Covariance with np.cov() in NumPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the covariance matrix of data using numpy.

NumPy

import numpy as np

data = np.array([[1, 2, 3], [4, 5, 6]])
cov_matrix = np.cov([1])
print(cov_matrix)

Drag options to blanks, or click blank then click option'

Anp.mean

Bnp.array

Cnp.cov

Ddata

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the function np.cov instead of the data array.

Passing np.mean which calculates mean, not covariance.

The np.cov() function calculates the covariance matrix of the input data array. Here, we pass the variable data to it.

2fill in blank

medium

Complete the code to calculate covariance matrix with rows as variables.

NumPy

import numpy as np

data = np.array([[1, 2, 3], [4, 5, 6]])
cov_matrix = np.cov(data, rowvar=[1])
print(cov_matrix)

Drag options to blanks, or click blank then click option'

ATrue

BFalse

CNone

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Setting rowvar=False when rows are variables.

Passing None which is invalid for rowvar.

By default, rowvar=True means each row represents a variable. Here, data rows are variables, so we set rowvar=True.

3fill in blank

hard

Fix the error in the code to get covariance matrix of columns as variables.

NumPy

import numpy as np

data = np.array([[1, 2, 3], [4, 5, 6]])
cov_matrix = np.cov(data, rowvar=[1])
print(cov_matrix)

Drag options to blanks, or click blank then click option'

ATrue

BFalse

C1

DNone

Attempts:

3 left

💡 Hint

Common Mistakes

Using rowvar=True when columns are variables.

Passing invalid values like None or 1.

When columns represent variables, set rowvar=False so covariance is calculated between columns.

4fill in blank

hard

Fill both blanks to create a covariance matrix from data with variables in columns and normalize by N (not N-1).

NumPy

import numpy as np

data = np.array([[1, 2, 3], [4, 5, 6]])
cov_matrix = np.cov(data, rowvar=[1], bias=[2])
print(cov_matrix)

Drag options to blanks, or click blank then click option'

AFalse

BTrue

CNone

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using rowvar=True when variables are columns.

Setting bias=False which normalizes by N-1.

Set rowvar=False because variables are in columns. Set bias=True to normalize by N instead of N-1.

5fill in blank

hard

Fill the blanks to compute covariance matrix of variables in rows, with bias normalization, and print shape.

NumPy

import numpy as np

data = np.array([[7, 8, 9], [10, 11, 12]])
cov_matrix = np.cov(data, rowvar=[1], bias=[2])
print(cov_matrix.shape, cov_matrix)

Drag options to blanks, or click blank then click option'

ATrue

BFalse

Attempts:

3 left

💡 Hint

Common Mistakes

Using rowvar=False when variables are rows.

Setting bias=False which normalizes by N-1.

Set rowvar=True because variables are rows. Set bias=True to normalize by N. The shape printed shows the covariance matrix size.

Practice

1. What does the np.cov() function compute in numpy?

easy

A. The standard deviation of each variable

B. The correlation coefficient between two variables

C. The mean of the dataset

D. The covariance matrix showing how variables vary together

Solution

1. Step 1: Understand covariance concept

   Covariance measures how two variables change together, indicating relationship strength and direction.

2. Step 2: Identify np.cov() function purpose

   The np.cov() function calculates the covariance matrix from input data arrays.

3. Final Answer:

   The covariance matrix showing how variables vary together -> Option D

4. Quick Check:

   np.cov() = covariance matrix [OK]

Hint: np.cov() always returns covariance matrix [OK]

Common Mistakes:

• Confusing covariance with correlation
• Thinking np.cov() returns mean or std deviation
• Assuming np.cov() returns a single number

2. Which of the following is the correct way to call np.cov() on two 1D arrays x and y?

easy

A. np.cov(x, axis=1)

B. np.cov(x, y)

C. np.cov(x + y)

D. np.cov([x + y])

Solution

1. Step 1: Understand input format for np.cov()

   np.cov(m, y=None) where for two variables, pass two 1D arrays as m=x, y=y.

