Trend lines on scatter plots in Matplotlib

Introduction

Trend lines help us see the general direction or pattern in scattered data points. They make it easier to understand relationships between two variables.

You want to see if there is a positive or negative relationship between two measurements.

You want to predict values based on existing data points.

You want to summarize a large set of scattered points with a simple line.

You want to explain data trends in presentations or reports.

Syntax

Matplotlib

import numpy as np
import matplotlib.pyplot as plt

# Create scatter plot
plt.scatter(x, y)

# Calculate trend line coefficients
coefficients = np.polyfit(x, y, degree)

# Create polynomial function from coefficients
poly_fn = np.poly1d(coefficients)

# Plot trend line
plt.plot(x, poly_fn(x), color='red')

plt.show()

Use np.polyfit to find the best fit line or curve. The degree is usually 1 for a straight line.

np.poly1d creates a function you can use to get y-values for any x.

Examples

This example draws a scatter plot and a straight trend line (degree 1) through the points.

Matplotlib

import numpy as np
import matplotlib.pyplot as plt

x = [1, 2, 3, 4, 5]
y = [2, 3, 5, 7, 11]

plt.scatter(x, y)
coefficients = np.polyfit(x, y, 1)
poly_fn = np.poly1d(coefficients)
plt.plot(x, poly_fn(x), color='red')
plt.show()

This example fits a cubic (degree 3) curve to noisy sine wave data.

Matplotlib

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 10, 50)
y = np.sin(x) + np.random.normal(0, 0.2, 50)

plt.scatter(x, y)
coefficients = np.polyfit(x, y, 3)  # cubic trend line
poly_fn = np.poly1d(coefficients)
plt.plot(x, poly_fn(x), color='green')
plt.show()

Sample Program

This program creates a scatter plot of points and draws a red straight trend line showing the general upward trend.

Matplotlib

import numpy as np
import matplotlib.pyplot as plt

# Sample data
x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
y = np.array([2.3, 2.9, 3.8, 4.5, 5.1, 5.8, 6.5, 7.1, 7.9, 8.4])

# Scatter plot
plt.scatter(x, y, label='Data points')

# Calculate linear trend line
coefficients = np.polyfit(x, y, 1)  # degree 1 for straight line
poly_fn = np.poly1d(coefficients)

# Plot trend line
plt.plot(x, poly_fn(x), color='red', label='Trend line')

# Add labels and legend
plt.xlabel('X values')
plt.ylabel('Y values')
plt.title('Scatter Plot with Trend Line')
plt.legend()
plt.grid(True)
plt.show()

Important Notes

Make sure your x values are sorted or use np.sort(x) before plotting the trend line for a smooth line.

Higher degree polynomials can fit data better but may cause wiggly lines that overfit.

Always label your plots to make them easy to understand.

Summary

Trend lines show the overall direction in scattered data.

Use np.polyfit and np.poly1d to create trend lines.

Plot the trend line on top of scatter points to visualize relationships.

Practice

1. What is the main purpose of adding a trend line to a scatter plot in matplotlib?

easy

A. To show the overall pattern or relationship between variables

B. To add colors to the scatter points

C. To change the size of scatter points

D. To remove outliers from the data

Solution

1. Step 1: Understand scatter plot purpose

   A scatter plot shows individual data points but may not clearly show the overall trend.

2. Step 2: Role of trend line

   A trend line summarizes the general direction or pattern in the data, making relationships easier to see.

3. Final Answer:

   To show the overall pattern or relationship between variables -> Option A

4. Quick Check:

   Trend line = overall pattern [OK]

Hint: Trend lines summarize data direction, not point details [OK]

Common Mistakes:

• Thinking trend lines change point colors
• Confusing trend lines with point size changes
• Believing trend lines remove data points

2. Which of the following code snippets correctly fits a linear trend line to data arrays x and y using numpy?

easy

A. coefficients = np.polyfit(x, y, 1)

B. coefficients = np.polyfit(y, x, 1)

C. coefficients = np.polyfit(x, y, 2)

D. coefficients = np.polyfit(x, y)

Solution

1. Step 1: Understand np.polyfit parameters

   np.polyfit(x, y, degree) fits a polynomial of given degree to x and y data.

