Normal rosepodil - the widest type of rosepodil. It should be discussed with him in the analysis of deaths, control of technological processes and regimes, as well as in the analysis of predictive events in biology, medicine and other knowledge.

The term "normal rozpodil" zastosovuєtsya in the mental sense as a commonplace in the literature, though not far away. So, firmness, which, as a sign under the normal law of a rose, does not mean the presence of any inviolable norms, which must lie at the basis of a phenomenon, as a sign of a sign that is seen, but under other laws, the nature of a rose is not signified.

The main feature of a normal rozpodіlu is in the fact that it is borderline, to which other rozpodіli are approaching. Normal rozpodіl vіdcrito Moivre 1733 rock. The normal law is subject to less than uninterrupted fluctuations. Shchіlnіst normal law rozpodіlu maє vyglyad.

Mathematical ochіkuvannya for the normal law rozpodіlu dorіvnyuє. The dispersion is good.

The main power of a normal rozpodіlu.

1. The function of the width of the subdivision is assigned on the entire numerical axis Oh , to skin significance X vіdpovіdaє tsіlkom pvne znachennya funktsії.

2. For all values X (both positive and negative) the width function gains positive values, so that the normal curve is spread over the veins Oh .

3. Intermediate functions of space with non-intermediated growth X up to zero, .

4. The function of the width of the normal distribution of the point at the point is maximum .

5. The graph of the function of width is symmetrical and straight.

6. The curve can be subdivided into two points of inflection with coordinates і .

7. The mode and the median of the normal distribution are adjusted to mathematical scaling a .

8. The shape of the normal curve does not change when changing the parameter a .

9. The coefficients of asymmetry and the excess of the normal distribution are equal to zero.

The importance of the calculation of these coefficients for the empirical rows is obvious;

The movability of falling into the interval is determined by the formula , de unpaired tabulated function.

Significantly the imovirnist of what is normally distributed vipadic value diverges from its mathematical scaling to a smaller value, so we know the immovability of the effect of unevenness or the imovirnіst of the underlying nervousness. Submitting the formula, we take

Virazivshi v_dhilennya vipadkovoї magnitude X in parts of the mid-square deviation, which, if left in the rest, is taken away.

Todi when taken away,

when taken,

when taken.

From the rest of the unevenness, it is obvious that the practical distribution of a normally distributed vertical magnitude is laid on a distance. The possibility that the vipadka value cannot be spent on tsyu plot is even small, and the most expensive is 0.0027, so that it can be less for three vipadkas out of 1000. Gruntuetsya on the hovering of the mirrors three sigma rule, which is formulated by the coming rank: even though the variable value may be normal, then the deviation of the value from the mathematical scaling in absolute value does not exceed the loss of the average quadratic deviation.

Example 28 . The part, prepared automatically, is respected by the adventitious one, so that the controlled size does not exceed the design size by 10 mm. Vipadkovі vіdkhilennya kontrolirovannogo razmіru vіd proektnym pіdokladkovanі normal law rozpodіlu іz srednі svіdnymi іdіlіnіm mm і mathematical ochіkuvannyam. How many attachments do you make the machine?

Solution. Let's take a look at the vipadian value X - Vіdhilennya rozmіru vіd project. The detail will be recognized as adventitious, so that the value should lie down to the interval. The ease of preparation of the accessory part is known by the formula . Also, the number of attached parts, which are automatically prepared, is 95.44% more expensive.

Binomial rozpodil

Binomial є rozpodіl ymovіrnosti appeared m number of pods P independent tests, in the skin z some symptoms appear under R . The mobility of a possible number is calculated using the Bernoulli formula:,

de. Postiyni P і R , which enters up to the first virase, the parameters of the binamic law. The binomial rozpodіl describes the rozpodіl ymovіrnosti discrete vypadkovy value.

Basic numerical characteristics bіnomnogo rozpodіlu. Mathematically ochіkuvannya one thing. Dispersion is more expensive . Coefficients of asymmetry and excess equal . With an unrestricted increase in the number of tests BUT і E Pragivat zero, then, you can allow that the binomial rozpodіl converge to the normal zі growth of the number of trials.

Example 29 . There are independent trials with the same ability to appear under the same BUT in skin testing. Know ymovіrnіst appear podії BUT in one trial, so the variance of the number of appearances in three trials is as high as 0.63.

Solution. For bіnomnogo rozpodіlu . Let's imagine the value, take it away stars or same .

