Cumulative histograms in Matplotlib - Step-by-Step Execution

Concept Flow - Cumulative histograms

Start with data array

↓

Calculate histogram bins

↓

Count data points per bin

↓

Calculate cumulative sum of counts

↓

Plot cumulative histogram

↓

End

The flow shows how data is binned, counts are accumulated cumulatively, and then plotted as a cumulative histogram.

Execution Sample

Matplotlib

import matplotlib.pyplot as plt
import numpy as np

data = np.array([1,2,2,3,4,5,5,5,6])
plt.hist(data, bins=6, cumulative=True)
plt.show()

This code plots a cumulative histogram of the data array with 6 bins.

Execution Table

Step	Data Points Processed	Bin Edges	Counts per Bin	Cumulative Counts	Plot Action
1	[]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[0, 0, 0, 0, 0, 0]	[0, 0, 0, 0, 0, 0]	Initialize bins and counts
2	[1]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 0, 0, 0, 0, 0]	[1, 0, 0, 0, 0, 0]	Count 1 in first bin
3	[1,2,2]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 0, 0, 0, 0]	[1, 3, 3, 3, 3, 3]	Count two 2s in second bin, cumulative sums updated
4	[1,2,2,3]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 1, 0, 0, 0]	[1, 3, 4, 4, 4, 4]	Count 3 in third bin, cumulative sums updated
5	[1,2,2,3,4]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 1, 1, 0, 0]	[1, 3, 4, 5, 5, 5]	Count 4 in fourth bin, cumulative sums updated
6	[1,2,2,3,4,5,5,5]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 1, 1, 3, 0]	[1, 3, 4, 5, 8, 8]	Count three 5s in fifth bin, cumulative sums updated
7	[1,2,2,3,4,5,5,5,6]	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 1, 1, 3, 1]	[1, 3, 4, 5, 8, 9]	Count 6 in last bin, cumulative sums updated
8	All data processed	[1.0, 1.83, 2.67, 3.5, 4.33, 5.17, 6.0]	[1, 2, 1, 1, 3, 1]	[1, 3, 4, 5, 8, 9]	Plot cumulative histogram and show

💡 All data points counted and cumulative sums calculated; histogram plotted.

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	After 6	After 7	Final
data processed	[]	[1]	[1,2,2]	[1,2,2,3]	[1,2,2,3,4]	[1,2,2,3,4,5,5,5]	[1,2,2,3,4,5,5,5,6]	[1,2,2,3,4,5,5,5,6]	[1,2,2,3,4,5,5,5,6]
counts per bin	[0,0,0,0,0,0]	[1,0,0,0,0,0]	[1,2,0,0,0,0]	[1,2,1,0,0,0]	[1,2,1,1,0,0]	[1,2,1,1,3,0]	[1,2,1,1,3,1]	[1,2,1,1,3,1]	[1,2,1,1,3,1]
cumulative counts	[0,0,0,0,0,0]	[1,0,0,0,0,0]	[1,3,3,3,3,3]	[1,3,4,4,4,4]	[1,3,4,5,5,5]	[1,3,4,5,8,8]	[1,3,4,5,8,9]	[1,3,4,5,8,9]	[1,3,4,5,8,9]

Key Moments - 2 Insights

Why do cumulative counts keep increasing even if some bins have zero counts?

Because cumulative counts add all previous bin counts, zeros do not reduce the sum but keep it the same, as shown in execution_table rows 5 and 6.

Why does the last bin count increase when data points equal the upper edge?

Matplotlib includes the right edge in the last bin, so data points equal to the max value fall into the last bin, as seen in step 7 of execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the cumulative count for the second bin?

A 1

B 3

C 2

D 4

💡 Hint

Check the 'Cumulative Counts' column at step 3 in execution_table.

At which step does the cumulative count for the last bin first increase?

A Step 4

B Step 5

C Step 7

D Step 6

💡 Hint

Look at the 'Cumulative Counts' column for the last bin in execution_table rows.

If the data array had no values equal to 6, how would the final cumulative count for the last bin change?

A It would be 4 instead of 9

B It would be 5 instead of 9

C It would remain 9

D It would be 0

💡 Hint

Refer to variable_tracker for counts per bin and consider the last bin counts.

Concept Snapshot

Cumulative histograms count data points in bins and add counts cumulatively.
Use plt.hist(data, bins, cumulative=True) to plot.
Bins divide data range; cumulative sums show running totals.
Useful to see distribution accumulation over range.

Full Transcript

This visual execution traces how a cumulative histogram is created using matplotlib. Starting with a data array, bins are defined to split the data range. Each data point is counted into its bin. Then counts are summed cumulatively from the first bin to the last. The cumulative counts are plotted as a histogram that shows how data accumulates across bins. The execution table shows each step of counting and summing. The variable tracker follows how counts and cumulative sums change after processing each data point. Key moments clarify why cumulative sums increase even with zero counts in some bins and why the last bin includes the maximum data value. The quiz questions test understanding of cumulative counts at specific steps and effects of data changes. The snapshot summarizes the concept and usage in a few lines.

Practice

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

easy

A. It shows the total counts accumulated up to each bin.

B. It changes the histogram to a bar chart.

C. It normalizes the histogram to show probabilities.

D. It colors the bars based on their height.

Solution

1. Step 1: Understand the cumulative parameter

   Setting cumulative=True in plt.hist() changes the histogram to show accumulated counts.

2. Step 2: Effect on histogram output

   Instead of showing counts per bin, it shows the sum of counts up to that bin, making the histogram cumulative.

