Accuracy and Precision – The Art of Measurement

Introduction

Accuracy and Precision play an important role while measuring anything. It is not possible for measurements to be completely error-free. Thus, when we are performing scientific calculations, our answers will carry certain errors, which can vary each time we repeat the experiment. For example, the acceleration due to gravity on Earth is 9.80665 m/s², but if you measured it experimentally, you would get a different value. Hence, errors are inevitable. The inevitability of errors necessitates ways to describe them. Accuracy and precision are two essential terms that help us understand the error in our measurement. In this article, we will discuss them in detail.

Define Accuracy

Whatever quantity you are measuring must have a true value. For instance, the value of the refractive index of water is 1.33333.

The accuracy of your measurement describes how near or far off you are from the true value of the quantity you are measuring. If you found the refractive index of water to be 1.31, you are quite close to the true value; thus, the measurement is accurate. Note that accuracy is defined individually for a single reading. Even in a set of readings, different readings will have different accuracy. As a whole, the set is accurate if its mean is close to the true value.

What is Precision?

While performing experiments, you usually take multiple readings to ensure better results. The term “precision” describes how close these readings are to each other. For example, consider the following measurements of the length of a stick −

• 1.01 m

• 1.02 m

• 0.99 m

• 1.02 m

As you can see, these readings do not deviate too much and thus, are said to be precise. As opposed to accuracy, precision is defined for a set of readings, not for each individual reading.

Accuracy and Precision Examples

To further clarify the meaning of accuracy and precision, let us take a couple of examples −

Example 1: A girl is 1.76 m tall. While measuring her height, her five friends record it as

• 1.7 m

• 1.78 m

• 1.73 m

• 1.72 m

• 1.75 m.

These values are very precise since they do not deviate much from each other. And in terms of accuracy, the last friend is the most accurate since she is only 0.01 m away from the true value. On the other hand, the first friend is the least accurate among this set.

Example 2: Once again, we take the same girl and this time, five other students record her height. The outcome is as follows −

• 1.51 m

• 1.61 m

• 1.88 m

• 1.72 m

• 1.80 m

Notice how far apart these values are from each other, i.e., how imprecise they are. At the same time, reading 4 is the most accurate, while reading 1 is once again, the least accurate.

Distinguish between Accuracy and Precision

Accuracy	Precision
Accuracy is a measure of the closeness of a value to a standard or true value.	Precision describes the variation of several values from each other.
Accuracy is defined for each individual reading, even if you have a set.	Precision has no meaning for one single reading. It is only defined for a set.
Accuracy measures agreement with a standard value.	Precision measures agreement of readings with one another.
If the results of the reading are consistently accurate, they must be precise. In other words, consistent accuracy requires precision.	Precision does not necessarily guarantee accuracy. One can have a highly precise set of values that are significantly far off from the true value.

Table-1: Difference between Accuracy and Precision

What is False Precision?

There are various instances in which, data can be misleading. Data presented in a format that makes it appear to be more precise than it truly is said to carry false precision.

An interesting example of false precision occurs when we convert between units. For example, imagine a car traveling at 40 mph. In terms of km/h, this corresponds to 64.3738 km/h, which has four significant digits after the decimal point, even though the original reading had none. False precision can also arise in the following cases −

• Adding unnecessary zeros in measurement leads to false precision. Mathematically, 1.00 m and 1 m are the same, but if our instrument cannot measure beyond one-tenth of a meter, the second zero in 1.00 m implies false precision.

• When high-precision and low-precision data are combined together, it can lead to false precision.

Quantification of Data

“Quantification” refers to the description of some value in numeric terms. Saying that a person is tall is an incomplete statement in science. We must specify their height as a number and only then, will it make scientific sense.

Quantification is essential since computers only handle numbers and thus, analysis done via computers requires data in numeric format. Quantification also allows us to perform statistical analysis, which is useful for machine learning and artificial intelligence.

Practice Questions

Q1. Discuss the accuracy and precision of the following set of readings of the refractive index of water (1.3333)

• 1.32

• 1.54

• 1.11

• 1.61

• 1.22

Ans. This set of readings is very imprecise since it diverts significantly from each other. The first reading (1.32) is the most accurate, while the fourth reading (1.61) has the worst accuracy.

Q2. A student uses a metre-scale and measures the length of his pencil to be 0.1237 m. Her teacher awards him zero marks for this. What could be the reason for that?

Ans. A metre-scale can only measure up to one-thousandth of a metre. This means that the fourth digit after the decimal place can’t possibly be measured by it. The data, therefore, has false precision and the student gets a zero.

Q3. Why is the following set of readings for the acceleration due to gravity not a good set?

• 9.805 m/s²

• 9.005 m/s²

• 10.610 m/s²

• 10.100 m/s²

• 9.512 m/s²

Ans. Even though the average of the above set of readings is 9.8064 m/s², which is only 0.002% inaccurate, it is still a bad set since it is very imprecise, i.e., the values diverge significantly from each other.

Conclusion

Errors while performing measurements are impossible to avoid and thus, we need ways to describe them. Accuracy and precision are two important tools in understanding the errors we encounter in scientific experiments. Accuracy is a measure of the diversion of a value from a true or standard value. It is defined separately for each reading, even for a set of readings. Precision on the other hand is a measure of the variation of readings from each other in a set and is not defined for a single reading. Accuracy and precision are independent, and it is possible to have one without the other. Sometimes, data appears to be more precise than it truly is due to the way it is presented. This is known as false precision and is encountered, for example, while converting between units. Combining data of varying precision can also lead to false precision. We need data from around us to be in numeric format to perform various computations, calculations, and analyses. The conversion of data into numbers is known as quantification and is essential for artificial intelligence and machine learning.

FAQs

Q1. How do you find the percentage error from a known value?

Ans. Use the following formula to find the percentage error.

$$\mathrm{e=\frac{experimental\: value-true\:value}{true\:value}\times 100\%}$$

Q2. How many significant digits should we use while performing calculations?

Ans. You should only use as many significant digits as you are initially given in the problem. While performing experiments, only use as many significant digits as your instrument can provide.

Q3. Of accuracy and precision, which is more important?

Ans. Both factors carry equal importance. Good scientific data must be both precise and accurate.

Q4. Discuss accuracy and precision in terms of statistical analysis.

Ans. A data set is deemed accurate when its mean is close to the true value, and it is called precise when its standard deviation is small.

Q5. How do we decide if our data is accurate enough?

Ans. Unless otherwise specified, there is a small rule of thumb you can use. Take the place value of the least significant digit and divide it by 2. This will give you the error margin you are allowed.

For example, if the true value is stated as 43.71 m, the least significant digit has a place value $\mathrm{\frac{1}{100}=0.01m}$. Thus, if you are ±0.005 m near the true value, you are accurate.