In practical activities, a person has to measure various quantities, take into account materials and products of labor, produce various calculations. The results of various measurements, counts and calculations are numbers. The numbers obtained as a result of the measurement, only approximately, with a certain degree of accuracy, characterize the desired values. Accurate measurements are not possible due to inaccuracies measuring instruments, the imperfections of our organs of vision, and the measured objects themselves sometimes do not allow us to determine their magnitude with any accuracy.

So, for example, it is known that the length of the Suez Canal is 160 km, the distance along railway from Moscow to Leningrad 651 km. Here we have the results of measurements made with an accuracy of up to a kilometer. If, for example, the length rectangular area 29 m, width 12 m, then, probably, the measurements were made with an accuracy of a meter, and fractions of a meter were neglected,

Before making any measurement, it is necessary to decide with what accuracy it needs to be performed, i.e. which fractions of the unit of measurement should be taken into account, and which should be neglected.

If there is some value a, the true value of which is unknown, and the approximate value (approximation) of this value is equal to X, they write a x.

With different measurements of the same quantity, we will obtain different approximations. Each of these approximations will differ from the true value of the measured value, equal, for example, a, by some amount, which we will call error. Definition. If the number x is an approximate value (approximation) of some quantity, the true value of which is equal to the number a, then the modulus of the difference of numbers, a and X called absolute error given approximation and denoted a x: or simply a. Thus, by definition,

a x = a-x (1)

From this definition it follows that

a = x a x (2)

If it is known what quantity we are talking about, then in the notation a x index a is omitted and equality (2) is written as follows:

a = x x (3)

Since the true value of the desired value is most often unknown, it is impossible to find the absolute error in the approximation of this value. You can only indicate in each specific case a positive number, greater than which this absolute error it can not be. This number is called the limit of the absolute error of the approximation of the quantity a and denoted h a. Thus, if x is an arbitrary approximation of the value a for a given procedure for obtaining approximations, then

a x = a-x h a (4)

It follows from the above that if h a is the boundary of the absolute error of the approximation of the quantity a, then any number greater than h a, will also be the boundary of the absolute error of the approximation of the quantity a.

In practice, it is customary to choose the smallest number that satisfies inequality (4) as the limit of the absolute error.

Solving the inequality a-x h a we get that a contained within the boundaries

x-h a a x + h a (5)

A more rigorous concept of the absolute error boundary can be given as follows.

Let be X- many possible approximations X quantities a for a given procedure for obtaining an approximation. Then any number h, satisfying the condition a-x h a for any xX, is called the boundary of the absolute error of approximations from the set X. Denote by h a smallest known number h. This number h a and is chosen in practice as the boundary of the absolute error.

The absolute approximation error does not characterize the quality of measurements. Indeed, if we measure any length with an accuracy of 1 cm, then in the case when we are talking about determining the length of a pencil, it will be poor accuracy. If, with an accuracy of 1 cm, determine the length or width of the volleyball court, then this will be a high accuracy.

To characterize the measurement accuracy, the concept of relative error is introduced.

Definition. If a a x: there is an absolute approximation error X some quantity, the true value of which is equal to the number a, then the ratio a x to the modulus of a number X is called the relative error of approximation and is denoted a x or x.

Thus, by definition,

The relative error is usually expressed as a percentage.

Unlike absolute error, which is most often a dimensional quantity, relative error is a dimensionless quantity.

In practice, it is not the relative error that is considered, but the so-called relative error limit: such a number E a, which cannot be greater than the relative error of the approximation of the desired value.

Thus, a x E a .

If a h a-- limit of the absolute error of approximations of the quantity a, then a x h a and hence

Obviously, any number E, satisfying the condition, will be the boundary of the relative error. In practice, some approximation is usually known X quantities a and the absolute error limit. Then the number

1. Numbers are exact and approximate. The numbers we encounter in practice are of two kinds. Some give the true value of the quantity, others only approximate. The first is called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact one, especially since in many cases exact number generally impossible to find.