3. Final Answer:

   It shows the total counts accumulated up to each bin. -> Option A

4. Quick Check:

   cumulative=True means accumulation [OK]

Hint: Remember cumulative=True sums counts up to each bin [OK]

Common Mistakes:

• Thinking it normalizes the histogram
• Confusing cumulative with coloring
• Assuming it changes chart type to bar chart

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

easy

A. plt.hist(data, cumulative=True)

B. plt.hist(data, cumulative=0)

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

D. plt.hist(data, cumulative)

Solution

1. Step 1: Check correct parameter name and usage

   The correct parameter to make a histogram cumulative is cumulative=True.

2. Step 2: Validate syntax correctness

   plt.hist(data, cumulative=True) uses the exact correct syntax. plt.hist(data, cumulative=0) uses 0 instead of True (no cumulative effect). plt.hist(data, cumulate=True) misspells the parameter. plt.hist(data, cumulative) is invalid syntax.

3. Final Answer:

   plt.hist(data, cumulative=True) -> Option A

4. Quick Check:

   Use cumulative=True exactly [OK]

Hint: Use cumulative=True exactly for cumulative histograms [OK]

Common Mistakes:

• Misspelling 'cumulative' parameter
• Using cumulative without assignment
• Using cumulative=0 (disables cumulative)

3. What will be the output counts of the cumulative histogram bins for data = [1, 2, 2, 3, 4] with bins [1, 2, 3, 4, 5] using plt.hist(data, bins=[1,2,3,4,5], cumulative=True)?

medium

A. [0, 1, 3, 5]

B. [1, 3, 4, 5]

C. [1, 1, 2, 3]

D. [1, 2, 3, 4]

Solution

1. Step 1: Count data points in each bin

   Bins: [1-2), [2-3), [3-4), [4-5). Counts per bin: 1 in first, 2 in second, 1 in third, 1 in fourth.

2. Step 2: Calculate cumulative counts

   Cumulative sums: 1, 1+2=3, 3+1=4, 4+1=5.

3. Final Answer:

   [1, 3, 4, 5] -> Option B

4. Quick Check:

   Cumulative sums of counts = [1,3,4,5] [OK]

Hint: Add counts bin by bin for cumulative totals [OK]

Common Mistakes:

• Counting bins inclusively instead of half-open
• Not summing counts cumulatively
• Miscounting data points per bin

4. The code below is intended to plot a cumulative histogram but does not produce the expected cumulative effect. What is the error?

import matplotlib.pyplot as plt
data = [1,2,2,3,4]
plt.hist(data, bins=4, cumulative=False)
plt.show()

medium

A. The bins parameter must be a list, not an integer.

B. plt.show() is missing and must be added.

C. The parameter cumulative should be set to True, not False.

D. Data list must be sorted before plotting.

Solution

1. Step 1: Identify the cumulative parameter usage

   The code sets cumulative=False, so the histogram is not cumulative.

2. Step 2: Correct the parameter to get cumulative histogram

   Changing cumulative=False to cumulative=True will produce the cumulative histogram as intended.

3. Final Answer:

   The parameter cumulative should be set to True, not False. -> Option C

4. Quick Check:

   cumulative=True needed for cumulative histogram [OK]

Hint: Set cumulative=True to enable cumulative histogram [OK]

Common Mistakes:

• Leaving cumulative=False by mistake
• Thinking bins must be a list always
• Believing data must be sorted
• Forgetting plt.show() (though present here)

5. You have two datasets: data1 = [1,2,3,4,5] and data2 = [2,3,4,5,6]. You want to plot their cumulative histograms on the same plot to compare their distributions. Which code snippet correctly does this?

hard

A. plt.hist([data1, data2], bins=6, cumulative=False, label=['Data1', 'Data2']); plt.legend(); plt.show()

B. plt.hist(data1 + data2, bins=6, cumulative=True); plt.show()

C. plt.hist(data1, bins=6); plt.hist(data2, bins=6); plt.show()

D. plt.hist([data1, data2], bins=6, cumulative=True, label=['Data1', 'Data2'], histtype='step'); plt.legend(); plt.show()

Solution

1. Step 1: Understand plotting multiple cumulative histograms

   To compare cumulative histograms, pass both datasets as a list, set cumulative=True, and use histtype='step' for clarity.

2. Step 2: Check options for correct parameters

   plt.hist([data1, data2], bins=6, cumulative=True, label=['Data1', 'Data2'], histtype='step'); plt.legend(); plt.show() correctly uses plt.hist([data1, data2], bins=6, cumulative=True, label=['Data1', 'Data2'], histtype='step') with legend and show. plt.hist(data1 + data2, bins=6, cumulative=True); plt.show() incorrectly combines the datasets into one histogram. plt.hist(data1, bins=6); plt.hist(data2, bins=6); plt.show() is not cumulative. plt.hist([data1, data2], bins=6, cumulative=False, label=['Data1', 'Data2']); plt.legend(); plt.show() is not cumulative.

3. Final Answer:

   plt.hist([data1, data2], bins=6, cumulative=True, label=['Data1', 'Data2'], histtype='step'); plt.legend(); plt.show() -> Option D

4. Quick Check:

   Multiple datasets + cumulative=True + histtype='step' = plt.hist([data1, data2], bins=6, cumulative=True, label=['Data1', 'Data2'], histtype='step'); plt.legend(); plt.show() [OK]

Hint: Use list of datasets and cumulative=True with histtype='step' [OK]

Common Mistakes:

• Combining datasets into a single histogram
• Forgetting cumulative=True
• Not using histtype='step' for clear lines
• Using cumulative=False by mistake