Estimation

Overview

Estimation is the act of approximating an attribute such as value, size, or amount. Estimation has applications in all walks of life—from how much salt to put on popcorn, to using complex mathematical methods to predict the economy. Many common ideas, such as estimating the distance from Earth to the Sun, would be inconceivable without mathematical estimation.

Fundamental Mathematical Concepts and Terms

Estimation is an essential tool in many mathematical situations. For example, many people have to set an alarm clock to make sure that they wake up in time to get ready and get to school. The first time they set their alarm, they probably had to think about the different tasks they needed to accomplish in the morning in order to get to school on time. Those tasks may be broken down in the following manner:

School starts at 8:00 a.m.
It takes about 20 minutes to walk to school.
It takes about 30 minutes to make and eat breakfast.
It takes 20 minutes to shower, brush their teeth, and get dressed.
They may plan to press the snooze bar twice, so they set the alarm for about 20 minutes before they actually need to get up.

Adding all these amounts of time together gives an estimation that it takes about an hour-and-a-half to get ready and get to the destination. According to this estimation, these people need to set the alarm for about 6:30 a.m to be able to arrive on time. The ability to make a good estimate considerably depends on known information and past experience. Final estimations also often depend on previously established estimations. For instance, the amounts of time that they spend performing the various tasks are estimates in themselves. In this case, the final estimation is based on:

Information provided—classes begin at 8:00 in the morning.
Past experience—people have showered and brushed their teeth before; they have made and eaten breakfast before; they know if they are snoozers and that they usually push the button twice before they get out of bed.
Rough estimation—they know how far school is from their house and about how long it takes to walk that distance; because they prefer to get to school early rather than late, so they overestimate the time just a bit.

Estimates are usually refined as they are tested. Each time a person wakes up, gets ready, and walks to school, they may make adjustments to their estimate. For instance, sometimes they may want to get to school earlier, so they set their alarm even earlier for that day. On the other hand, they may have the first class of the day canceled, and they decide to set the alarm for later so that they can sleep a little longer.

Defining acceptable levels of error is another key concept in making meaningful estimates. There are many different theories and methods for analysis of error in estimation.

Throughout history, multiple methods of estimation have been proposed and scrutinized. Controversy exists among the many estimators and analysts about which methods yield the most accurate results.

A Brief History of Discovery and Development

The word estimate is derived from a late sixteenth century Latin word meaning to determine, appraise, or value. However, various methods of estimation have been used throughout history.

An early account of mathematical estimation involves a question posed by Greek mathematician Archimedes (born c. 287 B.C.), in which he contemplated how many grains of sand would be needed to match the volume of Earth. In ancient times, the issue of understanding and labeling very large (and small) numbers posed a serious problem that hindered the capabilities of mathematicians. This issue is at least partly attributed to the limitations of the existing numbering system, which was much like the Roman numeral system. By utilizing his own numbering system (similar to the exponential numbering system adopted later), Archimedes was able to grasp numbers large enough to approximate key values for determining a reasonable estimation of the amount of sand required to fill the volume of the planet. His new notation also allowed him to convey his ideas to his peers and to effectively convince them of the relative accuracy of his estimate.

Archimedes employed existing geometric theory (the equation for finding the volume of a sphere), a commonly accepted estimation (an approximate value of the radius of Earth, the distance from the center to the crust), and an observed measurement (he estimated that an average grain of sand was basically spherical and had a diameter of about 1/100th of an inch). Using these tools, Archimedes was able to estimate that it would take over 10³² grains of sand to match Earth's volume. He was aware of many imperfections in his calculations, including the fact that every grain of sand is not perfectly spherical. Earth is not a perfect sphere either—it has mountains and canyons (and is squashed at the poles). The values for the radius of Earth and the average diameter for a grain of sand are also only approximate values. Nevertheless, he was able to derive an estimate that was substantial enough to give a manageable account of the magnitude of the answer to his question. This was an important step toward mankind becoming comfortable with previously unfathomable numbers.

In 1773, Benjamin Franklin found that a drop of oil placed on the surface of water will spread out across the water until it forms a layer that has the thickness of a single molecule of the oil (known as a monomolecular film). In addition to Franklin's immediate observations from this experiment (his notes describe the oil spreading quickly and causing wavy water to become calm almost immediately), his discoveries would have a profound influence in the first estimation of the thickness of a molecule more than a century later. Scientists first estimated the thickness of a molecule by recording the volume of a drop of oil and then placing the drop onto the surface of some water. Once the oil had spread to a thickness of a single molecule, the surface area of the floating film was estimated using an approximate value for the radius of the somewhat circular layer of film. Then the formula for volume was used to determine the thickness of the film. The volume of the oil is equal to the area of the film multiplied by the thickness of the film. So to determine the thickness of the film, which is the thickness of a single molecule, the volume of the drop of oil was divided by the approximate area of the film.

