Random error is an error in measurements that is uncontrollable and very difficult to predict. This is due to the fact that there are a huge number of parameters that are beyond the control of the experimenter, which affect the final performance. Random errors cannot be calculated with absolute accuracy. They are not caused by immediately obvious sources and take a long time to figure out the cause of their occurrence.
How to determine the presence of a random error
Unpredictable errors are not present in all measurements. But in order to completely exclude its possible influence on the measurement results, it is necessary to repeat this procedure several times. If the result does not change from experiment to experiment, or changes, but by a certain relative number, the value of this random error is zero, and you can not think about it. And vice versa, if the obtained measurement resulteach time is different (close to some average but different) and the differences are vague, hence being affected by an unpredictable error.
Example of occurrence
The random component of the error arises due to the action of various factors. For example, when measuring the resistance of a conductor, it is necessary to assemble an electrical circuit consisting of a voltmeter, an ammeter and a current source, which is a rectifier connected to the lighting network. The first step is to measure the voltage by recording the readings from the voltmeter. Then shift your gaze to the ammeter to fix its data on the strength of the current. After using the formula where R=U / I.
But it may happen that at the time of taking readings from the voltmeter in the next room, the air conditioner was turned on. This is a pretty powerful device. As a result, the network voltage decreased slightly. If you didn't have to look away at the ammeter, you could see that the voltmeter readings had changed. Therefore, the data of the first device no longer correspond to the previously recorded values. Due to the unpredictable activation of the air conditioner in the next room, the result is already with a random error. Drafts, friction in the axes of measuring instruments are potential sources of errors in measurements.
How it manifests
Suppose you need to calculate the resistance of a round conductor. To do this, you need to know its length and diameter. In addition, the resistivity of the material from which it is made is taken into account. When measuringthe length of the conductor, a random error will not manifest itself. After all, this parameter is always the same. But when measuring the diameter with a caliper or micrometer, it turns out that the data differ. This is because a perfectly round conductor cannot be made in principle. Therefore, if you measure the diameter in several places of the product, then it may turn out to be different due to the action of unpredictable factors at the time of its manufacture. This is a random error.
Sometimes it is also called the statistical error, since this value can be reduced by increasing the number of experiments under the same conditions.
Nature of occurrence
Unlike systematic error, simply averaging multiple totals of the same value compensates for random measurement errors. The nature of their occurrence is determined very rarely, and therefore is never fixed as a constant value. Random error is the absence of any natural patterns. For example, it is not proportional to the measured value, or never remains constant over multiple measurements.
There can be a number of possible sources of random error in experiments, and it depends entirely on the type of experiment and the instruments used.
For example, a biologist who studies the reproduction of a particular strain of bacteria may encounter an unpredictable error due to a small change in temperature or lighting in the room. However, whenthe experiment will be repeated for a certain period of time, it will get rid of these differences in the results by averaging them.
Random error formula
Let's say we need to define some physical quantity x. To eliminate random error, it is necessary to carry out several measurements, the result of which will be a series of results of N number of measurements - x1, x2, …, xn.
To process this data:
- For the measurement result x0 take the arithmetic mean x̅. In other words, x0 =(x1 + x2 +… + x) / N.
- Find the standard deviation. It is denoted by the Greek letter σ and is calculated as follows: σ=√((x1 - x̅)2 + (x2 -х̅)2 + … + (хn -х̅)2 / N - 1). The physical meaning of σ is that if one more measurement (N + 1) is carried out, then with a probability of 997 chances out of 1000 it will fall into the interval x̅ -3σ < xn+1 < s + 3σ.
- Find the bound for the absolute error of the arithmetic mean х̅. It is found according to the following formula: Δх=3σ / √N.
- Answer: x=x̅ + (-Δx).
The relative error will be equal to ε=Δх /х̅.
Calculation example
Formulas for calculating random errorquite cumbersome, therefore, in order not to get confused in the calculations, it is better to use the tabular method.
Example:
When measuring the length l, the following values were obtained: 250 cm, 245 cm, 262 cm, 248 cm, 260 cm. Number of measurements N=5.
| N n/n | l, see | I cf. arithm., cm | |l-l cf. arithm.| | (l-l compare arithm.)2 | σ, see | Δl, see |
| 1 | 250 | 253, 0 | 3 | 9 | 7, 55 | 10, 13 |
| 2 | 245 | 8 | 64 | |||
| 3 | 262 | 9 | 81 | |||
| 4 | 248 | 5 | 25 | |||
| 5 | 260 | 7 | 49 | |||
| Σ=1265 | Σ=228 |
The relative error is ε=10.13 cm / 253.0 cm=0.0400 cm.
Answer: l=(253 + (-10)) cm, ε=4%.
Practical benefits of high measurement accuracy
Note thatthe reliability of the results is higher, the more measurements are taken. To increase the accuracy by 10 times, you need to take 100 times more measurements. This is quite labor intensive. However, it can lead to very important results. Sometimes you have to deal with weak signals.
For example, in astronomical observations. Suppose we need to study a star whose brightness changes periodically. But this celestial body is so far away that the noise of electronic equipment or sensors that receive radiation can be many times greater than the signal that needs to be processed. What to do? It turns out that if millions of measurements are taken, then it is possible to single out the necessary signal with very high reliability among this noise. However, this will require a huge number of measurements. This technique is used to distinguish weak signals that are barely visible against the background of various noises.
The reason random errors can be solved by averaging is that they have an expected value of zero. They are really unpredictable and scattered around the average. Based on this, the arithmetic mean of errors is expected to be zero.
Random error is present in most experiments. Therefore, the researcher must be prepared for them. Unlike systematic errors, random errors are not predictable. This makes them harder to detect but easier to get rid of as they are static and are removedmathematical method such as averaging.