The results of operations with numbers give: with approximate numbers approximate numbers. For example. During the epidemic, 60% of St. Petersburg residents get the flu. This is approximately 3 million people. with exact numbers exact numbers Eg. There are 65 people in the audience at a lecture on mathematics. approximate numbers Eg. Average body temperature of the patient during the day 37.3: morning: 37.2; day: 36.8 ; evening38.

The theory of approximate calculations allows: 1) knowing the degree of accuracy of the data, to assess the degree of accuracy of the results; 2) take data with an appropriate degree of accuracy, sufficient to ensure the required accuracy of the result; 3) rationalize the calculation process, freeing it from those calculations that will not affect the accuracy of the result.

1) if the first (left) of the discarded digits is less than 5, then the last remaining digit is not changed (rounding down); 2) if the first discarded digit is greater than 5 or equal to 5, then the last remaining digit is increased by one (rounding up). Rounding: a) to tenths 12.34 12.3; b) up to hundredths 3.2465 3.25; 1038.79. c) up to thousandths 3.4335 3.434. d) up to thousands; This takes into account the following:

The quantities most commonly measured in medicine: mass m, length l, process speed v, time t, temperature t, volume V, etc. To measure a physical quantity means to compare it with a homogeneous quantity taken as a unit. 9 Units of measurement of physical quantities: Basic Length - 1 m - (meter) Time - 1 s - (second) Mass - 1 kg - (kilogram) Prod ucts Volume - 1 m³ - (cubic meter) Velocity - 1 m/s - (meter per second)

Prefixes to the names of units: Multiple prefixes - increase by 10, 100, 1000, etc. times g - hecto (×100) k - kilo (× 1000) M - mega (×) 1 km (kilometer) 1 kg (kilogram) 1 km = 1000 m = 10³ m 1 kg = 1000 g = 10³ g decrease by 10, 100, 1000, etc. times d - deci (×0.1) s - centi (× 0.01) m - milli (× 0.001) 1 dm (decimeter) 1dm = 0.1 m 1 cm (centimeter) 1cm = 0.01 m 1 mm (millimeter) 1mm = 0.001 m

For the diagnosis, treatment, prevention of diseases in medicine, various measuring medical equipment is used.

Thermometer. First, you need to take into account the upper and lower limits of measurement. The lower limit is the minimum and the upper limit is the maximum measurable value. If the expected value of the measured value is unknown, it is better to take the device with a "margin". For example, temperature measurement hot water do not carry out with a street or room thermometer. It is better to find a device with an upper limit of 100 ° C. Secondly, you need to understand how accurately the quantity should be measured. Since the measurement error depends on the division value, for more accurate measurements the instrument with the smallest scale interval is selected.

Measurement errors. To measure various diagnostic parameters, you need your own device. For example, length is measured with a ruler, and temperature with a thermometer. But rulers, thermometers, tonometers and other devices are different, so in order to measure any physical quantity, you need to choose a device that is suitable for this measurement.

The price of division of the device. The temperature of the human body must be determined accurately, drugs should be administered in a strictly defined amount, therefore the price of divisions of the scale of the measuring device is an important characteristic of each device. The rule for calculating the price division of the device. To calculate the price of divisions of the scale, you need to: a) select the two nearest digitized strokes on the scale; b) count the number of divisions between them; c) Divide the difference in values ​​around the selected strokes by the number of divisions.

The price of division of the device. Division value (50-30)/4=5 (ml) Division value: (40-20)/10=2 km/h, (20-10)/10= 1gm, (39-19)/10=2 LITR , (8-4)/10=0.4 psi, (90-50)/10= 4 temp, (4-2)/10=0.2 s

Determine the price of division of devices: 16

Absolute measurement error. Errors are bound to occur in any measurement. These errors are due to various factors. All factors can be divided into three parts: errors caused by the imperfection of instruments; errors caused by the imperfection of measurement methods; errors due to the influence of random factors that cannot be eliminated. When measuring any value, one wants to know not only its value, but also how much this value can be trusted, how accurate it is. To do this, it is necessary to know how much the true value of a quantity can differ from the measured one. For these purposes, the concept of absolute and relative errors is introduced.

