Accuracy and Precision

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: This article defines the relationship between accuracy and precision in physics. Accuracy is how well a method of measurement can determine the real value of a property. Precision is how consistent the measurements of that property are. A method that is accurate is not always precise, and measurements that are precise are not always accurate.

Principal Terms

• bias: the extent to which a measurement differs from the real value of the property being measured, or an intrinsic factor in a method that consistently causes such deviation.
• discrepancy: the difference between the measured value of a property and its real value, or between nonidentical measurements of the same property.
• measurement: the numerical value of a physical property according to a standard relative scale, or the act of determining that value.
• random error: a measurement error that is due to unpredictable and inconsistent factors that do not affect all measurements equally.
• systematic error: a measurement error that is due to intrinsic, mechanical, or environmental factors that affect all measurements equally.

Accuracy versus Precision

The terms "accuracy" and "precision" are often incorrectly used to mean the same thing. While it may seem logical to think that to be accurate and to be precise are the same thing, they are not. In physics, every property has one real value. That value can usually be determined, or at least estimated, by some form of measurement using a standard scale. Temperature, for example, is usually measured using an accepted standard temperature scales, such as kelvin, Celsius, or Fahrenheit.

Accuracy refers to how close the measured value of a property such as temperature comes to the real value of that property. The closer a measured value is to the real value, the more accurate the measurement is, and thus the method used to take that measurement. Precision, on the other hand, is a measure of the discrepancy between measurements of the same property. The closer those values are to one another, the more precise they are. Ideally, measurements should both precise and accurate, so that different measurements of the same property will have very similar values that are all very close to the property’s real value.

The difference between accuracy and precision can be explained using the example of an archer shooting arrows at the bull’s-eye of a target. The "real value" being measured is represented by the exact center of the bull’s-eye. The closer the arrows land to the bull’s-eye, the more accurate they are. A tight cluster of arrows within a small circle around the center would be a very accurate and precise result. A larger circle around the center, with the arrows farther apart, would be less accurate and less precise. Imagine the archer fires several arrows directly at the bull’s-eye when a strong wind blows across the field, causing the arrows to land slightly off-center but just as clustered as before. In that case, the precision of the shots is just as high, because they are still very close to one another. However, the accuracy of the shots is low, because they are not near the center. Thus, arrows fired with high precision may not be accurate, and arrows that are accurate may not be precise.

In all physical systems and methods of measurement, both systematic errors and random errors occur. In the above example, the wind is a systematic error: it affects all of the shots equally, shifting each arrow roughly the same distance in the same direction away from the bull’s-eye. This represents the bias of the system. The wind drives the shots toward their new impact point instead of their intended target. A random error, on the other hand, might be if the archer sneezes just as an arrow is released. Another would be if the archer fails to draw back the bowstring the same distance each time, making each arrow travel at a slightly different velocity. A systematic error affects all arrows in the same way, while a random error might affect only one arrow or several arrows in different ways.

Measurement and Error Limits

The goal of measurement is to find the real value of the property being measured. In science, every effort is made to prevent both systematic and random error. Researchers design experiments so that the value of one, and only one, factor can be measured. They calibrate, or adjust, the tools and methods to be used in the experiment against known standard values and outcomes. This ensures that the measured results will be both as accurate and as precise as possible.

There are physical limits on how well the value of any particular measurement in any situation can be known. This normally depends on the researcher’s visual perception—specifically, on the eye’s ability to estimate small distances. When reading the graduated scale of a measuring device, the standard rule of thumb is that the measurement can be read accurately only to one-tenth of the smallest scale division. For example, the length of an object is measured with a ruler whose smallest units are centimeters, the length can only be accurately determined to the nearest tenth of a centimeter, or one millimeter (1 mm). If the ruler is marked in millimeters, the length can only be determined to the nearest tenth of a millimeter (0.1 mm).

Digital electronic devices have greatly improved both precision and accuracy. However, because electronic circuits and sensors control the process by which devices take measurements, they have their own unique requirements for ensuring that the device is functioning properly. Electronic measuring devices must be calibrated often to specific standards, mainly because their parts are dynamic systems. Those parts are subject to drift as they are affected by environmental factors. Thus, the physical limits of the parts themselves, rather than the visual abilities of their users, determine the error limits in digital measurements.

Calibration and Calculating Accuracy

To maintain both the accuracy and the precision of various measuring devices, those devices must be calibrated by measuring properties whose values are both known and stable. Such standards serve as the reference base for the values taken from different devices. Comparing measurements against standard reference values may enable the measuring device to be calibrated and made more accurate. Or, it may simply allow the accuracy and error limits of the measuring device to be defined so that future measurements may be corrected. For example, if the time given by a wall clock is five minutes ahead of that shown on an atomic clock (the world’s most accurate timepiece), the owner of the wall clock may set it back five minutes so that it is more accurate, or simply remember to subtract five minutes from the displayed time in future.

Sample Problem

A very accurate device for measuring the pH of water-based solutions is used to measure the pH of ten different samples. Because the device is known to be accurate, these measurements are used as reference standards. The standard pH measurements from the first device are then used to test the accuracy of a second such device used outside of the laboratory. Calculate the accuracy of the second device (as a percentage), given the following data:

Answer:

To calculate the accuracy of a measurement, first subtract the standard or true value from the measured value. Then divide the absolute value of the difference by the standard value and multiply the result by 100. This calculation gives a value known as the percent error. The percent error can then be subtracted from 100 to determine the percent accuracy. For example, the percent accuracy of sample 1 is calculated as follows:

The measurement of the pH of sample 1 is 97.436 percent accurate. Performing the same calculation for the other samples gives the results below:

1. 97.436
2. 97.476
3. 97.430
4. 97.477
5. 97.507
6. 97.472
7. 97.445
8. 97.397
9. 97.467
10. 97.411

Thus, the accuracy of the second device is between 97.4 and 97.5 percent. However, because all measurements made with the same device can only be considered as accurate as the least accurate measurement, the lowest percent-accuracy value represents the functional accuracy of the device. Thus, the device can be said to be 97.397 percent accurate.

Accuracy and Precision

It is important to remember that accuracy and precision are never the same thing. Each is a different property of a single set of measurement events. A single measurement is precise by default because there are no other measurements whose values may vary. However, that same single measurement can be anything from perfectly accurate to wildly inaccurate, depending on the discrepancy between the measured and real values. For example, a device used to measure the temperature of an ice-water mixture may give a precise reading of 10.633 degrees Celsius. That would be a very inaccurate reading if the mixture has a real temperature of 0 degrees Celsius.

The terms "accuracy" and "precision" are most useful when applied to a set of measurements of the same property taken under the same conditions. In such cases, accuracy and precision are given number values and descriptions calculated through statistical analysis. Statistical terms, such as mean, median, and skew, describe the relationships between the different measurements in the set. This in turn makes it possible to determine the accuracy and precision of both the measurements and the methods used to take them.

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