An Introduction to Measurements
Our day-to-day activities would be incomplete without Measurements. Measurements need the use of both conventional and unconventional equipment, and different people may get different results from the same instrument. As a result, value measurements can be precise or accurate. Any measured value's accuracy is determined by how close it is to the true value. Precision relates to the degree to which the various measured quantities are connected.
For students who are searching for a deeper understanding of the concept of Precision in mathematics, Vedantu's experts have discussed and elaborated on it below. Vedantu offers the best education options for students who are studying for examinations and want to improve their marks in their respective subjects.
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 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 trials, 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 to 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 was 31, 33, 30, 34.
Group 2 – The age estimated by the four contestants was 39, 19, 55, 47.
Group 3 – The age estimated by the four contestants was 44, 47, 45, 48.
Afterward, they went up to the stranger and enquired about his actual age. He revealed that he was 33 years old. The correct answer is accurate. 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, 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.
Formula For Precision
The proportion of accurately classified occurrences or samples among those classed as positives is measured by precision. As a result, the precision formula is as follows:
Precision = True positives/ (True positives + False positives)
In the same fashion, students can write the formula of Accuracy,
Accuracy = (True positives + True Negatives)/ (True positives + True negatives + False positives + False negatives)
Bias
When we measure something several times and the results are close, there's a chance they're all faulty if there's a "Bias."
Bias is a systematic (inbuilt) inaccuracy that causes all measurements to be slightly off.
Bias in action
When there is nothing on the scales, they read "1 kilogram."
You always measure your height in thick-soled shoes.
When you click the stopwatch, it takes half a second to stop.
Every measurement is off by the same amount in each case. That is a type of bias.
Precision is Often Separated into Repeatability and Reproducibility
Repeatability is the variation arising when repeated measurements are taken during a short period under identical conditions.
Reproducibility is the variation arising when repeated measurements are taken over longer periods 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 a similar value with fewer digits after the decimal point, like 1.2334.
For example, the value of pi is approximately 3.14159265359.
A number that is quite accurate but not precise is 3.141. This is because there is no closer value to pi than it with the same number of digits.
A number that 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 that 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 a decimal or the 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.
FAQs on Precision in Math
1. What is the definition of precision in mathematics and science?
In mathematics and science, precision refers to the level of consistency and repeatability in a set of measurements. It describes how close multiple measurements of the same quantity are to each other, regardless of whether they are close to the actual or 'true' value. For example, if you measure the length of a pencil five times and get 4.51 cm, 4.52 cm, and 4.51 cm, your measurements are considered highly precise because they are very close together.
2. What is the main difference between accuracy and precision?
The main difference lies in their reference points. Accuracy measures how close a measurement is to the correct or true value. Precision measures how close a series of measurements are to each other. A simple way to remember this is:
- Accuracy: Closeness to the bullseye (the correct answer).
- Precision: Closeness of the shots to one another (consistency).
3. Can you provide an example of measurements that are precise but not accurate?
Yes. Imagine a weighing scale that is not calibrated correctly and consistently adds 0.5 kg to every measurement. If you weigh a 10 kg dumbbell three times and the scale shows 10.5 kg, 10.51 kg, and 10.49 kg, your measurements are precise because they are clustered together. However, they are not accurate because they are all significantly off from the true weight of 10 kg. This type of consistent error is known as a systematic error or bias.
4. How does the number of decimal places in a value relate to its precision?
The number of decimal places indicates the instrument's level of precision. A value with more decimal places, like 9.806 m/s², is more precise than a value like 9.8 m/s². This is because the instrument used to get the first measurement can distinguish between smaller increments, providing a more detailed and reproducible reading. Therefore, more significant figures or decimal places usually imply a higher degree of precision.
5. Is it possible for a set of measurements to be accurate on average but not precise? Why?
Yes, it is possible, though often due to random chance. This happens when measurements are widely scattered (low precision), but their average happens to be very close to the true value. For instance, if the true length of a rope is 20 metres, and three measurements are 18m, 22m, and 20m, the set is not precise because the values are far apart. However, the average (18+22+20)/3 = 20m, is perfectly accurate. This scenario is typically caused by random errors rather than a consistent issue with the measurement process.
6. How does systematic error (bias) affect the accuracy and precision of measurements?
A systematic error, or bias, is a consistent, repeatable error in a measurement system. It primarily affects accuracy by causing all measurements to be consistently higher or lower than the true value. For example, a clock running five minutes fast will always give an inaccurate time. However, systematic error does not necessarily affect precision. If the clock is always exactly five minutes fast, consecutive time readings would still be consistent relative to each other, thus remaining precise.
7. In the context of data classification, what does the formula for precision mean?
In fields like statistics and machine learning, precision has a specific formula: Precision = True Positives / (True Positives + False Positives). In this context, it answers the question: 'Of all the items we identified as positive, how many were actually positive?' It measures the reliability of the positive classifications. A high precision score means that when the model predicts something is positive, it is very likely to be correct. This is different from the concept of measurement precision but is a key metric for evaluating classification models.
