# Advice 1: How to determine the error of the instrument

The implementation of many educational and scientific-research works connected with carrying out a variety of measurements of physical quantities. After receiving the results of the instrumental measurements usually follows the treatment. For exact match the results of a calculation pattern of the experiment is required to take into account the error margin of measurements. Determining the error of measuring instruments is carried out by special methods. Instruction
1
Use to determine absolute instrumental error determined by the design of the instrument, a special table, the errors of measuring instruments. For example, the drawing line length is 500 mm and the interval is 1 mm absolute instrumental error of plus or minus 1 mm; and for micrometer with measuring range of 25 mm and 0.01 mm, this value will be plus or minus 0.005 mm.
2
Determine the absolute error of reference. It is derived from not very accurate readings, starting with measuring instruments and devices. In most cases it is equal to the half the price of division of the instrument scale. When measuring time absolute error of reference take is equal to the division value of the stopwatch (hours).
3
Calculate the maximum absolute error of direct observation. It is defined as the sum of the absolute instrumental error and the absolute error count (if other types of errors can be neglected):

A' = AI + AO where

A' – maximum absolute error of direct observations;
AI – absolute instrumental error;
AO absolute accuracy of the countdown.
4
When determining absolute error of measurement instrument, we round it to one significant digit. The numerical value of the result of the measurement procedure are rounded so that its last digit were in the same category as the digit error.
5
If there is a need for instrument re-measurement undertaken in the same controlled conditions, then the margin of error, referred to here as the random, determine the arithmetic average of errors of all measurements.
6
To define an absolute sampling error of electrical measuring instrument, find out its accuracy. It usually indicate the scale of the device or technical data sheet (description).

# Advice 2 : How to calculate the error

The error is a value which determines possible deviations of the data obtained from the exact value. There are concepts of relative and absolute error. Their presence is one of the tasks of mathematical analysis. In practice, however, more important is to calculate the error of the scatter of any measured indicator. Physical devices have their own margin of error. But not only it is necessary to consider the indicator. For the calculation of error scatter σ necessary to carry out several measurements of this magnitude. You will need
• A device for measuring the desired value
Instruction
1
Measure device or other means of measuring the desired quantity. Repeat the measurement several times. The greater the obtained value, the higher the accuracy of determining the error of the scatter. Usually spend 6-10 measurements. Note the resulting set of values of the measurand.
2
If all values are equal, therefore, the error of variation is equal to zero. If a row has different values, compute the error of the scatter. For its determination, there is a special formula. 3
According to the formula, and calculate the first average value <x> of the resulting values. To do this, add up all the values and their sum divide by the number of measurements n.
4
Determine alternately the difference between each value and the mean value <x>. Record the results of the obtained differences. Then lift all the difference in the square. Find the sum of these squares. Save the last received result.
5
Compute the expression n(n-1), where n is the number of your measurements. Divide the result from previous calculations on the resulting value.
6
Take the square root of the private from division. This is the error dispersion σ measured you size.

# Advice 3 : How to find the error

Through measurement, we can not guarantee the accuracy of, any instrument gives a certain error. To know the measurement accuracy or the accuracy class of the instrument, it is necessary to determine the absolute and relative error. You will need
• - several measurement results or other sample;
• calculator.
Instruction
1
Measure at least 3-5 times to be able to calculate the actual value of the parameter. Total the results and divide them by the number of measurements you got a valid value, which is used in the problems is true (it is impossible to determine). For example, if the measurement gave the result 8, 9, 8, 7, 10, the actual value will be (8+9+8+7+10)/5=8,4.
2
Find the absolute error of each measurement. From this measurement subtract the actual value, the signs of neglect. You will receive 5 of the absolute error, one for each dimension. In the example they are equal 8-8,4 = 0,4, 9-8,4 =0,6, 8-8,4=0,4, 7-8,4 =1,4, 10-8,4=1,6 (taken modules results).
3
To find the relative error of each measurement, divide the absolute error to the actual (true) value. Then multiply the result by 100%, it's usually at a percentage measured by this value. In the example locate the relative error is therefore: δ1=0.4 a/8,4=0,048 (or 4.8%), δ2=0,6/8,4=of 0.071 (or 7.1 %), δ3=0.4 a/8,4=0,048 (or 4.8%), δ4=1,4/8,4=0,167 (or 16.7%), δ5=1,6/8,4=to 0.19 (or 19%).
4
In practice, for the most accurate display errors using the standard deviation. To find it, erected in the square of the absolute error measure and add together. Then divide this number by (N-1), where N is the number of measurements. Calculating the root of the result, you will get a standard deviation that characterizes the error of measurement.
5
To find the limit of the absolute error, find the minimum number obviously exceeding the absolute error or equal to it. In the example just select the highest value of 1.6. It is also sometimes necessary to find the maximum relative error, in this case, find the number that is greater than or equal to the relative error in the example it is equal to 19%.