Absolute and relative errors. The absolute error shows how much the real value physical quantity different from the measured one. It depends on the device itself (instrumental error) and on the measurement process (reading error on the scale). The instrumental error must be indicated in the instrument's passport (as a rule, it is equal to the scale division of the instrument). The reading error is usually taken equal to half the division value. The absolute error of an approximate value is the difference Δ x \u003d | x - x 0 |, where x 0 is an approximate value, and x is the exact value of the measured value, or sometimes instead of x they use A ΔA \u003d | A - A 0 |.

Absolute and relative errors. Example. It is known that -0.333 is an approximate value for -1/3. Then by definition of absolute error Δ x= |x – x 0 |= | -1/3+0.333 | = | -1/3+33/1000 | = | -1/300 | = 1/300. In many practically important cases, it is impossible to find the absolute error of the approximation due to the fact that the exact value of the quantity is unknown. However, you can specify a positive number, more than which this absolute error cannot be. This is any number h that satisfies the inequality | ∆x | h It is called the absolute error limit.

In this case, they say that the value of x is approximately up to h equal to x 0. x \u003d x 0 ± h or x 0 - h x x 0 + h

Absolute instrumental errors of measuring instruments

Estimation of instrumental errors of measured quantities. For most measuring instruments, the error of the instrument is equal to its scale division. The exception is digital instruments and dial gauges. For digital devices, the error is indicated in their passport and is usually several times higher than the scale division of the device. For pointer measuring instruments, the error is determined by their accuracy class, which is indicated on the scale of the instrument, and the measurement limit. The accuracy class is indicated on the scale of the device as a number that is not surrounded by any frames. For example, in the figure shown, the accuracy class of the pressure gauge is 1.5. The accuracy class shows how many percent the error of the device is from the limit of its measurements. For a pointer pressure gauge, the measurement limit is 3 atm, respectively, the pressure measurement error is 1.5% of 3 atm, that is, 0.045 atm. It should be noted that for most pointer devices, their error turns out to be equal to the division value of the device. As in our example, where the division price of the barometer is 0.05 atm.

Absolute and relative errors. The absolute error is needed to determine the range in which the true value can fall, but for assessing the accuracy of the result as a whole, it is not very indicative. After all, measuring a length of 10 m with an error of 1 mm is certainly very accurate, at the same time, measuring a length of 2 mm with an error of 1 mm is obviously extremely inaccurate. The absolute measurement error is usually rounded up to one significant figure ΔA 0.17 0.2. The numerical value of the measurement result is rounded so that its last digit is in the same digit as the error figure A=10.332 10.3

Absolute and relative errors. Along with the absolute error, it is customary to consider the relative error, which is equal to the ratio of the absolute error to the value of the quantity itself. The relative error of an approximate number is the ratio of the absolute error of an approximate number to this number itself: E = Δx. 100% x 0 The relative error shows how many percent of the value itself an error could occur and is indicative when assessing the quality of the experimental results.

Example. When measuring the length and diameter of the capillary, l = (10.0 ± 0.1) cm, d = (2.5 ± 0.1) mm were obtained. Which of these measurements is more accurate? When measuring the length of the capillary, an absolute error of 10mm per 100mm is allowed, therefore the absolute error is 10/100=0.1=10%. When measuring the capillary diameter, the permissible absolute error is 0.1/2.5=0.04=4% Therefore, the measurement of the capillary diameter is more accurate.

In many cases, no absolute error can be found. Hence the relative error. But you can find the limit of the relative error. Any number δ satisfying the inequality | ∆x | / | x o | δ, is the limit of the relative error. In particular, if h is the absolute error limit, then the number δ= h/| x o |, is the boundary of the relative error of the approximation x o. From here. Knowing the border rel.p-i. δ, one can find the limit of the absolute error h. h=δ | x o |

Example. It is known that 2=1.41… Find the relative accuracy of the approximate equality or the limit of the relative error of the approximate equality 2 1.41. Here x \u003d 2, x o \u003d 1.41, Δ x \u003d 2-1.41. Obviously 0 Δ x 1.42-1.41=0.01 Δ x/ x o 0.01/1.41=1/141, Absolute error limit is 0.01, relative error limit is 1/141

