## What is an estimate?

An estimate is an approximation of a quantity obtained by analyzing measurements or by existing hypotheses. In estimation, we calculate an approximate value of a quantity or a parameter using statistical analysis or mathematical expressions already existing.

## History of estimates

Science, scientists, and engineers deal with different quantities and their values. How much? How heavy? Everything  has to be either measured experimentally or calculated from the existing measurements. Most of the time, we express the result in terms of a number and quantity or both. For that, some true value is assumed beforehand from the physical laws and hypotheses that already exist. During experiments, the instrument used has a limit to how precisely it can measure the given quantity. To understand precision, let us assume that  we have a centimeter-scale (with only centimeters marked). We need to measure the length of a rope. This scale will give its value only in centimeters. Now consider this, if we assume the actual length is 2.56 cm, and we try to measure the length with the given centimeter scale, the results of the experiment would give us values of integers in centimeters only. Repeated measurements would give two or three values depending upon the trails. Now, the instrument is limited to give a certain value within 1 cm. This does not give a precise result. In such a case, a millimeter-scale is more precise than a centimeter scale.

Now, it's not always possible to get the most precise instruments to measure different quantities, like the radius of an atom, the size of a galaxy. In even more daily life experiments, the results are not the exact assumed values, even when everything we know about the system is running as predicted by our hypothesis. There is a deviation of the results from the exact true value of the parameter being measured. This difference is called an error. There are certain avoidable and unavoidable errors in nature. Given this situation, we see that just direct/indirect measurement of a value by an experiment or an instrument does not give the exact value of it. The errors we get in the trails can even propagate through when we try to calculate the required quantity using the ones we can measure.

So, we don't exactly measure the value of a parameter or quantity, but we estimate it. An estimate is an approximate outcome of an analysis of previous information or knowledge. This can be a set of data coming from repeated trials of a given experiment or measurement, or a group of laws that we know about the system.

If we look at ancient times, many quantities were estimated indirectly. The calculation of the circumference of the earth by Eratosthenes would be a very good example. There are several others to mention. Thus, estimation has been a part of science, where it has also become a way to test out the validity of the laws of physics that we know. For example, using Kepler's laws and Newton's law of universal gravitation, we have estimations of orbits of different heavenly bodies. This gives us a strong base to say that the law of gravitation is valid. But then a little difference in behavior is shown by the planet mercury, which paved the way for looking into deeper aspects of gravitation. Thus, estimations have helped scientists both to calculate different parameters and test our hypotheses about the laws of the universe.

## Estimations in experiments

In experiments, as we already discussed before, the results tend to deviate from the true value of the parameter. The difference between the true value and the experimental result is called the 'Absolute error'. But to calculate it, we need to know the true value of the parameter. But, when an experiment is conducted, an error inevitably creeps in, from different sources such as instruments, environment, etc. In direct measurement, mostly, we represent the instrumental error as the error of the estimate.
And the average value or mean value of all measurements is considered the best estimate of the quantity in many cases.
Average value or mean value=  Sum of measurements / Number of measurements

For example, if the time of fall of a ball from a given height is measured in three trials with a second' stop clock. The measured values are given in the table below.

 Trail number Measurement (seconds) I 3 II 4 III 3

Now, the estimate of the value of the time of fall is given by the average of the given measurements given by,

T=(3+4+3)/3 =3.33 s

Now the error of the estimation is given by the instrumental error of the stop clock, that is,

δt=±1s

Now, the estimate of the time of fall is given by,

Time of fall = T±δt

= 3.33 ± 1s

## Estimations in Physics

As discussed above, many hypotheses are tested in physics using estimations, making predictions and then measuring the values and see if it comes out in the same order. In many situations, mathematical formulae form the base for estimating different quantities. The quantities which we know within limited precisions are used as inputs for the mathematical formulae. These mathematical formulae can come from statistical analysis or dimensional analysis. Sometimes experience helps in forming mathematical equations relating seemingly unrelated quantities to estimate the given quantity up to certain accuracy.

## Context and Applications

This topic is relevant to the people doing laboratory or observatory research. The knowledge of estimates and errors is very much used in fields like

• Astronomy
• Bachelor in Physics
• Masters in Physics
• Bachelor in Technology
• Masters in Technology

## Practice Problem

Q. 1 What is the exact value of a quantity called?

1. True value
2. Approximate value
3. Ultimate value
4. Given value

Answer: (a)- True value

Explanation- The exact value of a quantity is called true value.

Q. 2. Which of the following is the most precise instrument?

1. Vernier calipers
2. Screw gauge
3. Millimeter scale
4. Our elbow

Answer: (c)- Millimeter scale

Explanation- A more precise instrument measures the given quantity within lower bounds. Among the given instruments, the screw gauge has an error bound of 0.01 mm.

Q. 3. Which of the following can be used to estimate the value of a quantity?

1. Experimental values
2. Formula
3. Statistical analysis
4. All of these

Answer: (d)- All of these

Explanation- An estimate of a quantity can be obtained using experimental values, formulas, and statistical analysis.

Q. 4. What is the average of the measurements of a quantity done in multiple trails considered as?

1. Estimate
2. Error
3. True value
4. None of these

Explanation- Most of the time, the average of all the measurements of a quantity is considered the best estimate.

Q. 5. Can we say, 'Measurement is error-free'?

1. True
2. False
3. Can’t say
4. None of these

Explanation- Error inevitably creeps into our measurement through different sources.

### Key Points

1. An estimate is not the accurate value but a close approximation to the value of the quantity.
2. Errors are inevitable
3. Errors propagate through our calculations.

## Formulas

• The average value or mean value of all the measurements is given by,

Average value or mean value= Sum of measurements / Number of measurements

• Error analysis
• Measure theory
• Statistics
• Standard deviation
• Fermi calculation
• Approximations

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