Rozpodil Poisson

The law of rozpodіlu rіdkіsnih yavisch

Rozpodil Poisson describes the number of pods m , which are due for the same intervals of an hour for a wash, which are independently one in the same with a constant average intensity. When tsimu kіlkіst viprobuvan P is great, and the ability to appear under skin testing R small. That is why Poisson was called the law of rare manifestations, or the simplest flow. The parameter of the Poisson subdivision is the value that characterizes the intensity of the appearance of subdivisions P samples. Poisson's subdivision formula .

The Poissonian rozpodil well describes the amount of money that could be paid for insurance sums for the river, the number of calls that came to the telephone station for the song hour, the number of items in the case of testing for vain, the number of defects and so on.

The main numerical characteristics of Poisson's subdivision. Mathematically clearing one variance and one a . Tobto . Tse є dictatorial peculiarity whom I have broken. Coefficients of asymmetry and excess are obviously equal.

Example 30 . The average number of payments of insurance sums for a day is more than two. Know the ability to pay in five days: 1) 6 sums insured; 2) less than six soums; 3) not less than six. or exponential Rozpodіl.

Tsej rozpodіl often poderіgaєtsya schodo terms of service of various outbuildings, an hour of bezvіdmovnої work of okremih elementіv, parts of the system and the system in a haze, under an hour of looking at the vipadkovy promizhkіv hour between the appearances of two last rare occasions.

The width of the display rozpodіlu is determined by the parameter, which is called the intensity of the waves. This term is related to a specific area of ​​addition - the theory of supremacy.

Virase of the integral function of the illustrative rozpodіlu can be calculated, vicorist power of the differential function:

Mathematical scaling of the display distribution, variance, rms scaling. In such a rank, for which it is characteristic that the average quadratic deviation is numerically superior to the mathematical one. With any value of the parameter of the coefficient of asymmetry and excess - a constant value.

Example 31 . The middle hour of the TV work until the first anniversary of 500 years. Know the efficiency of the fact that the TV set has been propagated without breakdowns for more than 1000 years.

Solution. Oskіlki middle hour of work until the first time is 500, then . Shukanu ymovirnist is known by the formula.

The law of normal distribution, so called Gaus's law, is one of the most extensive laws. Tse fundamental law in the theory of imovirnosti and in її zastosuvanni. The normal rozpodіl is most often seen in the development of natural and social and economic phenomena. Otherwise, the greater number of statistical aggregates in nature and the order are subject to the law of normal distribution. Obviously, it can be said that corruption great number great for obsyagom vibirok pіdryadkovuyutsya to the law of normal rozpodіlu. Those of the aggregates, which seem to be in the normal rozpodіlu after the special transformations, can be close to the normal. At the link with this is a reminder that the principle of the specialty of which law is completely different from the law of the cordon, to which the other laws are closer to the law of the singing (typical) minds.

It should be noted that the term "normal rozpodil" can be clever as a general acceptance in mathematical and statistical and mathematical literature terms. The confirmation that the third sign of whether a phenomenon is subject to the law of a normal subordination does not mean the inviolability of the norms, nor does it mean that the power of the existing phenomenon, but the introduction of the rest to a different kind of law means an abnormality of this phenomenon. For whom the whole sense of the term "normal rose" is not familiar in the distance.

Normal distribution (Gaus-Laplace law) is a type of continuous expansion. De Moivre (1773, France) vivіv the normal law of razpodіlu imovіrnosti. The main ideas of the first opinion were victorious in the pardon theory by K. Gauss (1809, Niechchyna) and A. Laplace (1812, France), as they made a theoretical contribution to the development of the law itself. Zokrema, K. Gauss, in his studies, came out of the knowledge of the most significant values ​​of the variable value - the arithmetic mean. Zagalnі umovi vyniknennya normal rozpodіlu having installed A.M. Lyapunov. It was brought to light that the sign was added as a result of the total influx of various factors, skin with some few influxes with more others, and the influx of the skin factor on the last result is richly overlapped by the total influx of all other factors, then rose to normal close.

The normal name is the distribution of fluctuations of an uninterrupted depression of magnitude, which may be:

1 +1 (& #) 2

/ (x, x,<т) = - ^ е 2 st2

de x - mathematical ochіkuvannya chi mean value. As you can see, the normal distribution is characterized by two parameters: x and °. To install a normal distribution, it is enough to know the mathematical grading or the mean and mean quadratic deviation. Two values ​​determine the center of the grouping that shape

curve on the graph. The graph of the function i (xx, c) is called a normal curve (Gauss curve) with parameters x and b (Fig. 12).

