Problem Solving > Average Value of a Function

## 1. Average Definition

The **average** is one measure of the center of a set of data.

A simple **formula**, which works for most situations, is:

average = total sum of all the numbers / number of items in the set.

More formally, the formula is written as:

The summation sign (Σ) means to “add up”. Here, the letter *n* is used to represent the number of items. You can use this formula to get to the exact same answer as in the problem above:

Average = ⅓ * (2 + 4 + 6) = 12.

**Example: **Find the average: 210, 230, 240, 260, 280, 290, 300, 310, 330.

**Solution**:

- Sum up the numbers: 210 + 230 + 240 + 260 + 280 + 290 + 300 + 300 + 330 = 2450
- Divide Step 1 by the number of items in your set (9): 2450 / 9 = 271.11

The arithmetic mean has the distinct disadvantage of being extremely influenced by outliers —very small, or very large data points. For example, let’s say you have: 11, 12, 22, 40, 70 and 15,000.

The mean is: (11 + 12 + 22 + 40 + 70 + 15,000) / 6 = 2526, a poor reflection of the center point.

The **median **is resistant to outliers; the middle of this set is 26: in between the 22 and 40. Whether you use the mean or the median depends on lots of factors (including your instructor’s preferences), but in general, **don’t use the average if you have outliers.**

## Find the Mean on the TI 89 Graphing Calculator

**Example problem:** Find the mean: 12, 23, 78, 98, 121, 342, 88, 7, 27.

**Step 1:** Press the HOME button.

**Step 2:** Press CATALOG.

**Step 3:** Press ALPHA , then 5. This should bring you to the letter M in the catalog.

**Step 4:** Scroll to “mean(” and press ENTER.

**Step 5:** Press 2nd then (.

**Step 6:** Enter your data. Each number should be separated by a comma.

**Step 7:** Press the 2nd key, then ).

**Step 8:** Press ENTER to get **796/9**.

Note: For a decimal solution, press the diamond key, then press ENTER. The decimal solution is **88.4444**.

Lost your guidebook? You can download a new one from the TI website here.

## How to Find Average Value of a Function (with Integrals)

When it comes to finding the average value of a function, **the simple formula given above doesn’t work.** That’s mainly because the formula works for discrete variables, and a function is usually continuous. However, finding the average value of a function is relatively simple, if you’re comfortable with integration. Note that in order for the formula to work, you *must *have an integrable function that is also a continuous function.

Let’s say you wanted to find the average height of a graph of a function (f) over an interval from *a* to* b.Â Â *The formula is:

The integral is really just the **area under a curve**. The idea is that you’re taking infinitely many slices of this area under a curve and finding a tiny sliver that represents the average.

## Average Value of a Function: Example

**Example question:** Find the average value of a function f(x) = 3x^{2} – 2x on the closed interval [2, 3].

Step 1: **Insert the given function into the formula**:

- The function itself (3x
^{2}– 2x) replaces the*f(x)*on the right side of the equation. - The given upper limit (4) and lower limit (1) replace the
*b*and*a*in two places: the denominator on the left side of the equation and the upper and lower integral on the right side.

Step 2: **Solve the integral using the usual rules for integration**. Solve this particular integral using the power rule for integrals.

For 3x^{2}, the constant moves to the front of the integral symbol and integrates to zero, leaving ∫x^{2} = x^{3} / 3.

Step 3: **Evaluate the function at the given limits of integration (3 and 2). **

(27 – 9) – (8 – 4) = 18 – 4** 14.**

This step is easier than it looks:

- Plug the upper limit of integration (3) into the formula you obtained in Step 2.
- Repeat for the lower limit of integration (2).
- Subtract the two terms as shown.

## References

Larson, R. (2011). Calculus 1 with Precalculus. Cengage Learning; 3 edition.