np.std() and np.var() for spread in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does np.std() calculate in a dataset?

np.std() calculates the standard deviation, which measures how spread out the numbers are from the average (mean). A small value means data points are close to the mean, and a large value means they are more spread out.

beginner

What is the difference between np.var() and np.std()?

np.var() calculates variance, which is the average of squared differences from the mean. np.std() is the square root of variance. So, standard deviation is in the same units as the data, while variance is in squared units.

intermediate

How does changing the ddof parameter affect np.std() and np.var()?

The ddof parameter stands for 'Delta Degrees of Freedom'. Setting ddof=1 calculates the sample standard deviation or variance (dividing by N-1), which is used when data is a sample. The default ddof=0 calculates population values (dividing by N).

beginner

Why is standard deviation often preferred over variance to describe spread?

Standard deviation is preferred because it is in the same units as the original data, making it easier to understand and compare. Variance is in squared units, which can be harder to interpret.

beginner

If a dataset has values [2, 4, 4, 4, 5, 5, 7, 9], what is the approximate standard deviation using np.std() with default settings?

The mean is 5. The squared differences average to 4, so variance is 4. The standard deviation is the square root of 4, which is 2.

What does np.var() measure in a dataset?

A Average squared distance from the mean

B Square root of the average distance from the mean

C Median of the dataset

D Sum of all data points

np.var() calculates the variance, which is the average of squared differences from the mean.

Which function returns a value in the same units as the original data?

A np.std()

B np.var()

C np.mean()

D np.sum()

np.std() returns the standard deviation, which is in the same units as the data.

What does setting ddof=1 do in np.std()?

A Calculates population standard deviation

B Ignores the first data point

C Calculates median instead

D Calculates sample standard deviation

Setting ddof=1 calculates the sample standard deviation by dividing by N-1.

If variance is 9, what is the standard deviation?

A 9

B 3

C 81

D 1

Standard deviation is the square root of variance, so sqrt(9) = 3.

Why might variance be harder to interpret than standard deviation?

A It is always smaller than standard deviation

B It is not related to the mean

C It is in squared units, not original units

D It ignores outliers

Variance is in squared units, which can make it less intuitive than standard deviation.

Explain in your own words what np.std() and np.var() tell us about a dataset.

Think about how spread out the data points are and how these functions measure that.

Describe how changing the ddof parameter affects the calculation of spread using np.std() or np.var().

Consider why we sometimes divide by N-1 instead of N.

Practice

1. What does np.std() measure in a dataset?

easy

A. The maximum value in the dataset

B. The sum of all data points

C. The average distance of data points from the mean in original units

D. The total number of data points

Solution

1. Step 1: Understand the purpose of np.std()

   np.std() calculates the standard deviation, which shows how spread out the data is around the mean in the same units as the data.

2. Step 2: Compare options to definition

   Only The average distance of data points from the mean in original units correctly describes this as the average distance from the mean in original units.

3. Final Answer:

   The average distance of data points from the mean in original units -> Option C

4. Quick Check:

   Standard deviation = average spread in original units [OK]

Hint: Standard deviation = spread in original units [OK]

Common Mistakes:

• Confusing standard deviation with variance
• Thinking it counts data points
• Mixing it with sum or max values

2. Which of the following is the correct syntax to calculate variance of array arr using NumPy?

easy

A. np.std(arr)

B. np.variance(arr)

C. arr.var()

D. np.var(arr)

Solution

1. Step 1: Recall the NumPy function for variance

   The correct NumPy function to calculate variance is np.var().

2. Step 2: Check each option

   np.var(arr) uses np.var(arr) which is correct. np.variance(arr) uses a non-existent function np.variance(). arr.var() uses arr.var() which works only if arr is a NumPy array, but the question asks for NumPy function syntax. np.std(arr) calculates standard deviation, not variance.

3. Final Answer:

   np.var(arr) -> Option D

4. Quick Check:

   Variance function = np.var() [OK]

Hint: Variance uses np.var(), not np.variance() [OK]

Common Mistakes:

• Using np.variance() which does not exist
• Confusing variance with standard deviation function
• Using method on list instead of NumPy array

3. What is the output of the following code?

import numpy as np
arr = np.array([2, 4, 4, 4, 5, 5, 7, 9])
print(np.var(arr))

medium

A. 2.0

B. 4.0

C. 2.5

D. 3.0

Solution

1. Step 1: Calculate the mean of the array

   The array is [2,4,4,4,5,5,7,9]. Sum = 40, count = 8, mean = 40/8 = 5.

2. Step 2: Calculate variance

   Variance = average of squared differences from mean:
   (2-5)^2=9, (4-5)^2=1, (4-5)^2=1, (4-5)^2=1, (5-5)^2=0, (5-5)^2=0, (7-5)^2=4, (9-5)^2=16
   Sum = 9+1+1+1+0+0+4+16 = 32
   Variance = 32/8 = 4.0

3. Step 3: Check code output

   NumPy's np.var() by default calculates population variance (divides by N). So output is 4.0.

4. Final Answer:

   4.0 -> Option B

5. Quick Check:

   Variance = 4.0 [OK]

Hint: Variance = mean of squared differences [OK]

Common Mistakes:

• Calculating sample variance (dividing by N-1) instead of population
• Mixing variance with standard deviation
• Incorrect mean calculation

4. The code below is intended to print the standard deviation of data. What is wrong?

import numpy as np
data = [1, 2, 3, 4, 5]
print(np.std(data, ddof=1))

medium

A. Nothing is wrong; code runs correctly

B. ddof is not a valid argument for np.std()

C. The code will raise a TypeError because data is a list, not an array

D. The code calculates variance, not standard deviation

Solution

1. Step 1: Check if ddof is valid for np.std()

   The ddof parameter (delta degrees of freedom) is valid for np.std() and adjusts divisor for sample standard deviation.

2. Step 2: Check if input type is acceptable

   np.std() accepts lists and converts them internally to arrays, so no error occurs.

3. Final Answer:

   Nothing is wrong; code runs correctly -> Option A

4. Quick Check:

   ddof works with np.std() and lists accepted [OK]

Hint: ddof works with np.std() for sample std [OK]

Common Mistakes:

• Thinking ddof is invalid for standard deviation
• Assuming lists cause errors in NumPy functions
• Confusing variance and standard deviation

5. You have two datasets:
data1 = np.array([10, 12, 14, 16, 18])
data2 = np.array([10, 10, 10, 10, 10])
Which statement is true about their standard deviation and variance?

hard

A. data1 has higher variance and standard deviation than data2

B. data2 has zero variance and standard deviation

C. data2 has higher variance but lower standard deviation than data1

D. Both datasets have the same variance and standard deviation

Solution

1. Step 1: Analyze data2 values

   All values in data2 are the same (10), so variance and standard deviation must be zero because there is no spread.

2. Step 2: Analyze data1 values

   data1 values vary from 10 to 18, so variance and standard deviation are greater than zero, meaning data1 has higher spread.

3. Final Answer:

   data1 has higher variance and standard deviation than data2 -> Option A

4. Quick Check:

   Constant data variance = 0; varying data variance > 0 [OK]

Hint: Constant data has zero variance and std [OK]

Common Mistakes:

• Assuming equal variance for different spread data
• Confusing variance with mean
• Thinking zero variance means error