Error and accuracy of approximation. Finding absolute and relative error

A) Absolute?

B) Relative?

A) The absolute error of the approximation is the magnitude of the difference between the true value of a quantity and its approximate value. |x - x_n|, where x is the true value, x_n is the approximate value. For example: The length of a sheet of A4 paper is (29.7 ± 0.1) cm. And the distance from St. Petersburg to Moscow is (650 ± 1) km. The absolute error in the first case does not exceed one millimeter, and in the second - one kilometer. The question is to compare the accuracy of these measurements.

If you think that the length of the sheet is measured more accurately because the absolute error does not exceed 1 mm. Then you are wrong. These values ​​cannot be directly compared. Let's do some reasoning.

When measuring sheet length absolute error does not exceed 0.1 cm by 29.7 cm, that is, as a percentage it is 0.1/29.7 * 100% = 0.33% of the measured value.

When we measure the distance from St. Petersburg to Moscow, the absolute error does not exceed 1 km per 650 km, which as a percentage is 1/650 * 100% = 0.15% of the measured value. We see that the distance between cities is measured more accurately than the length of an A4 sheet.

B) The relative approximation error is the ratio of the absolute error to the absolute value of the approximate value of a quantity.

mathematical error fraction

where x is the true value, x_n is the approximate value.

Relative error is usually expressed as a percentage.

Example. Rounding the number 24.3 to units gives the number 24.

The relative error is equal. They say that the relative error in this case is 12.5%.

5) What kind of rounding is called rounding?

A) With a disadvantage?

B) In excess?

A) Rounding down

When rounding a number expressed as a decimal fraction to the nearest 10^(-n), the first n decimal places are retained and the subsequent ones are discarded.

For example, rounding 12.4587 to the nearest thousandth, we get 12.458.

B) Excessive rounding

When rounding a number expressed as a decimal fraction to the nearest 10^(-n), the first n decimal places are retained in excess, and the subsequent ones are discarded.

For example, rounding 12.4587 to the nearest thousandth, we get 12.459.

6) The rule for rounding decimal fractions.

Rule. To round decimal to a certain digit of the integer or fractional part, all smaller digits are replaced by zeros or discarded, and the digit preceding the digit discarded during rounding does not change its value if it is followed by the numbers 0, 1, 2, 3, 4, and is increased by 1 (one) , if the numbers are 5, 6, 7, 8, 9.

Example. Round the fraction 93.70584 to:

ten thousandths: 93.7058

thousandths: 93.706

hundredths: 93.71

tenths: 93.7

whole number: 94

tens: 90

Conclusion

Despite the equality of absolute errors, because the measured quantities are different. The larger the measured size, the smaller the relative error while the absolute error remains constant.

The absolute calculation error is found by the formula:

The modulus sign shows that we do not care which value is greater and which is less. Important, how far the approximate result deviated from the exact value in one direction or another.

The relative error of calculations is found by the formula:
, or the same thing:

The relative error shows by what percentage the approximate result deviated from the exact value. There is a version of the formula without multiplying by 100%, but in practice I almost always see the above version with percentages.

After a short reference, let's return to our problem, in which we calculated the approximate value of the function using a differential.

Let's calculate the exact value of the function using a microcalculator:
, strictly speaking, the value is still approximate, but we will consider it accurate. Such problems do occur.

Let's calculate the absolute error:

Let's calculate the relative error:
, thousandths of a percent were obtained, so the differential provided just an excellent approximation.

Answer: , absolute calculation error, relative calculation error

The following example for an independent solution:

Example 4

at point . Calculate a more accurate value of the function at a given point, estimate the absolute and relative error of calculations.

An approximate sample of the final design and an answer at the end of the lesson.

Many people have noticed that roots appear in all the examples considered. This is not accidental; in most cases, functions with roots are actually proposed in the problem under consideration.

But for suffering readers, I dug up a small example with arcsine:

Example 5

Calculate approximately the value of a function using a differential at the point

This short but informative example is also for you to solve on your own. And I rested a little so that with renewed vigor I could consider the special task:

Example 6

Calculate approximately using differential, round the result to two decimal places.

Solution: What's new in the task? The condition requires rounding the result to two decimal places. But that’s not the point; I think the school rounding problem is not difficult for you. The fact is that we are given a tangent with an argument, which is expressed in degrees. What should you do when you are asked to solve a trigonometric function with degrees? For example , etc.

