# Calculating Error in Algebraic Approximation: \Delta f(x,y)

• whozum
In summary, the conversation discusses two equations for determining error in experimental data, one being the standard deviation of a function and the other being the chi square test for goodness of fit. The standard deviation equation includes a correction for independent variables, while the chi square equation compares observed and expected values to determine if they are within the expected range. The two equations are different and serve different purposes in analyzing experimental data.
whozum
For a function f(x,y): The error is:

$$\Delta f(x,y) = \sqrt{(\frac{df}{dx})^2+(\frac{df}{dy})^2}$$

Is this a form of the approximation in algebraic error determination:

$$\Delta f(x,y) = \sqrt{f(x+\Delta x,y) + f(x,y+\Delta y)}$$?

Now I was trying to do this in my bio lab for genetics and probability distribution. We had 27 samples, predicted 75% (20.25) to be purple and 25% (6.75) to be white. We observed that 23 were purple and 4 were white, so to calculate the deviation I used the above function and got:

$$\frac{df}{dx} = \frac{23-20.25}{20.25}, \frac{df}{dy} = \frac{4-6.75}{6.75}$$

*For some reason latex doesn't want to show it, but I plugged in my numbers into the first equation above.

So the observed sample was 18.44% off?

They gave us an equation to find the deviation of a population from the expected value which is similar to the one above:

$$\chi^2 = {\frac{(Obs_p - Exp_p)}{Exp_p}^2+\frac{(Obs_w - Exp_w)}{Exp_w}^2}$$

$$\chi^2 = {\frac{(2.75)}{20.25}^2+\frac{(2.75)}{6.75}^2} = 1.494$$

It claims that "If the value for chi squared is less than or equal to 3.841, then your sample is within the expected range."

Is this a standard deviation?

This is pretty similar to the first equation, except we're squaring the numerator instead of the whole fraction. Did I get my error equation wrong, or are these truly diferent and unlinked?

Last edited:
$$\Delta f(x,y) = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}$$

isn't quite right. It's

$$\Delta f(x,y) = \sqrt{\left(\frac{\partial f}{\partial x}\Delta x\right)^2+\left(\frac{\partial f}{\partial y}\Delta y\right)^2}$$

this error formula is indeed just the standard deviation of $f$, assuming $\Delta x$ and $\Delta y$ are standard deviations of the respective variables, and that $x$ and $y$ are independent.

Last edited:
Your corrected formula is the standard deviation of f? Or are you talking about my Chi formula? Are they the same?

The equation I posted gives the standard deviation of $f$. Your chi formula is an example of the chi square test for goodness of fit of experimental data to a certainly distribution (the one that the expected values come from). The best online resource that I found to explain it (and it's not very good...) is here: http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm

## 1. What is error in algebraic approximation?

Error in algebraic approximation is the difference between the actual value of a function and its approximate value. It is a measure of how accurate the approximation is.

## 2. How is error calculated in algebraic approximation?

The error in algebraic approximation is calculated using the formula ∆f(x,y) = f(x,y) - fapprox(x,y), where f(x,y) is the actual value of the function and fapprox(x,y) is the approximate value.

## 3. What is the purpose of calculating error in algebraic approximation?

The purpose of calculating error in algebraic approximation is to determine the accuracy of the approximation and to make improvements if necessary. It also helps in identifying areas where the approximation may need to be adjusted.

## 4. Can error in algebraic approximation be negative?

Yes, error in algebraic approximation can be negative. This means that the approximation is greater than the actual value of the function. A negative error indicates an overestimation of the function.

## 5. How can error in algebraic approximation be minimized?

Error in algebraic approximation can be minimized by using more accurate and precise methods of approximation, such as increasing the number of terms in a Taylor series or using a smaller step size in numerical methods. It is also important to carefully choose the points at which the function is approximated.

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