2. Step 2: Check correct input structure

   Passing x, y directly uses the two-argument form of np.cov().

3. Final Answer:

   np.cov(x, y) -> Option B

4. Quick Check:

   np.cov(x, y) for two 1D arrays [OK]

Hint: np.cov(x, y) for two 1D arrays [OK]

Common Mistakes:

• Passing summed arrays
• Using wrong axis parameter
• Passing list containing summed array

3. What is the output of this code?

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
cov_matrix = np.cov([x, y])
print(cov_matrix)

medium

A. [[1.5 1.5]
[1.5 1.5]]

B. [[0.66666667 0.66666667]
[0.66666667 0.66666667]]

C. [[1. 1.]
[1. 1.]]

D. [[2. 2.]
[2. 2.]]

Solution

1. Step 1: Calculate covariance of x and y

   x mean=2, deviations=[-1,0,1], sum of squared deviations=2, np.cov uses ddof=1 by default (unbiased estimator), so variance = 2/(3-1) = 1. Same for covariance.

2. Step 2: Understand np.cov() output

   np.cov([x, y]) returns 2x2 matrix with 1.0 everywhere.

3. Final Answer:

   [[1. 1.] [1. 1.]] -> Option C

4. Quick Check:

   np.cov uses ddof=1 by default (unbiased) [OK]

Hint: np.cov() divides by (n-1) by default [OK]

Common Mistakes:

• Using biased variance (/n=3)
• Misreading np.cov() output shape
• Assuming default normalization is biased

4. Identify the error in this code snippet:

import numpy as np
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
z = np.array([7, 8, 9])
cov_matrix = np.cov(x, y, z)
print(cov_matrix)

medium

A. np.cov() requires a single array or a list of arrays, not separate arrays

B. x and y must be reshaped to 2D arrays before np.cov()

C. np.cov() cannot handle integer arrays

D. Missing axis parameter in np.cov() call

Solution

1. Step 1: Check np.cov() input requirements

   np.cov() expects either a single 2D array or a list/array of variables, not multiple separate arrays as arguments.

2. Step 2: Identify correct input format

   Passing x, y, z as separate arguments causes TypeError; correct is np.cov([x, y, z]).

3. Final Answer:

   np.cov() requires a single array or a list of arrays, not separate arrays -> Option A

4. Quick Check:

   Input must be single array or list [OK]

Hint: Pass variables as list, not separate args to np.cov() [OK]

Common Mistakes:

• Passing multiple arrays as separate arguments
• Assuming np.cov() auto-stacks inputs
• Ignoring error messages about input shape

5. You have two datasets:

data1 = np.array([2, 4, 6, 8])
data2 = np.array([1, 3, 5, 7])

You want to find the covariance matrix but only for the first three elements of each dataset. Which code correctly computes this?

hard

A. np.cov(np.vstack((data1[:3], data2[:3])))

B. np.cov([data1[:3], data2])

C. np.cov(data1[0:3], data2)

D. np.cov(data1, data2)[:3, :3]

Solution

1. Step 1: Select first three elements from each dataset

   Use slicing data1[:3] and data2[:3] to get first three elements.

2. Step 2: Prepare input for np.cov()

   np.cov() expects 2D array with variables as rows; stacking with np.vstack() creates correct shape.

3. Step 3: Compute covariance matrix

   Calling np.cov(np.vstack((data1[:3], data2[:3]))) computes covariance matrix for sliced data correctly.

4. Final Answer:

   np.cov(np.vstack((data1[:3], data2[:3]))) -> Option A

5. Quick Check:

   Stack sliced arrays then np.cov() [OK]

Hint: Stack sliced arrays with vstack before np.cov() [OK]

Common Mistakes:

• Passing separate arrays without stacking
• Slicing after calling np.cov()
• Using incorrect slicing syntax