In 1801, Carl Frederick Gauss, also commonly viewed as one of the most important figures in the history of mathematics, made the first applicable estimation of the orbit of planets. His first subject was a newly discovered planet named Ceres. Using his method of least square, which remains an important contribution to the development of estimation methods, Gauss was able to enhance prior theories about the orbits of planets, incorporating calculations that represent imperfections in orbital paths due to factors such as interference caused by other celestial bodies.

From tracking and maintaining populations of endangered species, to predicting genetic abnormalities in unborn babies, estimation remains an essential concept in many mathematical and scientific procedures and discoveries.

Real-life Applications

BUYING A USED CAR

For most people, the purchase of their first car can be an overwhelming event. There is much more to consider than whether they like the color or the rims on the wheels. With so many factors that can affect the value of an automobile, people have to be careful that they don't get cheated.

Often, the seller will begin by asking for much more money than the car is actually worth, or keep any problems a secret until they officially sell the car. So, how can one know if they are getting a good deal or being cheated? First, one should be aware that issues might exist, such as engine trouble, damaged upholstery, or body damage that lower the true value of the car. Similarly, any enhancements that can raise the value must be considered, including custom parts, limited edition features, audio or video equipment, global positioning system (GPS) tracking devices, safety features, and so on.

The most significant obstacle in coming to an agreement between the seller and the buyer is that there is no true value of a used car. Too many factors influence each car to be defined generally. Luckily, one thing that the buyer and the seller will probably agree on is that they both want to finalize the transaction as quickly and smoothly as possible. Therefore, both people have to use estimation involving available information to find an agreeable price range. Most buyers and sellers alike refer to one of many periodically released publications, such as the Kelley Blue Book, as a basis for what the car should cost depending on the year, make, model, mileage, and general condition. Using this base price, both parties attempt to factor in as many positive and negative characteristics as they can find to determine what they think the car is worth and to arrive at lower and upper bounds for an acceptable price. From there, everything depends on keen negotiation skills.

GUMBALL CONTEST

Jen has been entered into the critical-thinking contest at the annual mathematics fair, in which the top math students from around the region compete to solve difficult problems. The first problem posed to the contestants involves the estimation of the number of gumballs contained in a glass case that is 4 feet long, 4 feet wide, and 8 feet tall. Each contestant is expected to use mathematical reasoning to decide whether they think that the number of gumballs inside the glass case is less than or greater than 25,000.

Jen examines the glass case and thinks about how she can make a good approximation of the number of gumballs inside. The first thing she does is collect as much information as she can about the problem at hand. Jen takes note of the following information:

The glass case is transparent, so Jen can approximate the size of the gumballs. As far as she can see, the gumballs are all about the same size—somewhere between 1 inch and 2 inches in diameter.
The volume of the glass case is equal to the product of its dimensions. Since she will be estimating the volume of the gumballs in cubic inches, Jen multiplies each dimension of the glass case by 12 to convert to inches. The glass case is 48 inches by 48 inches by 96 inches. Multiplying these values together, she finds that the volume of the glass case is 221,184 cubic inches.

Her estimate of the diameter of each gumball consists of an upper bound and a lower bound. In this way, she hopes to simplify the problem without concerning herself too much with the true size of one gumball—let alone all of them! If the total estimated volume that she finds using the lower bound for the size of a gumball is more than the volume of the case, then she will know that 25,000 gumballs will not fit into the case. If the estimate she finds using the upper bound for the size of a gumball is less than the volume of the case, then she can safely conclude that 25,000 gumballs will fit into the glass case. However, if the volume of the glass case is between her lower and upper estimates, then she cannot make a confident conclusion using this information, and she will have to attempt to refine her estimate of the size of a gumball.

Jen knows that she needs the size of the gumballs to be expressed in terms of volume so that she can compare the volume of 25,000 gumballs to the volume of the glass case. She needs to find the volume of a single gumball and then multiply this value by 25,000. Using the formula for the volume of sphere, she finds that a gumball that is 1 inch in diameter (having a half-inch radius) has a volume of approximately 2.09 cubic inches. Similarly, she finds that a gumball that is 2 inches in diameter (having a one-inch radius) has a volume of 4.19 cubic inches. At this point, she feels confident that the volume of a single gumball is somewhere between these two estimated bounds. To find bounds for the total estimated volume of 25,000 gumballs, she multiplies the bounds for the volume of a single gumball by 25,000. She is reasonably certain that the volume of 25,000 gumballs is between 52,359 and 104,750 cubic inches. Since the upper bound for her estimate of the total volume of the gumballs is less than the volume of the glass case, Jen could decide at this point that she is convinced that 25,000 gumballs will in fact fit into the case. However, just before she turns her response in for evaluation, she becomes aware of a major flaw in her reasoning.