Example. When reading the reading from the scale, it is important that your gaze falls perpendicular to the scale of the instrument, while the error will be less. To determine the thermometer reading: 1. determine the number of divisions, 2. multiply them by the division price 3. take into account the error 4. write down the final result. t = 20 °C ± 1.5 °C This means that the temperature is between 18.5° and 21.5°. That is, it can be, for example, 19, and 20 and 21 degrees Celsius. To increase the accuracy of measurements, it is customary to repeat them at least three times and calculate the average value of the measured value

N A C O R D E N I A A N E D E N G O N I N I O N I Measurement results C 1 \u003d 34.5 C 2 \u003d 33.8 C 3 \u003d 33.9 C 4 \u003d 33 .5 C 5 \u003d 54.2 a) Let's find the average value of four quantities with cp \u003d (c 1 + c 2 + c 3 + c 4): 4 c cf \u003d (34.5 + 33.8 + 33.9 + 33 ,5):4 = 33.925 33.9 b) Find the deviation of the value from the average value Δс = | c-cp | ∆c 1 = | c 1 – c cp | = | 34.5 – 33.9 | = 0.6 ∆c 2 = | c 2 – c cp | = | 33.8 – 33.9 | = 0.1 ∆c 3 = | c 3 – c cp | = | 33.9 – 33.9 | = 0 ∆c 4 = | c 4 – c cp | = | 33.5 – 33.9 | = 0.4

C) Find the absolute error Δc \u003d (c 1 + c 2 + c 3 + c 4): 4 Δc \u003d (0.6 + 0.4): 4 \u003d 0.275 0.3 g) Find the relative error δ \u003d Δc: s SR δ = (0.3: 33.9) 100% = 0.9% e) Write down the final answer c = 33.9 ± 0.3 δ = 0.9%

HOMEWORK Prepare for practical lesson based on the lecture. Perform a task. Find the mean value and error: a 1 = 3.685 a 2 = 3.247 a 3 = 3.410 a 4 = 3.309 a 5 = 3.392. Create presentations on the topics: “Rounding of values ​​in medicine”, “Measurement errors”, “Medical measuring equipment”

Introduction

Absolute error- is an estimate of the absolute measurement error. Computed different ways. The calculation method is determined by the distribution of the random variable. Accordingly, the magnitude of the absolute error, depending on the distribution of the random variable, may be different. If is the measured value and is the true value, then the inequality must hold with some probability close to 1. If random value distributed according to the normal law, then usually its standard deviation is taken as the absolute error. Absolute error is measured in the same units as the value itself.

There are several ways to write a quantity along with its absolute error.

· The notation with the ± sign is usually used. For example, the 100m record set in 1983 is 9.930±0.005 s.

· To record values ​​measured with very high accuracy, another notation is used: the numbers corresponding to the error of the last digits of the mantissa are added in brackets. For example, the measured value of the Boltzmann constant is 1.380 6488 (13)?10?23 J/K, which can also be written much longer as 1.380 6488?10?23 ±0.000 0013?10?23 J/K.

Relative error- measurement error, expressed as the ratio of the absolute measurement error to the actual or average value of the measured quantity (RMG 29-99):.

Relative error is a dimensionless quantity, or is measured as a percentage.

Approximation

Too much and too little? In the process of calculations, one often has to deal with approximate numbers. Let be BUT- the exact value of a certain quantity, hereinafter called the exact number a. Under the approximate value of the quantity BUT, or approximate numbers called a number a, which replaces the exact value of the quantity BUT. If a a< BUT, then a is called the approximate value of the number And for lack. If a a> BUT,- then in excess. For example, 3.14 is an approximation of the number R by deficiency, and 3.15 by excess. To characterize the degree of accuracy of this approximation, the concept is used errors or errors.

Error D a approximate number a is called the difference of the form

D a = A-a,

where BUT is the corresponding exact number.

The figure shows that the length of the segment AB is between 6 cm and 7 cm.

This means that 6 is the approximate value of the length of the segment AB (in centimeters)\u003e with a deficiency, and 7 is with an excess.