The solution algorithm is fundamentally the same, that is, it is necessary, as in previous examples, to apply the formula

Let's write an obvious function

The value must be presented in the form . Will provide serious assistance table of values ​​of trigonometric functions . By the way, for those who have not printed it out, I recommend doing so, since you will have to look there throughout the entire course of studying higher mathematics.

Analyzing the table, we notice a “good” tangent value, which is close to 47 degrees:

Thus:

After preliminary analysis degrees must be converted to radians. Yes, and only this way!

IN in this example directly from the trigonometric table you can find out that. Using the formula for converting degrees to radians: (formulas can be found in the same table).

What follows is formulaic:

Thus: (we use the value for calculations). The result, as required by condition, is rounded to two decimal places.

Answer:

Example 7

Calculate approximately using a differential, round the result to three decimal places.

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

As you can see, there is nothing complicated, we convert degrees to radians and adhere to the usual solution algorithm.

Approximate calculations using the total differential of a function of two variables

Everything will be very, very similar, so if you came to this page specifically for this task, then first I recommend looking at at least a couple of examples of the previous paragraph.

To study a paragraph you must be able to find second order partial derivatives , where would we be without them? In the above lesson, I denoted a function of two variables using the letter . In relation to the task under consideration, it is more convenient to use the equivalent notation.

As in the case of a function of one variable, the condition of the problem can be formulated in different ways, and I will try to consider all the formulations encountered.

Example 8

Solution: No matter how the condition is written, in the solution itself to denote the function, I repeat, it is better to use not the letter “zet”, but .

And here is the working formula:

In fact, before us elder sister formulas of the previous paragraph. The variable has only increased. What can I say, myself the solution algorithm will be fundamentally the same!

According to the condition, it is required to find the approximate value of the function at the point.

Let's represent the number 3.04 as . The bun itself asks to be eaten:
,

Let's represent the number 3.95 as . The turn has come to the second half of Kolobok:
,

And don’t look at all the fox’s tricks, there is a Kolobok - you have to eat it.

Let's calculate the value of the function at the point:

We find the differential of a function at a point using the formula:

From the formula it follows that we need to find partial derivatives first order and calculate their values ​​at point .

Let's calculate the first order partial derivatives at the point:

Total differential at point:

Thus, according to the formula, the approximate value of the function at the point:

Let's calculate the exact value of the function at the point:

This value is absolutely accurate.

Errors are calculated using standard formulas, which have already been discussed in this article.

Absolute error:

Relative error:

Answer: , absolute error: , relative error:

Example 9

Calculate the approximate value of a function at a point using a total differential, estimate the absolute and relative error.

This is an example for you to solve on your own. Anyone who takes a closer look at this example will notice that the calculation errors turned out to be very, very noticeable. This happened for the following reason: in the proposed problem the increments of arguments are quite large: .

General pattern that's how it is a - the larger these increments in absolute value, the lower the accuracy of the calculations. So, for example, for a similar point the increments will be small: , and the accuracy of the approximate calculations will be very high.

This feature is also true for the case of a function of one variable (the first part of the lesson).

Example 10

Solution: Let's calculate this expression approximately using the total differential of a function of two variables:

The difference from Examples 8-9 is that we first need to construct a function of two variables: . I think everyone understands intuitively how the function is composed.

The value 4.9973 is close to “five”, therefore: , .
The value 0.9919 is close to “one”, therefore, we assume: , .

Let's calculate the value of the function at the point:

We find the differential at a point using the formula:

To do this, we calculate the first order partial derivatives at the point.

The derivatives here are not the simplest, and you should be careful:

;

.

Total differential at point:

Thus, the approximate value of this expression is:

Let's calculate a more accurate value using a microcalculator: 2.998899527

Let's find the relative calculation error:

Answer: ,

Just an illustration of the above, in the problem considered, the increments of arguments are very small, and the error turned out to be fantastically tiny.

Example 11

Using the complete differential of a function of two variables, calculate approximately the value of this expression. Calculate the same expression using a microcalculator. Estimate the relative calculation error as a percentage.

This is an example for you to solve on your own. An approximate sample of the final design at the end of the lesson.

As already noted, the most common guest in this type of task is some kind of roots. But from time to time there are other functions. And a final simple example for relaxation:

Example 12

Using the total differential of a function of two variables, calculate approximately the value of the function if

The solution is closer to the bottom of the page. Once again, pay attention to the wording of the lesson tasks, in various examples in practice, the formulations may be different, but this does not fundamentally change the essence and algorithm of the solution.

