## Estimating Unknown Quantities

### What Is It?

Estimation is the process of guessing the approximate value of a number. An estimate is useful when:

An exact value is impossible or impractical to obtain (e.g., the number of stars in our galaxy)

An approximate value is adequate (e.g., the number of people who will attend a party)

An approximation serves as a rough check of the accuracy of a measurement (e.g., your scale says you weigh 1,850 pounds)

Estimation is an important aspect of quantitative thinking—and a critical life skill in a world in which we often need to make decisions on the basis of inexact or undefined information. Students at every grade level, from kindergarten to high school, should learn increasingly sophisticated estimation skills.

**Note**: Students are often tested on estimation skills using multiple-choice items, such as "Which of these estimates most closely represents the total?"

11 + 17 + 15 = ?

- 20
- 30
- 40
- 50

This type of problem is more closely related to rounding than to estimating. In computations in which a quantity can be found, rounding numbers to reach a reasonable estimation is an important skill.

### Types of Estimation

There are at least three different ways of estimating unknown quantities:

#### Eyeball Estimates

An "eyeball estimate" is an educated guess about a quantity based on sight and using some known benchmarks as guides. For example, based on our memories of how long a 12" ruler looks and how high 10' looks (the height of a basketball net), we can make an estimate about how high a ceiling is.

Depending on our life experiences, the benchmarks that we use in making eyeball estimates may include the length of a football field (300 feet, 100 yards), a gallon of milk, Larry Bird's height (6' 9"), the size of a sweater that will fit you, the length of the top part of your thumb (about 1"), and so on. To be able to make eyeball estimates, children must have developed their own set of benchmark quantities for standard measures of length, volume, weight, time, money, and so on.

#### Sampling

Sampling is another estimation skill that is used when it is easier to count "parts" than "wholes." We count one part, estimate the amount of parts in the whole, and then multiply. For example, in trying to estimate the number of candies in a five-gallon jug, we might use a formula like this:

Total candies = candies per handful x handfuls per layer x layers per jug

To find out how many times a human heart beats in a lifetime, we can count the number of heartbeats in a minute, multiply by 60 to get the number in an hour, and so on.

To find the number of hairs on someone's head, we might try to count the number of hairs in one square inch, make an assumption about the number of square inches in total, and multiply to find an estimate of the total.

A refinement of this method is to pull several samples and count the items in each, find an average, and then multiply using this number. For example, to find the number of candies in a jar, we can count the number of candies in five handfuls, then take the average as an estimate of the number of candies in a handful, and then multiply by the number of handfuls we estimate are in the jar.

#### Estimate by Analysis

An even more sophisticated kind of estimation involves dividing a complex problem into small parts, then working out estimates for each part separately. In these kinds of problems where there is no correct solution, spreadsheets are extremely valuable because students can create formulas for calculations, change assumptions easily, and immediately see the implications of their changes.

For example, consider the following budget for a simple party.

**Assumptions**guests 25 glasses of punch per person 2 sandwiches per person 1 cost of one liter of punch 1.25 servings per liter 8 cost of loaf of bread 2.50 sandwiches per loaf 8 cost of pound of cold cuts 4.50 sandwiches per pound 6 **Calculations****Items****Cost/Item****Total Cost**liters of punch 7 1.25 8.75 loaves of bread 4 2.50 10.00 lb. of cold cuts 5 4.50 22.50 Total 41.25 Cost/person 1.65 It's impossible to measure the exact amount of food or drink you need for a party since it can only be measured after the event has occurred. However, you can predict the amount that you will need before the event, and create a budget for the party. The estimated budget is based on past experiences and assumptions. Break down the budget into its component parts ("itemizing"), then estimate quantities and prices for each part to make an educated guess about how much the party will cost.

With further research (e.g., a visit to the supermarket), the estimate can be improved by refining some of the assumptions. Other ways to improve the estimate might include taking actual measurements (how many slices of bread per loaf) or consulting experts.

#### Multiple Methods

One way to improve the accuracy of an estimate is to try more than one method of calculation. For example, to estimate the number of supermarkets in the United States, one method would be to count the number of supermarkets in one city. Estimate the population of the city, and use that number to divide to find an estimate of the number of people per supermarket. To calculate a rough estimate of the number of supermarkets in the United States, divide the number of people in the United States (300 million in 2007) by this number.

Another method method might be to visit a supermarket and try to estimate the number of customers in the store at that time. Use this number to estimate the total number of people who might be shopping

*at that same time*everywhere in the United States. Calculate how many supermarkets it would take to accommodate this many people by dividing 300 million by the number of customers in your local market. Taken alone, each method produces a very rough estimate. However, used in combination, they may serve as a check on each other, giving an estimate that we may be more confident in than if we used only one method.