Denoting the length of the segment with the letter y, we get: 6< у < 1. Если a < х < b, то а называют приближенным значением числа х с недостатком, a b - приближенным значением х с избытком. Длина segment AB (see Fig. 149) is closer to 6 cm than to 7 cm. It is approximately equal to 6 cm. They say that the number 6 was obtained by rounding the length of the segment to integers.

Absolute value differences between the approximate and exact (true) value of a quantity is called absolute error approximate value. for example if the exact number 1,214 rounded to tenths, we get an approximate number 1,2 . In this case, the absolute error of the approximate number will be 1,214 – 1,2 = 0,014 .

But in most cases, the exact value of the quantity under consideration is unknown, but only approximate. Then the absolute error is also unknown. In these cases indicate border which it does not exceed. This number is called boundary absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the boundary error. for example, number 23,71 is the approximate value of the number 23,7125 up to 0,01 , since the absolute approximation error is equal to 0,0025 and less 0,01 . Here, the boundary absolute error is equal to 0,01 .*

(* Absolute error is both positive and negative. for example, 1,68 ≈ 1,7 . Absolute error is 1 ,68 – 1,7 ≈ - 0,02 . Boundary error is always positive).

The boundary absolute error of the approximate number " a » is denoted by the symbol Δ a . Recording

x ≈ a ( Δ a)

should be understood as follows: the exact value of the quantity X is in between a +Δ a and a –Δ a, which are named respectively bottom and upper bound X and denote H G X and AT G X .

for example, if X≈ 2,3 ( 0,1), then 2,2 < X < 2,4 .

On the contrary, if 7,3 < X < 7,4, then X≈ 7,35 ( 0,05).

Absolute or boundary absolute error not characterize the quality of the measurement. The same absolute error can be considered significant and insignificant, depending on the number that expresses the measured value.

for example, if we measure the distance between two cities with an accuracy of one kilometer, then such an accuracy is quite sufficient for this measurement, while at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable.

Therefore, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. So the measure of accuracy is the relative error.

Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the boundary absolute error to the approximate number is called boundary relative error; denote it like this: Δ a/a. Relative and boundary relative errors are usually expressed in percentages.

for example if measurements show that the distance between two points is greater than 12.3 km, but less 12.7 km, then for approximate its meaning is accepted average these two numbers, i.e. them half sum, then boundary the absolute error is semi-difference these numbers. In this case X≈ 12,5 ( 0,2). Here is the boundary absolute the error is 0.2 km, and the boundary

For modern tasks it is necessary to use a complex mathematical apparatus and developed methods for solving them. In this case, one often encounters problems for which the analytical solution, i.e., a solution in the form of an analytical expression linking the initial data with the required results is either impossible at all, or is expressed in such cumbersome formulas that it is impractical to use them for practical purposes.

In this case, numerical solution methods are used, which make it possible to quite simply obtain a numerical solution to the problem. Numerical methods are implemented using computational algorithms.

The whole variety of numerical methods is divided into two groups:

Exact - they assume that if the calculations are carried out accurately, then with the help of a finite number of arithmetic and logical operations, the exact values ​​\u200b\u200bof the desired quantities can be obtained.

Approximate - which, even under the assumption that the calculations are carried out without rounding, allow you to obtain a solution to the problem only with a given accuracy.

1. value and number. A quantity is something that can be expressed as a number in certain units.

When they talk about the value of a quantity, they mean a certain number, called the numerical value of the quantity, and its unit of measurement.

Thus, a quantity is a characteristic of a property of an object or phenomenon, which is common to many objects, but has individual values ​​for each of them.

The values ​​can be constant or variable. If, under certain conditions, the value takes only one value and cannot change it, then it is called constant, if it can take various meanings, then is a variable. Yes, acceleration free fall body in this place the earth's surface is a constant value, taking on a single numerical value g = 9.81 ... m / s2, while the path s, traversed material point during its movement, is a variable.

2. approximate values ​​of numbers. The value of the quantity, the truth of which we do not doubt, is called exact. Often, however, when looking for the value of a quantity, only its approximate value is obtained. In the practice of calculations, one often has to deal with approximate values ​​of numbers. So, p is an exact number, but due to its irrationality, only its approximate value can be used.

