Normalized histograms in Matplotlib - Time & Space Complexity

Time Complexity: Normalized histograms

O(n)

Understanding Time Complexity

We want to understand how the time to create a normalized histogram changes as the data size grows.

How does the work increase when we have more data points?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import matplotlib.pyplot as plt
import numpy as np

n = 1000  # example data size

data = np.random.randn(n)  # n data points
plt.hist(data, bins=50, density=True)
plt.show()

This code creates a histogram with 50 bins and normalizes it so the area sums to 1.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Counting how many data points fall into each of the 50 bins.
• How many times: Each of the n data points is checked once to find its bin.

How Execution Grows With Input

As the number of data points grows, the time to count them into bins grows roughly the same.

Input Size (n)	Approx. Operations
10	About 10 checks to place points in bins
100	About 100 checks
1000	About 1000 checks

Pattern observation: The work grows directly with the number of data points.

Final Time Complexity

Time Complexity: O(n)

This means the time to create the normalized histogram grows linearly with the number of data points.

Common Mistake

[X] Wrong: "The number of bins affects the time complexity a lot, so more bins means much slower."

[OK] Correct: The number of bins is usually fixed and small compared to data size, so it does not change the main growth pattern.

Interview Connect

Understanding how data size affects histogram creation helps you explain performance in data visualization tasks clearly and confidently.

Self-Check

"What if we increased the number of bins to grow with the data size? How would the time complexity change?"

Practice

1. What does setting density=True in plt.hist() do to the histogram?

easy

A. It changes the histogram bars to a different color.

B. It sorts the data before plotting the histogram.

C. It normalizes the histogram so the area sums to 1, showing probability density.

D. It plots the histogram with logarithmic scale on the y-axis.

Solution

1. Step 1: Understand the role of density=True

   Setting density=True scales the histogram so the total area under the bars equals 1.

2. Step 2: Interpret the effect on histogram

   This scaling means the histogram shows probability density instead of raw counts, useful for comparing distributions.

3. Final Answer:

   It normalizes the histogram so the area sums to 1, showing probability density. -> Option C

4. Quick Check:

   Normalized histogram = density=True [OK]

Hint: Remember: density=True makes area = 1, not counts [OK]

Common Mistakes:

• Thinking density=True changes colors
• Assuming density=True sorts data
• Confusing density=True with log scale

2. Which of the following is the correct syntax to plot a normalized histogram of data array data using matplotlib?

easy

A. plt.hist(data, scale='density')

B. plt.hist(data, density=True)

C. plt.hist(data, norm=True)

D. plt.hist(data, normalize=True)

Solution

1. Step 1: Recall matplotlib histogram parameters

   The correct parameter to normalize histogram counts is density=True.

2. Step 2: Check each option

   The other options use incorrect or nonexistent parameters for normalization.

3. Final Answer:

   plt.hist(data, density=True) -> Option B

4. Quick Check:

   Use density=True for normalized histogram [OK]

Hint: Use density=True exactly, no shortcuts [OK]

Common Mistakes:

• Using normalize=True which is invalid
• Trying norm=True instead of density=True
• Passing scale='density' which is not a parameter

3. What will be the output of this code snippet?

import matplotlib.pyplot as plt
import numpy as np
np.random.seed(0)
data = np.random.randint(1, 5, size=100)
counts, bins, patches = plt.hist(data, bins=4, density=True)
print(counts.round(2))

medium

A. [0.27 0.25 0.23 0.25]

B. [0.25 0.25 0.25 0.25]

C. [25 25 25 25]

D. [0.20 0.30 0.25 0.25]

Solution

1. Step 1: Understand data and bins

   Data has 100 integers between 1 and 4, bins=4 means bins cover 1-2, 2-3, 3-4, 4-5.

2. Step 2: Calculate normalized counts

   Using density=True, counts represent probability density, roughly proportional to frequency divided by bin width. The printed counts are close to [0.27, 0.25, 0.23, 0.25].

3. Final Answer:

   [0.27 0.25 0.23 0.25] -> Option A

4. Quick Check:

   Normalized counts sum to ~1 area [OK]

Hint: density=True scales counts to sum area 1 [OK]

Common Mistakes:

• Expecting raw counts instead of densities
• Confusing counts with percentages
• Ignoring bin width effect on density

4. Identify the error in this code that tries to plot a normalized histogram:

import matplotlib.pyplot as plt
data = [1, 2, 2, 3, 3, 3]
plt.hist(data, norm=True)
plt.show()

medium

A. plt.show() is missing and needed to display the plot.

B. The data list must be converted to a numpy array first.

C. Bins parameter is missing, causing an error.

D. The parameter 'norm' is invalid; should be 'density'.

Solution

1. Step 1: Check parameter names in plt.hist()

   The correct parameter to normalize histogram is density=True, not norm=True.

2. Step 2: Verify other parts

   Data can be a list, plt.show() is present, and bins are optional. So no error from those.

3. Final Answer:

   The parameter 'norm' is invalid; should be 'density'. -> Option D

4. Quick Check:

   Use density=True, not norm=True [OK]

Hint: Use density=True, not norm=True [OK]

Common Mistakes:

• Using norm=True instead of density=True
• Thinking data must be numpy array
• Forgetting plt.show() (though present here)
• Assuming bins are mandatory

5. You have two datasets with different sizes: data1 has 1000 points, data2 has 200 points. You want to compare their distributions fairly using histograms. Which approach is best?

hard

A. Plot histograms with density=True for both datasets.

B. Plot histograms with default counts for both datasets.

C. Plot histograms with different bin sizes for each dataset.

D. Plot histograms using cumulative counts.

Solution

1. Step 1: Understand the problem of different sizes

   Raw counts histograms will show higher bars for the larger dataset, making comparison unfair.

2. Step 2: Use normalized histograms

   Setting density=True scales histograms to probability densities, allowing fair comparison regardless of sample size.

3. Final Answer:

   Plot histograms with density=True for both datasets. -> Option A

4. Quick Check:

   Normalize histograms to compare different sizes [OK]

Hint: Normalize histograms to compare different sample sizes [OK]

Common Mistakes:

• Comparing raw counts directly
• Using different bin sizes causing bias
• Using cumulative counts which show totals, not densities