The total estimated volume of the gumballs definitely gives her a better feeling for this problem, but she quickly realizes that this will not yield conclusive results because she has not considered the air space in between all of the gumballs. What she has really figured out is that if she were to chew up 25,000 gumballs and press them into the glass case, the huge blob of gum would fit (especially if the gumballs are hollow). Little does Jen know that this is an example of a sphere-packing problem—a classic problem in mathematics for which there is no standard solution. Nevertheless, she is on the right track to making a fairly good estimate.

While thinking of a way to refine her estimation methods, she imagines having each gumball wrapped perfectly into a little box. She understands that this idea will lead to an overestimate of the amount of airspace in the glass case because the gumballs do not stack on top of each other perfectly. She will use the values found using this method as upper bounds for the total volume of the gumballs taking airspace into account.

Each gumball-wrapping box would have length, width, and height equal to the diameter of the gumball. For a gumball with a 1-inch diameter, the surrounding box would have a volume of 1 cubic inch. For a gumball with a 2-inch diameter, the box would have a volume of 8 cubic inches. The question now is whether or not 25,000 of these surrounding boxes will fit into the glass case. Multiplying by 25,000, she finds lower and upper bounds for the total volume of all of the wrapping boxes. The total volume of the boxes is between 25,000 and 200,000 cubic inches.

At this point, Jen stops to think about her progress so far. Since she can show that her largest estimate of 200,000 cubic inches—found by packing gumballs into boxes that overcompensate for airspace—is smaller than the volume of the glass case (221,184 cubic inches), she feels fairly certain that 25,000 gumballs fit into the glass case. (Note that if any of Jen's estimates were greater than 221,184 cubic inches, then she would have had to either refine her estimate of the diameter of a gumball or come up with a more accurate way to account for air space.)

POPULATION SAMPLING

Wildlife conservationists are often confronted with the task of estimating how many members of a certain species of animal are living in a given area. For example, suppose that a team of conservationists needs to estimate the number of fish in a small lake (without draining the lake and counting all of the fish). This may seem a daunting task because fish move around the lake, reproduce, and die. However, the team will be able to use population sampling techniques to find an estimate that is suitably accurate for their needs.

To begin, the team catches a sample of 300 fish. Each of these fish is tagged and returned to the lake. The team then makes a simplifying assumption that will be critical to the estimation process: over time, all of the fish in the lake move about at random. This is a reasonable assumption based on previous studies about these fish.

After waiting a week for the fish to redistribute themselves, the team again catches 300 fish and finds that 25 fish out of this sample are tagged. This time, the team members must do their best to select the fish at random from the total population of the lake. To ensure that they collect a random sample, they may collect the sample from various areas of the lake.

Next, the team uses the basic sampling principle as it applies to their situation: the proportion of tagged fish in the second sample should reflect the proportion of tagged fish in the entire lake population, as long as the sample size is reasonably large.

In the second sample, the team found that 25 out of 300 fish were tagged, so the proportion of tagged fish in the second sample is 25 divided by 300, or 1/12. The team also knows that there were 300 tagged fish in the lake (barring any fatalities among the first sample), so the proportion of tagged fish in the entire lake is 300 out of the total population of fish, the value that the team is attempting to estimate. In accordance with the basic sampling principle, the formula 1/12 ≈ 300/N, where N is the total number of fish in the lake helps team members find an estimate for the total population of fish in the lake.

After dividing both sides by 300 and simplifying, the team finds that N ≈ 3,600. The team of conservationists now has a rough estimate of 3,600 fish living in the lake. Depending on the requirements of the study, the team may or may not need to take more samples and find an average value. The team would not replace the fish after each sample, so that the fish are not counted twice. A relatively consistent number of tagged fish in each sample would be a good indication that the estimations are sufficiently accurate. Using a larger sample will usually result in higher accuracy as well.

DIGITAL IMAGING

A digital image is an arrangement of tiny square regions called pixels. In the case of a gray-scale (black-and-white with shades of gray) image, the brightness of each pixel is determined by a numeric value. A typical gray-scale image contains values ranging from 0 to 255, with 0 representing black, 255 representing white, and intermediate values representing shades of gray.

A color image can be represented using different mixtures of red, green, and blue. The color of each pixel in the digital image is usually determined by a set of three numbers, one representing red, one representing green, and one representing blue. These values each range from 0 to 255, where 0 indicates that none of that color is present in that pixel and 255 indicates a maximum amount of that color. When a digital image is magnified many times, the pixels can be seen clearly. If only part of a magnified image is visible, it may look like nothing more than different colored squares.

Estimation is a key concept in digital image compression processes. The goal of digital image compression is to reduce the size of the image file (so that it can be efficiently stored and shared) without losing so much quality that the human eye will easily notice the change. The main difference between image formats is the way that they compress images. The graphics interchange format (GIF) and joint photographic experts group (JPEG or JPG) formats—two of the most common digital image formats—are good examples of the effects of the various image compression techniques.

