If the mean is 0 and standard deviation is 1, what is this normal distribution called?

A Exponential distribution

B Standard normal distribution

C Uniform distribution

D Binomial distribution

A normal distribution with mean 0 and std 1 is called the standard normal distribution.

Explain how to use scipy to find the probability that a value falls below a certain point in a normal distribution.

Think about the function that accumulates probability up to a value.

Describe the role of mean and standard deviation in shaping a normal distribution curve.

Consider how the curve moves and stretches.

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