To be honest, I was a little tired because the material was a bit boring. It was not pedagogical to say this at the beginning of the article, but now it’s already possible =) Indeed, the tasks computational mathematics usually not very complex, not very interesting, the most important thing, perhaps, is not to make a mistake in ordinary calculations.

May the keys of your calculator not be erased!

Solutions and answers:

Example 2:

Solution: We use the formula:
In this case: , ,

Thus:

Answer:

Example 4:

Solution: We use the formula:
In this case: , ,

Thus:

Let's calculate a more accurate value of the function using a microcalculator:

Absolute error:

Relative error:

Answer: , absolute calculation error, relative calculation error

Example 5:

Solution: We use the formula:

In this case: , ,

Thus:

Answer:

Example 7:

Solution: We use the formula:
In this case: , ,

Instructions

First of all, take several measurements with an instrument of the same value in order to be able to obtain the actual value. The more measurements are taken, the more accurate the result will be. For example, weigh on an electronic scale. Let's say you got results of 0.106, 0.111, 0.098 kg.

Now calculate the real value of the quantity (real, since the true value cannot be found). To do this, add up the results obtained and divide them by the number of measurements, that is, find the arithmetic mean. In the example, the actual value would be (0.106+0.111+0.098)/3=0.105.

Sources:

• how to find measurement error

An integral part of any measurement is some error. It represents a qualitative characteristic of the accuracy of the research. According to the form of presentation, it can be absolute and relative.

You will need

• - calculator.

Instructions

The second arise from the influence of causes, and random nature. These include incorrect rounding when calculating readings and influence. If such errors are significantly less than the scale divisions of this measuring device, then it is advisable to take half the division as the absolute error.

Miss or Rough error represents an observational result that differs sharply from all others.

Absolute error approximate numerical value– this is the difference between the result during measurement and the true value of the measured value. The true or actual value reflects the physical quantity being studied. This error is the simplest quantitative measure of error. It can be calculated using the following formula: ∆Х = Hisl - Hist. She can accept the positive and negative meaning. For a better understanding, let's look at . The school has 1205 students, rounded to 1200 absolute error equals: ∆ = 1200 - 1205 = 5.

There are certain calculations of the error values. First of all, absolute error the sum of two independent quantities is equal to the sum of their absolute errors: ∆(X+Y) = ∆X+∆Y. A similar approach is applicable for the difference between two errors. You can use the formula: ∆(X-Y) = ∆X+∆Y.

Sources:

• how to determine absolute error

Measurements physical quantities are always accompanied by one or another error. It represents the deviation of the measurement results from the true value of the measured value.

You will need

• -measuring device:
• -calculator.

Instructions

Errors may result from influence various factors. Among them are the imperfection of measurement tools or methods, inaccuracies in their manufacture, and failure to comply with special conditions when conducting research.

There are several classifications. According to the form of presentation, they can be absolute, relative and reduced. The first represent the difference between the calculated and actual value of a quantity. They are expressed in units of the measured phenomenon and are found by the formula: ∆x = hisl-hist. The second ones are determined by the ratio of absolute errors to the value of the true value of the indicator. The calculation formula is: δ = ∆x/hist. It is measured in percentages or shares.

Reduced error measuring instrument is found as the ratio of ∆x to the normalizing value xn. Depending on the type of device, it is taken either equal to the measurement limit or assigned to a certain range.

According to the conditions of occurrence, they distinguish between basic and additional. If measurements were carried out in normal conditions, then the first type appears. Deviations caused by values ​​going beyond normal limits are additional. To evaluate it, the documentation usually establishes standards within which the value can change if the measurement conditions are violated.

Also errors physical measurements are divided into systematic, random and rough. The first are caused by factors that act when measurements are repeated many times. The second arise from the influence of reasons and character. A miss is an observation that differs sharply from all others.

Depending on the nature of the measured value, they can be used various ways measurement error. The first of them is the Kornfeld method. It is based on calculating a confidence interval ranging from the minimum to the maximum result. The error in this case will be half the difference between these results: ∆x = (xmax-xmin)/2. Another method is to calculate the mean square error.

Measurements can be carried out with to varying degrees accuracy. At the same time, even precision instruments are not absolutely accurate. Absolute and relative errors may be small, but in reality they are almost always present. The difference between approximate and exact values a certain quantity is called absolute error. In this case, the deviation can be both larger and smaller.

