Precision in Math

Precision - Concepts of Accuracy and Precision

The Merriam-Webster dictionary defines precision as the quality or state of being precise. Of course, while speaking of mathematics the Merriam-Webster is largely irrelevant. But in precision, the meaning in the English language, be it spoken or written, is not far off the mathematical definition.

Precision refers to how consistent any given values are with each other. It shows the closeness of the given values, or measurements are with each other. It is irrelevant of the ‘correct' or closest to an expected value of any given measurements. For example, if you measure the weight of an object five times, and you find that it is 245 grams each time, it is perfectly precise. If you find only a few grams of deviation, that is if you find it is 245 grams in 3 trials and find it 244 grams and 247 grams in the remaining trails, it is quite precise. It is irrelevant to the precision of the object whether the actual weight of the object is 250 grams or 500 grams.

The concepts of precision and accuracy are easily confused. But is possible to differentiate it after understanding the following:

Accuracy refers to the closeness of the value being measured to the actual measurement of the object, while precision is the consistency of multiple measurements of the same object with each other irrelevant of the actual measurement of the object.

Suppose there is a game being played among some people, and the rules of the game are to guess the age of a random stranger whom the contestants select. There are three groups of contestants with four members in each group. They select a random stranger and each contestant separately guesses the age of the selected stranger

After each of them guessed the age of the stranger, it was found that:

Group 1 – The age estimated by the four contestants were 31, 33, 30, 34.

Group 2 – The age estimated by the four contestants were 39, 19, 55, 47.

Group 3 – The age estimated by the four contestants were 44, 47, 45, 48.

Afterward, they went up to the stranger and enquired his actual age. He revealed that he was 33 years old. The correct answer is an accurate answer. Only a single contestant from the first group was perfectly accurate in his answer. The remaining people in the first group were close to the correct answer, but their answers were not the correct answer.

Group 1 – The four guesses were consistent, with one of them being correct while three others were close to being correct. This means one of the answers is perfectly accurate while the other three were fairly accurate. Since all the four guesses were close to each other, they can be considered precise.

Group 2 – The four guesses were all over the place, with none of them being close to the correct answer. Also, none of the guesses are close to each other. Hence, they are neither precise nor accurate.

Group 3 – The four guesses were incorrect, and quite far from the correct answer. But the four guesses are quite close to each other. The contestants in this group must have had some reason for their guess since their answers were close to each other. Nevertheless, the answers are not accurate, but they are precise.

In other words, accuracy is the degree of closeness between measurement and the true value of a measurement. Precision is the degree to which repeated measurements under the same conditions are unchanged.
Unlike accuracy, precision is not a definite term. In the above example, while 33 is the accurate value, and nothing is more accurate than it. Values like 32, 34 and 31 are quite close but not entirely accurate. But while speaking of precision, considering any value, if we measure it multiple times, and we get 33, 34, 35, and 32, the values are precise. If the measured values are 33, 34, 33 and 34, it is also precise, only more precise than the previous measurements. If the measured values are 33, 33, 33 and 33, it is perfectly precise. But generally speaking, all the three given measurements are precise.

Similarly, if there is a dart-board game, with three contestants throwing three darts, and the first contestant manages to land all the darts in the center of the board, he is both accurate and precise. If the second contestant lads the darts on the fringes of the board far away from each other, he is neither precise nor accurate, but if he lands all three bolts together on the fringes of the board, he is precise and accurate.

Precision is often separated into repeatability and reproducibility.

Repeatability is the variation arising when repeated measurements are taken during a short period of time under identical conditions.

Reproducibility is the variation arising when repeated measurements are taken over longer periods of time among different instruments and operators.

Precision can also be considered the amount of information conveyed by a value. A number with more digits after the decimal point, for instance, 1.233443322 is more precise than a number with similar value with fewer digits after the decimal point, like 1.2334.

For example, the value of pi is 3.14159265359 approximately.

A number which is quite accurate but not precise is 3.141. This is due to the fact that there is no closer value to pi than it with the same number of digits.

A number which is not accurate but precise is 1.14159265359. It is precisely because it conveys more information. It is not accurate because it is nowhere near the actual value of pi.

A number which is precise and accurate is 3.14159265359. It conveys the maximum information while being close to the value of pi.

We thus consider precision as the maximum number of significant digits after the decimal point in a decimal number.

Numerical problems related to precision:

Question 1: Find out the precision value from the given condition: 11 x 3.4

Solution:

The given condition is 11 x 3.4
The result of the multiplication is 37.4
Here the precision is 1 that is decimal or significant digit is 1.

Question 2: Find the accuracy and precision values for 8.213

Solution:
Value = 8.213
To find Accuracy and Precision,
Accuracy = 8.213 (Closest value with the same number of digits)
Precision = 8.2139999… (Value which gives maximum information)

The more approximate values are given for the number.
Thus, this is the difference between the accuracy and precision of the numbers.