2. Step 2: Correct usage for linear fit

   For a linear trend line, degree = 1, and x is independent variable, y is dependent variable.

3. Final Answer:

   coefficients = np.polyfit(x, y, 1) -> Option A

4. Quick Check:

   np.polyfit(x, y, 1) fits linear trend [OK]

Hint: Use degree=1 for linear trend with np.polyfit(x, y, 1) [OK]

Common Mistakes:

• Swapping x and y in np.polyfit
• Omitting degree parameter
• Using degree 2 for linear trend

3. Given the code below, what will be the output of print(trendline(3))?

import numpy as np
x = np.array([1, 2, 3, 4])
y = np.array([2, 4, 6, 8])
coefficients = np.polyfit(x, y, 1)
trendline = np.poly1d(coefficients)
print(trendline(3))

medium

A. Error due to incorrect polyfit usage

B. 5.0

C. 9.0

D. 6.0

Solution

1. Step 1: Analyze data and polyfit

   Data y doubles x (2,4,6,8), so linear fit slope ~2 and intercept ~0.

2. Step 2: Calculate trendline at x=3

   trendline(3) = 2*3 + 0 = 6.0

3. Final Answer:

   6.0 -> Option D

4. Quick Check:

   trendline(3) = 6.0 [OK]

Hint: Check if y is roughly 2*x; then trendline(3) = 6 [OK]

Common Mistakes:

• Confusing x and y order in polyfit
• Forgetting poly1d creates function
• Assuming error instead of numeric output

4. Identify the error in the following code that tries to plot a scatter plot with a trend line:

import numpy as np
import matplotlib.pyplot as plt
x = np.array([1, 2, 3, 4])
y = np.array([2, 3, 5, 7])
coeff = np.polyfit(x, y, 1)
trend = np.poly1d(coeff)
plt.scatter(x, y)
plt.plot(x, trend)
plt.show()

medium

A. np.polyfit arguments are in wrong order

B. plt.plot(x, trend) should be plt.plot(x, trend(x))

C. plt.scatter should be called after plt.plot

D. Missing plt.xlabel and plt.ylabel calls

Solution

1. Step 1: Check trend line plotting

   plt.plot expects y-values, but trend is a function, not values.

2. Step 2: Correct function call

   Call trend(x) to get y-values for plotting: plt.plot(x, trend(x))

3. Final Answer:

   plt.plot(x, trend) should be plt.plot(x, trend(x)) -> Option B

4. Quick Check:

   Plot function needs y-values, not function [OK]

Hint: Call trend(x) to get y-values for plt.plot [OK]

Common Mistakes:

• Passing function instead of values to plt.plot
• Swapping x and y in polyfit
• Thinking plot order affects output
• Missing axis labels is not an error

5. You have noisy data with outliers. Which approach best improves the trend line fit on a scatter plot using matplotlib and numpy?

hard

A. Plot scatter without trend line to avoid confusion

B. Use np.polyfit with degree 1 on all data points

C. Remove outliers before fitting trend line with np.polyfit

D. Use plt.plot instead of plt.scatter for all points

Solution

1. Step 1: Understand effect of outliers

   Outliers can distort the trend line fit, making it less representative.

2. Step 2: Improve fit by removing outliers

   Removing or filtering outliers before fitting helps get a better trend line.

3. Final Answer:

   Remove outliers before fitting trend line with np.polyfit -> Option C

4. Quick Check:

   Remove outliers for better trend line [OK]

Hint: Filter outliers before fitting trend line for accuracy [OK]

Common Mistakes:

• Fitting trend line on all data including outliers
• Skipping trend line when data is noisy
• Confusing plot types for trend line fitting