You will need

• - measurement data;
• - calculator.

Instructions

Before calculating the absolute error, take several postulates as initial data. Eliminate gross errors. Assume that the necessary corrections have already been calculated and applied to the result. Such an amendment may be a transfer of the original measurement point.

Take as a starting point that random errors are taken into account. This implies that they are less than systematic, that is, absolute and relative, characteristic of this particular device.

Random errors affect the results of even highly accurate measurements. Therefore, any result will be more or less close to the absolute, but there will always be discrepancies. Determine this interval. It can be expressed by the formula (Xizm- ΔХ)≤Xism ≤ (Xism+ΔХ).

Determine the value that is closest to the value. In measurements, the arithmetic is taken, which can be obtained from the formula in the figure. Accept the result as the true value. In many cases, the reading of the reference instrument is accepted as accurate.

Knowing the true value, you can find the absolute error, which must be taken into account in all subsequent measurements. Find the value of X1 - the data of a specific measurement. Determine the difference ΔХ by subtracting the smaller from the larger. When determining the error, only the modulus of this difference is taken into account.

note

As a rule, in practice it is not possible to carry out absolutely accurate measurements. Therefore, the maximum error is taken as the reference value. It represents the maximum value of the absolute error module.

Helpful advice

In practical measurements, half of the smallest division value is usually taken as the absolute error. When working with numbers, the absolute error is taken to be half the value of the digit, which is in the digit next to the exact digits.

To determine the accuracy class of an instrument, the ratio of the absolute error to the measurement result or to the scale length is more important.

Measurement errors are associated with imperfection of instruments, tools, and techniques. Accuracy also depends on the attentiveness and state of the experimenter. Errors are divided into absolute, relative and reduced.

Instructions

Let a single measurement of a quantity give the result x. The true value is denoted by x0. Then absolute errorΔx=|x-x0|. She evaluates absolute. Absolute error consists of three components: random errors, systematic errors and misses. Usually, when measuring with an instrument, half the division value is taken as an error. For a millimeter ruler this would be 0.5 mm.

The true value of the measured quantity in the interval (x-Δx ; x+Δx). In short, this is written as x0=x±Δx. It is important to measure x and Δx in the same units and write in the same format, e.g. whole part and three commas. So, absolute error gives the boundaries of the interval in which the true value is located with some probability.

Relative error the ratio of the absolute error to the actual value of the quantity: ε(x)=Δx/x0. This is a dimensionless quantity and can also be written as a percentage.

Direct and indirect measurements. In direct measurements, the desired value is immediately measured with the appropriate device. For example, bodies with a ruler, voltage with a voltmeter. In indirect measurements, a value is found using the formula for the relationship between it and the measured values.

If the result is a dependence on three directly measured quantities having errors Δx1, Δx2, Δx3, then error indirect measurement ΔF=√[(Δx1 ∂F/∂x1)²+(Δx2 ∂F/∂x2)²+(Δx3 ∂F/∂x3)²]. Here ∂F/∂x(i) are the partial derivatives of the function for each of the directly measured quantities.

Helpful advice

Errors are gross inaccuracies in measurements that occur due to malfunction of instruments, inattentiveness of the experimenter, or violation of the experimental methodology. To reduce the likelihood of such mistakes, be careful when taking measurements and describe the results obtained in detail.

Sources:

• Guidelines for laboratory work in physics
• how to find relative error

The result of any measurement is inevitably accompanied by a deviation from the true value. The measurement error can be calculated in several ways depending on its type, for example, by statistical methods of determining the confidence interval, standard deviation, etc.

You will need sugar per month. Sometimes blood samples are taken for analysis multiple times throughout the day, sometimes 1-2 times a week is enough. Self-monitoring is especially necessary for patients with type 1 diabetes.

Permissible error for a glucometer according to international standards

The glucometer is not considered a high-precision device. It is intended only as an approximate determination of blood sugar concentration.

The permissible error of a glucometer according to international standards is 20% for glycemia of more than 4.2 mmol/l.

For example, if during self-monitoring the sugar level is 5 mmol/l, then the real concentration value is in the range from 4 to 6 mmol/l.

The permissible error of a standard glucometer is measured in , not in mmol/l. The higher the indicators, the greater the error in absolute numbers. For example, if it reaches about 10 mmol/l, then the error does not exceed 2 mmol/l, and if sugar is about 20 mmol/l, then the result laboratory measurement can be up to 4 mmol/l.

