Random error is a measurement error that is uncontrolled and very difficult to predict. This is due to the fact that there are a huge number of parameters that are outside the control of the experimenter, which affect the final indicators. Random errors cannot be calculated with absolute accuracy. They are caused not immediately by obvious sources and require a lot of time to find out the cause of their occurrence.
How to determine the presence of random error
Unpredictable errors are not present in all dimensions. But in order to completely eliminate 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 if it changes, but by a certain relative number, the value of this random error is zero, and you can not think about it. And, on the contrary, if the obtained measurement result is different each time (close to some average value, but excellent), and the differences are uncertain, therefore, an unpredictable error affects it.
Occurrence example
The random component of the error occurs due to various factors. For example, when measuring the resistance of a conductor, it is necessary to assemble an electric 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 take a look at the ammeter to record its current strength data. After use the formula, where R = U / I.
But it may happen that at the time of reading the voltmeter in the next room turned on the air conditioner. This is a pretty powerful device. As a result, the mains voltage decreased slightly. If you did not have to look away at the ammeter, you could notice that the voltmeter readings had changed. Therefore, the data of the first device no longer matches the previously recorded values. Due to the unpredictable inclusion of the air conditioner in the next room, the result is already obtained with a random error. Drafts, friction in the axes of measuring instruments are potential sources of measurement errors.
How does it manifest
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 measuring the 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 is different. This is because a perfectly round conductor cannot be made in principle. Therefore, if you measure the diameter in several places of the product, 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 statistical error, since this value can be reduced by increasing the number of experiments under the same conditions for their implementation.
Nature of occurrence
In contrast to the bias, a simple averaging of several totals of the same value compensates for random errors in the measurement results. The nature of their occurrence is very rarely determined, and therefore it is never fixed as a constant. Random error is the absence of any natural laws. For example, it is not proportional to the measured value or never remains constant during several measurements.
There may be a number of possible sources of random errors in experiments, and it completely depends on the type of experiment and the instruments used.
For example, a biologist studying the propagation of a particular strain of a bacterium may encounter an unpredictable error due to a small change in temperature or lighting in the room. However, when the experiment is repeated over a certain period of time, it will get rid of these differences in the results by averaging them.
Random error formula
Suppose we need to determine some physical quantity x. To exclude a random error, it is necessary to conduct several measurements, the result of which will be a series of results N of the number of measurements - x 1, x 2, ..., x n.
To process this data:
- For the measurement result x 0 take the arithmetic mean x̅. In other words, x 0 = (x 1 + x 2 + ... + x n ) / N.
- Find the standard deviation. It is denoted by the Greek letter σ and is calculated as follows: σ = √ ((x 1 - x̅) 2 + (x 2 -x̅) 2 + ... + (x n - x̅) 2 / N - 1). The physical meaning of σ is that if one more measurement (N + 1) is taken, then with a probability of 997 chances out of 1000, it will fall in the interval x̅ -3σ <x n + 1 <c + 3σ.
- Find the boundary of the absolute error of the arithmetic mean x̅. It is found by the following formula: Δx = 3σ / √N.
- Answer: x = x̅ + (-Δx).
The relative error will be equal to ε = Δ / ̅.
Calculation example
The formulas for calculating the random error are rather 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. The number of measurements N = 5.
N p / p | l cm | I cf. arithm., cm | | ll cf. arithm. | | (ll cf. arithm. ) 2 | σ, cm | Δ l, cm |
1 | 250 | 253.0 | 3 | nine | 7.55 | 10.13 |
2 | 245 | eight | 64 |
3 | 262 | nine | 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%.
The practical benefits of high measurement accuracy
It should be borne in mind that the reliability of the results is higher, the more measurements are taken. To increase accuracy by 10 times, it is necessary to conduct 100 times more measurements. This is a rather laborious task. However, it can lead to very important results. Sometimes you have to deal with weak signals.
For example, in astronomical observations. Suppose you want to study a star whose brightness changes periodically. But this celestial body is so far away that the noise of electronic equipment or sensors receiving 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 isolate 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 between weak signals that are barely noticeable against a background of various noises.
The reason random errors can be solved by averaging is because they have a zero expected value. They are truly unpredictable and scattered on 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, random errors are not predictable. This makes them difficult to detect, but it is easier to get rid of, since they are statistical and can be removed using a mathematical method, such as averaging.