CURVE FITTING

Curve Fitting

Definition Of Curve Fitting

The curve which is obtained by plotting a set of data points is called Curve Fitting.

More About Curve Fitting

The curves like circular, elliptical, parabolic, and hyperbolic. Sine, cosine, etc. are also obtained by fitting data points.

Video Examples: Curve Fitting in Excel

Example of Curve Fitting

example of Curve Fitting

In the given figure, a curve is drawn through several points.

Solved Example on Curve Fitting

Ques: The table shows the number of workers in the industry, where t represents the
time in years since 2000. Draw a scatter plot and fit a curve for this data.

example of Curve Fitting

Choices:

A. W = 495(4)^t
B. W = 360t² + 1125t + 495
C. W = 1485t + 495
D. W = 360t² + 495 Correct Answer: B

Solution:

Step 1: Plot the points and join with a smooth curve.
example of Curve Fitting Step 2: The graph appears to be curved, not linear.
Step 3: As the ratios of consecutive number of workers is not same,
an exponential model does not fit the data.
step 4: W = at² + bt + c [Quadratic model.]
Step 5: Consider the three data pairs from the table (0, 495),
(1, 1980), and (2, 4185) and substitute in the model.
Step 6: The equations obtained are
c = 495 ------------ (1)
a + b + c = 1980 ----------- (2)
4a + 2b + c = 4185 --------- (3)
Step 7: a = 360, b = 1125, and c = 495 [Solve the simultaneous equations.]
Step 8: So, the number of workers in the industry modeled by W = 360t² + 1125t + 495.

Quick Summary

Curve fitting involves finding a mathematical function that closely matches a set of data points.
Different types of curves (linear, polynomial, exponential, etc.) can be used depending on the data.
The goal is to minimize the difference between the curve and the actual data points.

\[ y = f(x) \]

🍎 Teacher Insights

Emphasize the importance of visualizing the data and understanding the context of the problem. Use real-world examples to illustrate the applications of curve fitting. Discuss the trade-off between model complexity and generalization ability.

🎓 Prerequisites

Algebra
Graphing
Data Analysis
Basic Statistics

Check Your Knowledge

Q1: Which of the following is the primary goal of curve fitting?

To create aesthetically pleasing graphs To find a function that closely matches a set of data points To extrapolate data beyond its known range To simplify complex data sets

Q2: What is a potential drawback of overfitting data in curve fitting?

It leads to a simpler model It improves the accuracy of predictions for new data It captures noise and random variations, reducing generalizability It always results in a perfect fit

Frequently Asked Questions

Q: How do I choose the best curve for my data?
A: Examine the scatter plot of the data to identify trends. Consider the underlying physics or principles of the system being modeled.

Q: What is overfitting?
A: Overfitting occurs when the curve fits the data too closely, capturing noise and random variations instead of the underlying pattern. This leads to poor predictions for new data.