# Advice 4 : How to calculate measurement error

Measurements of physical quantities are always accompanied by one or another error. It represents the deviation of measurement results from the true value of the measurand. You will need
• measuring device:
• calculator.
Instruction
1
Errors can occur as a result of various factors. Among them are the imperfection of the means or methods of measurement inaccuracies in their manufacture, failure to comply with the special conditions in conducting research.
2
There are several classifications of errors. On the submission form they can be absolute, relative and given. The former represents the difference between the calculated and the actual value. Expressed in units of the measured phenomenon and are according to the formula:∆x = hisl - Hist. The second is determined by the ratio of the absolute error to the magnitude of the true value of the indicator.The calculation formula is:δ = ∆x/Hist. Measured in percentages or fractions.
3
Reduced error of the measuring device is as the ratio of ∆x to the fiducial value XH. Depending on the type of device it is taken either equal to the limit of measurement, either related to their specific range.
4
On the conditions of occurrence distinguish between basic and advanced. If the measurements were carried out in normal conditions, there is the first kind. Deviation due to output values outside the normal, is optional. For its evaluation documentation is typically a set of rules within which may change the magnitude of a breach of the conditions of measurement.
5
The errors physical measurements are divided into systematic, random and rough. The first is caused by factors that act at multiple repetition of the measurements. The second arises from the influence of reason and random. The mistake is the result of observation, which differs sharply from all the others.
6
Depending on the nature of the measured quantity can be used various ways to measure error. The first of these is the method of Kornfeld. It is based on the calculation of the confidence interval in the range from the minimum to the maximum result. The error in this case will represent half the difference of these results: ∆x = (hmag-xmin)/2. Another way is to calculate mean square error.

# Advice 5 : How to calculate measurement error

The result of any measurement is inevitably accompanied by a deviation from the true value. To calculate the measurement error in several ways depending on its type, for example, statistical methods for determining the confidence interval, standard deviation, etc. Instruction
1
There are several reasons that give rise to error of measurement. This instrument inaccuracy, imperfection techniques, as well as errors caused by inattention of the operator performing the measurements. In addition, often the true value of the parameter take its actual value, which actually is merely the most probable, based on the analysis of statistical sample results a series of experiments.
2
Accuracy is a measure of the deviation of the measured parameter from its true value. According to the method of Kornfeld, determine the confidence interval, which ensures a certain degree of reliability. They find the so-called confidence limits, which varies the magnitude error is calculated as half the sum of these values:∆ = (xmax - xmin)/2.
3
This is interval estimation error, which is carried out in a small volume of the statistical sample. Point estimation is to calculate the mathematical expectation and standard deviation.
4
Mathematical expectation is an integral sum of a number of works by the two parameters of the observations. It actually measured value and its probability at these points:M = Σxi•pi.
5
The classic formula for calculating the standard deviation involves the calculation of the average values of the analyzed sequence of values of the measurand, and also takes into account the volume series of the experiments:σ = √(∑(xi – khsr)2/(n - 1)).
6
According to the method of expression also distinguish absolute, relative and reduced error. The absolute error is expressed in the same units as the measured value, and is equal to the difference between the estimated and true value is:∆x = x1 – x0.
7
The relative error measurements associated with the absolute, but is more efficient. It has no dimension, sometimes expressed in percentages. It is the ratio of absolute error to the true or calculated value of the measured parameters:σx = ∆x/x0 or σx = ∆x/x1.
8
Given the margin of error is expressed as the ratio between absolute error and some conventionally accepted value x, which is constant for all measurements and is determined by the calibration of the instrument scale. If the scale starts from zero (one-sided), this normalizing value equal to its upper limit, and if a double – width of its range:σ = ∆x/xn.

# Advice 6 : How to determine the measurement error

The deviation from the true value inevitably arises when constructing probabilistic models for some parameter. This concept is applied in order to determine the error of measurement, compare the results of a series of experiments with the aim of obtaining the true value. Instruction
1
There are two ways to estimate the error of measurement: interval and point. This is due to the degree of reliability that you want to set. The first method involves finding the confidence interval, which obviously will block actual value of the measured parameter or its mathematical expectation.
2
The confidence interval represents the interval of possible values, i.e. a subset of the elements of the sample. The boundaries of the interval are called the confidence limits are on certain formulas. For example, the mathematical expectation they will be equal to:khsr – t•σ/√N < M(x) < khsr + t•σ/√N where:khsr – the arithmetic mean of the sample;σ – standard deviation;M(x) is the mathematical expectation;N is the sample size;t is the parameter of the function Laplace.
3
In the above formulas are two kinds of point of error: the standard deviation and mathematical expectation. They represent some value that is a measure of the deviation of the calculated values of a random variable from its true value. This is different from interval estimation, which involves a whole range of possible errors. The degree of reliability falling within this range is determined by the Laplace function.
4
The standard deviation, in turn, is calculated by three methods, the most common of them – classic, using the sample mean:σ = √(∑(XI – khsr)2/(N - 1)), where XI are the elements of the sample.
5
Mathematical expectation is the value, around which are distributed the elements of a sample. I.e. is the average of the expected values that can take the random variable. To calculate this kind of variation, you need to make sets of samples and their probabilities array of works of their couples and make all of the elements of the array:M(x) = Σхi•pi.
6
To determine another point , the error of measurement, variance, need to take the square root of the standard deviation or use the following formula relative to the expected value:D = (x – M(x))2 = Σpi•(XI – M(x))2.
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