GIF images only support 256 colors—not much compared to the millions of colors found in most color photographs. If an image is converted to the GIF format, a compression technique called dithering is used to compensate for any loss of color. Image dithering involves repeating a pattern of two or more available colors in order to trick the eye into seeing a color very close to the color found in the original photograph. For example, to represent a solid area of a shade of red that is not included in the available 256 colors, the dithering process may alternate every other pixel with the two closest available shades of red. As the image is magnified, the pattern becomes more and more apparent. The color patterns that result from the dithering process are determined by mathematical functions that perform operations for estimating unavailable colors. The GIF format is best suited for illustrations and graphics with large regions of solid color. On the other hand, when a photograph containing millions of colors is converted to GIF, it usually appears grainy because there are too many colors to be adequately represented by just 256 colors.

JPEG images are much better suited for photographs. The JPEG format supports millions of colors and its compression method is intended to handle quick changes in color from pixel to pixel. However, graphics with relatively large areas of solid color converted to JPEG images tend to display messy spots around the areas where colors change. For example, if a company logo consisting only of a blue word on a solid red background is saved as a JPEG image, it will most likely have fuzzy areas all around the border of the letters. These fuzzy areas are called compression artifacts and, as implied by their name, are results of the compression process. As with the dithering process of the GIF format, the compression process is heavily dependent on mathematical functions that attempt to reduce the file size while retaining the image as seen through the human eye.

CARBON DATING

One of the most influential concepts in the field of archeology is that of carbon-14 dating, which allows archeologists to estimate the age of fossils and human artifacts. The basic idea behind carbon-14 dating is that all living things, from plants to humans, contain the same ratio of carbon-14 and carbon-12 atoms at all times (for every carbon-14 atom, there are a certain number of carbon-12 atoms). In a living organism, both types of atoms are constantly being created and destroyed, but the ratio between the two remains constant.

As soon as a living organism dies, it stops producing new carbon atoms. The carbon-14 decays and is no longer replaced, while the carbon-12 does not decay at all. By comparing the ratio of carbon-14 to carbon-12 in a formerly living organism to that in a living organism, it is possible to estimate how long the former has been dead. This concept has allowed archeologists to uncover many important milestones in the history of humankind.

Potential Applications

THE HUBBLE SPACE TELESCOPE

The Hubble Space Telescope (HST), a high-powered telescope attached to a spacecraft, has revolutionized astronomy by allowing astronomers to view celestial sights that are billions of light years away. Due to the fact that the images are captured from billions of light years away, astronomers know that the events depicted took place billions of years ago! Breathtaking images that have been constructed using data transmitted from the HST can be found on the Internet, in books, magazines, and newspapers.

However, these images are not exact representations of what is truly out there. The HST is capable of detecting different types of light and heat, including visible light (that humans can see), ultraviolet light, and infrared light. The raw data transmitted by the HST are electronic black-and-white images that reveal very little detail. Astronomers must combine the data from the various images (created from the different types of light and heat) and interpret the overall picture. These interpretations require advanced estimation methods, as well as some imagination and creativity. Since its launch in 1990, the HST has undergone many revisions, including updates to its image-capturing tools. As space telescope technology is refined, astronomers are able to construct increasingly accurate representations of celestial activity, providing valuable insight into the vastness of the universe.

SOFTWARE DEVELOPMENT

Software developers strive to estimate the amount of time that it will take to complete a software development cycle. A single development cycle often involves a vast number of steps that can take anywhere from a few weeks to a few years. All of these steps must be accounted for in the development plan to ensure that the software is completed, tested, and revised in a specified amount of time. If the product is not ready on time, the software company may lose clients and funding. Some of the major steps in the development cycle include conception (coming up with initial ideas), planning (organizing ideas and time), design (working out the look and feel of the on-screen display and general functionality), coding (using a programming language to write the software), and testing (checking to make sure that things look and work correctly). All of these steps are made up of multiple smaller tasks. For example, design might be split into visual design and functional design. Visual design might be split into window design, menu design, and so on. If any part of testing fails, issues must be listed and categorized by severity. The development team must then go back to the development cycle. How far back in the cycle the development team must go depends on the issues found by the testing team.

For decades, people have tried to conceive a universally accepted method for estimating the time it will take to complete a software product. However, software companies continue to run into unforeseen snags in the process, causing them to miss deadlines. Compensating for less tangible aspects of the development process proves to be a difficult task. For example, the complexity of the project (the number of pages, the types of tasks the software performs, how information is processed, etc.) is a consistent source of error.

In spite of past limitations to software development strategies, the desire to streamline the development processes continues to grow. This is due to a steady increase in the demand for software products, a trend that is not expected to change in the near future.