Normalized histograms in Matplotlib - Step-by-Step Execution

Concept Flow - Normalized histograms

Start with data array

↓

Choose number of bins

↓

Calculate frequency counts per bin

↓

Normalize counts by total data points and bin width

↓

Plot histogram with normalized heights

↓

Show histogram where area sums to 1

This flow shows how data is divided into bins, counts are normalized to represent probabilities, and then plotted as a histogram.

Execution Sample

Matplotlib

import matplotlib.pyplot as plt
import numpy as np

data = np.random.randn(1000)
plt.hist(data, bins=30, density=True)
plt.show()

This code creates a normalized histogram of 1000 random numbers from a normal distribution.

Execution Table

Step	Action	Data/Variable	Result/Output
1	Generate data	data = 1000 random values	Array of 1000 floats
2	Set bins	bins = 30	Divide data range into 30 intervals
3	Count data in bins	counts per bin	Array of counts, sum ~1000
4	Normalize counts	density=True	Counts divided by (total points * bin width)
5	Plot histogram	plt.hist(..., density=True)	Bar heights sum to 1 area
6	Show plot	plt.show()	Histogram displayed on screen

💡 All steps complete, histogram normalized and displayed

Variable Tracker

Variable	Start	After Step 1	After Step 3	After Step 4	Final
data	None	Array of 1000 floats	Same	Same	Same
bins	None	None	30 intervals	Same	Same
counts	None	None	Counts per bin (~1000 total)	Normalized counts	Used for plotting

Key Moments - 2 Insights

Why does setting density=True change the histogram heights?

Setting density=True divides counts by total points and bin width, so the histogram shows probability density instead of raw counts, as seen in execution_table step 4.

What does the area under the normalized histogram represent?

The area sums to 1, representing the total probability of the data distribution, confirmed in execution_table step 5.

Visual Quiz - 3 Questions

Look at the execution table, what does the 'counts per bin' represent at step 3?

A The number of data points in each bin

B The total number of bins

C The normalized frequency of data

D The bin width

💡 Hint

Check the 'Data/Variable' column at step 3 in execution_table

At which step does the histogram get normalized so that the area sums to 1?

A Step 2

B Step 3

C Step 4

D Step 6

💡 Hint

Look for 'Normalize counts' and 'density=True' in execution_table

If we increase the number of bins, how would the 'counts per bin' change at step 3?

A Counts per bin would increase

B Counts per bin would decrease

C Counts per bin would stay the same

D Counts per bin would become negative

💡 Hint

More bins means data is split into smaller intervals, so fewer points per bin

Concept Snapshot

Normalized histograms show data distribution as probabilities.
Use plt.hist(data, bins, density=True) to normalize.
Counts are divided by total points and bin width.
Area under histogram sums to 1.
Useful to compare distributions regardless of sample size.

Full Transcript

This lesson shows how to create normalized histograms using matplotlib. First, data is generated or given. Then, the data range is divided into bins. We count how many data points fall into each bin. To normalize, counts are divided by the total number of points and the bin width, so the histogram represents a probability density. Finally, the histogram is plotted with bars whose heights correspond to these normalized values. The area under the histogram sums to 1, representing the total probability. This helps compare different datasets fairly. The execution table traces each step from data generation to plotting. Key moments clarify why normalization changes bar heights and what the area means. The quiz tests understanding of counts, normalization step, and effect of bin number.

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