The curve of a normal divergence may have an inflection point at X ± 1. If you want to show graphically, then X = + l i 1 \u003d -1 is 0.683 parts of the entire area of ​​the curve (tobto 68.3%). At the boundaries of X = + 2 and X-2 there are 0.954 areas (95.4%), and between X = + 3 and X = - 3 - 0.997 parts of the entire area of ​​the subfloor (99.7%). On fig. 13 illustrates the nature of a normal rosacea with one-, two-trisigmoid cordons.

With a normal spread, the mean is arithmetic, the mode of that median is equal to each other. The form of a normal curve may look like a single-vertex symmetrical curve, the needles being asymptotically approached to the abscissa axis. The largest ordinate of the curve is x = 0. At this point, on the abscissa axis, there is a numerical sign that is closer to the arithmetic mean, mod and median. On the offending sides at the top of the crooked її gіlka come, changing at the singing points the shape of the bulge to the curve. Qi points are symmetrical and correspond to the values ​​x = ± 1, that is, the values ​​of the signs, the deviation of any of the mean values ​​are numerically superior to the mean quadratic deviation. The ordinate, which shows the arithmetic mean, divides the entire area between the curve and the full abscissa. From the same time, the significance of the additional signs of greater and lesser ones appeared for the average

arithmetic value 0.50, then x, (~ ^ x) \u003d 0.50 V

Fig.12. Curve of normal distribution (Gaus curve)

The shape and position of the normal curve are measured by the mean value and the mean square value. Mathematically, it has been shown that changing the value of the mean (mathematical scaling) does not change the shape of the normal curve, but only leads to a shift in the abscissa axis. The curve collapses to the right, like ~ growth, and to the left, like ~ coming.

Fig.14. Curves of normal distribution with different values ​​of the parameterin

About changing the shape of the graph of the normal curve when changing

mean square deviation can be judged by the maximum

differential function of normal rozpodіlu, equal 1

As you can see, with the increase in the value of °, the maximum curve of the ordinate will change. Later, the curve of a normal rosette compresses to the abscissa axis and takes on a larger flat-topped shape.

I, on the other hand, when the parameter is changed, the normal curve turns in a positive direction along the ordinate axis, and the shape of the "ringing" becomes more peaked (Fig. 14). Significantly, it is independent in terms of the magnitude of the parameters ~ i in the area, it is surrounded by the abscissa and the curve, it is always the most common unit (the power of the space has been subdivided). First of all, I illustrate the graph (Fig. 13).

The names of the higher features of the manifestation of "normality" rozpodіlu allow you to see a low level of authority, like a curve of a normal rozpodіlu:

1) whether a normal curve reaches a point to a maximum (X= х) arrive without interruption right-handed and left-handed in the direction of the new one, step by step approaching the abscissa axis;

2) whether a normal curve is symmetrical in relation to a straight line,

parallel to the y-axis i pass through the point to the maximum (X= x)

the maximum ordinate is dorіvnyuє ^^^ i;

3) whether the curve is normal, may have the form of a "ringing", may be swelled, as if it is straight uphill to the point of maximum. At the points x ~ ° and x + v, the bulge changes, and the less a, the more "dzvіn", and the more a, the more sickly the top of the "dzvon" becomes (Fig. 14). Change of mathematical calculation (for fixed value

c) do not produce to change the shape of the curve.

When x \u003d 0 and ° \u003d 1, the normal curve is called the normalized curve or the normal division of the canonical view.

The normalized curve is described by the following formula:

Pobudov normal curve for empirical data to be carried out according to the formula:

pі 1 - "" = --- 7 = e

de i ™ is the theoretical frequency of the skin interval (group) of subdivision; - the sum of frequencies, equal obyagu sukupnostі; - crochet interval;

g - extending the length of the stake to її diameter, like folding

e - the basis of natural logarithms, which is 2.71828;

The other third part of the formula) is the function

normalized variation of the central frequency), how can one calculate for any X values. Tables of the value of the central frequency) should be called "tables of ordinates of the normal curve" (addition 3). When vikoristannі tsikh funktsіy robocha formula normal rozpodіl nabuvaet simple looking:

butt. Let's take a look at the trend of the normal curve from the butt of data about 57 practitioners for a daily wage (Table 42). Behind the data of tables 42, we know the arithmetic mean:

~ = ^ = I6 54 =

Razrakhovuemo mid-square deviation:

For the skin row of the table, the value of normalized ventilation is known

x i ~ x | 12 g => - = - ^ 2 = 1.92

a 6.25 (dd I first interval too).