In most cases, the glucometer overestimates blood glucose readings.

The standards allow exceeding the stated measurement error in 5% of cases. This means that every twentieth study can significantly distort the results.

Permissible error for glucometers from different companies

Glucometers are subject to mandatory certification. The documents accompanying the device usually indicate the permissible measurement error. If this item is not in the instructions, then the error corresponds to 20%.

Some manufacturers pay Special attention measurement accuracy. There are devices from European companies that have an acceptable error of less than 20%. The best figure today is 10-15%.

Error in the glucometer during self-monitoring

The permissible measurement error characterizes the operation of the device. Several other factors also affect the accuracy of the study. Incorrectly prepared skin, too small or large volume of blood drop received, unacceptable temperature regime- all this can lead to errors.

Only if all the rules of self-control are followed can one count on the stated permissible error of the study.

Rules for self-monitoring using a glucometer can be obtained from your doctor.

The accuracy of the meter can be checked at service center. Manufacturers' warranties provide free consultation and troubleshooting.

Introduction. Measurement and measurement accuracy If we need
measure any
the size we use
special
measuring instruments:

Strokes

Strokes

Scale division

For comparison:

The device is smaller than what is measured
quantities
The device is larger than what is measured
quantities

10.

Measurement error
is allowed in any case.
If it seems that the meaning
matches perfectly
with a stroke on the ruler, then
there is an error,
since the assessment by eye is not
can be perfectly accurate.

11.

Measurement error
equal to half the price
scale divisions
measuring instrument

12.

1.
3.
2. Water thermometer
1
2
3

13. Absolute error

Absolute error
or, in short, error
approximate number
called the difference between
this number and its exact
value (from a larger number
the lesser is subtracted)*.
Example 1. At an enterprise
1284 workers and employees. At
rounding this number to
1300 absolute error
is 1300 - 1284 = 16.
When rounded to 1280
absolute error
is 1284 - 1280 = 4.

14. Relative error

Relative error
approximate number
called relation
absolute error
approximate number to
this number itself.
Example 2. At school 197
students. Let's round it up
number up to 200. Absolute
the error is 200 197 = 3. Relative
the error is 3/197 or,
rounded, 3/197 = 1.5%.

15.

In most cases it is impossible to know the exact value
approximate number, and therefore the exact magnitude of the error. However
it is almost always possible to establish that the error (absolute or
relative) does not exceed a certain number.
Example 3. A seller weighs a watermelon on a scale. Set of weights
the smallest is 50 g. Weighing gave 3600 g. This number is approximate.
The exact weight of the watermelon is unknown. But the absolute error does not exceed
50 g. The relative error does not exceed 50/3600 ≈ 1.4%.
A number that obviously exceeds the absolute error (or at worst
case equal to it) is called the maximum absolute error.
A number that obviously exceeds the relative error (or at worst
case equal to it) is called the maximum relative error.
In example 3, the maximum absolute error can be taken as 50 g, and
for the maximum relative error - 1.4%.

16.

The magnitude of the maximum error is not entirely certain. So, in
example 3 can be taken as the maximum absolute error of 100 g, 150 g and
in general, any number greater than 50 g. In practice, it is taken whenever possible
smaller value of the maximum error. In cases where the exact
the magnitude of the error, this value simultaneously serves as the limit
error. For each approximate number, its
maximum error (absolute or relative). When she doesn't really
indicated, it is understood that the maximum absolute error is
half a unit of the last discharged digit. So, if given
an approximate number of 4.78 without indicating the maximum error, then
it is assumed that the maximum absolute error is 0.005.
As a result of this agreement, it is always possible to dispense with specifying the limit
number errors.
The maximum absolute error is designated Greek letterΔ (“delta”);
maximum relative error - the Greek letter δ (“small delta”).
If the approximate number is denoted by the letter a, then
δ = Δ/a.
Example 4. The length of a pencil is measured with a ruler with millimeter divisions.
The measurement showed 17.9 cm. What is the maximum relative error of this
measurements?
Here a = 17.9 cm; we can take Δ = 0.1 cm, since we can measure with an accuracy of 1 mm
pencil is not difficult, but can be significantly reduced, the maximum error cannot be
(with skill you can read 0.02 or even 0.01 cm on a good ruler, but
pencil edges can vary by a large amount). Relative
the error is 0.1/17.9. Rounding, we find δ = 0.1/18 ≈ 0.6%.