In many problems, due to the complexity, and often the impossibility of obtaining exact solutions, approximate solution methods are used, these include: approximate solution of equations, interpolation of functions, approximate calculation of integrals, etc.

The main requirement for approximate calculations is compliance with the specified accuracy of intermediate calculations and the final result. At the same time, both an increase in errors (errors) by unjustified coarsening of calculations, and the retention of redundant figures that do not correspond to actual accuracy are equally unacceptable.

There are two classes of errors resulting from calculations and rounding numbers - absolute and relative.

1. Absolute error (error).

Let us introduce the notation:

Let A be the exact value of some quantity, Record a » A We will read "a is approximately equal to A". Sometimes we will write A = a, keeping in mind that we are talking about approximate equality.

If it is known that a< А, то а называют approximate value of A with a disadvantage. If a > A, then a is called approximate value of A in excess.

The difference between the exact and approximate values ​​of a quantity is called approximation error and is denoted by D, i.e.

D \u003d A - a (1)

The error D of the approximation can be both positive and negative.

In order to characterize the difference between the approximate value of a quantity and the exact value, it is often sufficient to indicate the absolute value of the difference between the exact and approximate values.

The absolute value of the difference between the approximate a and accurate BUT number values ​​is called absolute error (error) of approximation and denoted by D a:

D a = ½ a – BUT½ (2)

Example 1 When measuring a line l used a ruler, the scale division value of which is 0.5 cm. We got an approximate value for the length of the segment a= 204 cm.

It is clear that during the measurement they could be mistaken by no more than 0.5 cm, i.e. the absolute measurement error does not exceed 0.5 cm.

Usually, the absolute error is unknown, since the exact value of the number A is unknown. Therefore, some assessment absolute error:

D a <= Da before. (3)

where D before. – marginal error (number, more zero), which is set taking into account the certainty with which the number a is known.

The limiting absolute error is also called margin of error. So, in the given example,
D before. = 0.5 cm.

From (3) we get: D a = ½ a – BUT½<= Da before. . and then

a-D a before. ≤ BUT≤ a+ D a before. . (4)

Means, a-D a before. will be an approximation BUT with a disadvantage and a + D a before – approximate value BUT in excess. They also use shorthand: BUT= a±D a before (5)

It follows from the definition of the limiting absolute error that the numbers D a before, satisfying inequality (3), there will be an infinite set. In practice, we try to choose possibly less from numbers D before, satisfying the inequality D a <= Da before.

Example 2 Let us determine the limiting absolute error of the number a=3.14, taken as an approximate value of the number π.

It is known that 3,14<π<3,15. Hence it follows that

|a – π |< 0,01.

The number D can be taken as the limiting absolute error a = 0,01.

However, if we take into account that 3,14<π<3,142 , then we get a better estimate :D a= 0.002, then π ≈3.14 ±0.002.

Relative error (error). Knowing only the absolute error is not enough to characterize the quality of the measurement.

Let, for example, when weighing two bodies, the following results are obtained:

P 1 \u003d 240.3 ± 0.1 g.

P 2 \u003d 3.8 ± 0.1 g.

Although the absolute measurement errors of both results are the same, the measurement quality in the first case will be better than in the second. It is characterized by a relative error.

Relative error (error) number approximation BUT is called the absolute error ratio D a approximation to the absolute value of the number A:

Since the exact value of a quantity is usually unknown, it is replaced by an approximate value and then:

Limiting relative error or limit of relative approximation error, called the number d and before.>0, such that:

d a<= d and before.

For the limiting relative error, one can obviously take the ratio of the limiting absolute error to the absolute value of the approximate value:

From (9) the following important relation is easily obtained:

∆and before. = |a| d and before.

The limiting relative error is usually expressed as a percentage:

Example. The base of natural logarithms for the calculation is taken equal to e=2.72. We took as the exact value e m = 2.7183. Find the absolute and relative errors of an approximate number.

D e = ½ e – e t ½=0.0017;

.

The value of the relative error remains unchanged with a proportional change in the most approximate number and its absolute error. So, for the number 634.7, calculated with an absolute error D = 1.3, and for the number 6347 with an error D = 13, the relative errors are the same: d= 0,2.