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You will need

• - several measurement results or another sample;
• - calculator.

Instructions

1

Measure at least 3-5 times to havethe ability to calculate the actual value of a parameter. Add the results and divide them by the number of measurements, you got a real value, which is used in tasks instead of true (it can not be determined). For example, if the measurements yield a result of 8, 9, 8, 7, 10, then the actual value will be (8 + 9 + 8 + 7 + 10) / 5 = 8.4.

2

Find the absolute error of each dimension. To do this, subtract the actual value from the measurement result, neglect the signs. You will get 5 absolute errors, one for each measurement. In the example, they will be 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 (the result modules are taken).

3

To know the relative error each dimension, divide the absolute error to the actual (true) value. Then multiply the result by 100%, usually in percent this value is measured. In the example, find the relative error in this way: δ1 = 0.4 / 8.4 = 0.048 (or 4.8%), δ2 = 0.6 / 8.4 = 0.071 (or 7.1%), δ3 = 0.4 / 8.4 = 0.048 (or 4.8%), δ4 = 1.4 / 8.4 = 0.167 (or 16.7%), δ5 = 1.6 / 8.4 = 0.19 (or 19%).

4

In practice, for the most accurate mappingThe errors use the standard deviation. To find it, square all the absolute measurement errors and add them together. Then divide this number by (N-1), where N is the number of measurements. Calculating the root of the obtained result, you will get the standard deviation, characterizing error measurements.

5

To find the absolute limit error, find the minimum number, certainly exceeding the absolute error or equal to it. In this example, simply select the largest value - 1.6. It is also sometimes necessary to find the relative relative error, in such a case, find a number greater than or equal to the relative error, in the example it is equal to 19%.

You will need

• - calculator.

Instructions

1

The errors in physical measurements are subdividedto systematic, random and gross. The former are caused by factors that act the same for repeated repetition of measurements. They are constant or regularly changing. They can be caused by improper installation of the instrument or imperfection of the selected measurement method.

2

The second arise from the influence of causes, and randomcharacter. These include incorrect rounding when calculating the readings and the effect of the environment. If such errors are much smaller than the scale division of this measuring instrument, then it is advisable to take half the division as the absolute error.

3

Slip or rough error is the result of observation, which is sharply different from all others.

4

Absolute error approximate numerical value is the differencebetween the result obtained during the measurement and the true value of the measured value. The true or actual value most accurately reflects the physical quantity being studied. This error is the simplest quantitative measureerrors. It can be calculated by the following formula: ΔХ = Hasl - HIST. It can take a positive and negative value. For a better understanding, consider an example. In the school there are 1205 students, with rounding up to 1200 absolute error is equal to: Δ = 1200 - 1205 = 5.

5

There are certain rules for calculating the error of quantities. First, the absolute error the sum of two independent quantities is equal to the sum of their absolute errors: Δ (X + Y) = ΔX + ΔY. A similar approach is applicable for the difference of two errors. One can use the formula: Δ (X-Y) = ΔX + ΔY.

6

The correction is an absolute error, taken with the opposite sign: Δп = -Δ. It is used to exclude a systematic error.

Instructions

1

There are two ways to calculate the error measurements: interval and point. This is due to the degree of reliability that needs to be set. The first method involves finding a confidence interval, which will certainly block the actual value of the measured parameter or its mathematical expectation.

2

The confidence interval isthe range of possible values, i.e. subset of sample elements. The boundaries of the interval are called the confidence limits and are found by certain formulas. For example, for a mathematical expectation, they will be equal: xcp - t • σ / √N <M (x) <xcp + t • σ / √N, where: xcp is the arithmetic mean of the sample elements, σ ​​is the standard deviation, M (x) - mathematical expectation, N - sample size, t - parameter of the Laplace function.

3

In the above formulas there are two typespoint error: root mean square deviation and mathematical expectation. They represent a certain value, which is a measure of the deviation of the calculated value of a random variable from its true value. This is in contrast to interval estimation, which involves a whole range of possible errors. The degree of reliability of falling into this range is determined by the Laplace function.

4

The root-mean-square deviation, in turn,is calculated by three methods, the most common of which is the classical one using the sample mean: σ = √ (Σ (xi-xsr) ² / (N-1)), where xi are the sample elements.

5

A mathematical expectation is a value,around which the elements of the sample are distributed. Those. this is the average of the expected values ​​that a random variable can take. To calculate this type of deviation, we need to compile an array of products of their pairs from the sample sets and their probabilities and add all the elements of the array: M (x) = Σxi • pi.

6

To define one more point error measurements, the variance, you need to extract the square root of the root-mean-square deviation or use the following formula for the mathematical expectation: D = (x - M (x)) ² = Σpi • (xi - M (x)) ².