Column 8 tab. For example, for the first interval X = 1.92, "1.9" is known against "2" (0.0632).

For the calculation of theoretical frequencies, so that the ordinates of the curve of the normal distribution, the multiplier is calculated:

* = ^ = 36,5 a 6.25

All known tabular values ​​of the function /(r) are multiplied by 36.5. So, for the first interval, we take 0.0632x36.5 = 2.31 tons.

frequencies (P "<5) combine (in our butt - the first two and the remaining two intervals).

Although the extreme theoretical frequencies are significantly offset from zero, the difference between the sums of empirical and theoretical frequencies can be significant.

The graph of the distribution of empirical and theoretical frequencies (normal curve) for the given butt, as you can see, is shown in figure 15.

Let's take a look at the example of the frequency distribution of the normal drop, as in the extreme intervals the daily frequency (Table 43). Here is empirical

X - standardized allowance, (c) a - mid-square deviation.

the frequency of the first interval is zero. Otriman's sum of unspecified frequencies is not equal to the sum of all empirical values ​​(56*57). And here the theoretical frequency is provided for the transfer of the value to the center of the interval, the normalized adjustment of that function.

In the table 43 qi values ​​are circled with a rectangle. When prompted, the graph of the normal curve is continued at the time of the theoretical curve. In this case, the normal curve will be continued in the case of negative results in the middle, the first frequency is not specified. For the sum of the clarified frequencies, they are taken from the empirical

Table 42

Table 43

Razrahunok of frequencies of normal distribution (virіvnjuvannya empirіchnyh frequencies according to the normal law)

Rice. 15. Empiric rozpodil(1) that normal curve (2)

The curve of the normal rozpodіlu after the last suupnіstyu can be induced in a different way (on the vіdmіnu vіd razglyany vishche). So, as it is necessary for the mother to come close to the occurrence of the actual normal occurrence, the calculation of the upcoming sequence. We designate the maximum ordinate, so that we give signs to the average size), then, having calculated the mean quadratic expansion, calculate the coordinates of the curve point of the normal distribution under the scheme, shown in tables 42 and 43. The middle beats the center of the fourth interval (25-27). Also, the frequency of this interval "20" can be taken (if the schedule is set) to the maximum ordinate). Let me calculate the variance (v = 2.41, see Table 43), calculate the value of the coordinates of all the necessary points of the curve of the normal distribution (Tables 44, 45). After subtracting the coordinates, we follow the normal curve (Fig. 16), taking the maximum frequency as the ordinate of the fourth interval.

The convenience of an empiric rozpodіlu from a normal one can also be established by the way of simplification of rozrahunkіv. So, as an extension of the indication of the degree of asymmetry (^) to its mean-square pardon w and "or as an indication of the excess (Ex) to its mean-square pardon, t& outweigh the number "3" for the absolute value,

BUT c E X

yakscho ™ A> 3 or sh e "> 3).

Є y іnshі, not trudomіstki priyomii vstanovlennja "normality" rozpodіlu: a) equalization of the arithmetic mean with the mode and the median; b) selection of Westergard numbers; c) placing a graphic image behind an additional napіvlogarithmic grid Turbine; d) calculation of special criteria for the satisfaction of that іn.

Table 44

Coordinates 7 point of the curve of normal distribution

Table 45

Calculation of coordinates of a point of a curve of a normal distribution

Fig.16. Curve of a normal distribution, induced by seven points

In fact, if the marriage is to please the її rozpodіlu with the normal one, it is often corrosive "rule 3sg".

Mathematically brought the one, the vidhilennya of the middle of the middle, the absolutely sorrowful of the Men, the Sredtelnoye quadratic vidhilen, the dorivnu, Tobto, the same, the value of the soreness of the Service of the Service of the Service is absolutely Vykhodyachi z the principle of the impossibility of small-moving podia, you can vvazhati practically impossibly "vipadok perevishchennya" 3 tbsp. Although the vipad value is distributed normally, the absolute value of її vіdhilennya vіd mathematical ochіkuvannya (mean) does not outweigh the trinary mean quadratic іdhіlennya.

In practical roses, they develop in such a way. As for the unknown nature of the difference, the value of the allowance for the increase in the average value of the increase should appear less value 3 CT, so let me know what sign, what to follow, it is normal. Although setting the parameter to override the numerical value of 3 ST, it should be noted that the difference between the following values ​​does not fit with the normal value.

Calculation of theoretical frequencies for the dosledzhuvannogo empiricheskogo row rozpodіlu accepted to call virіvnyuvannyam empirіchnyh curves for the normal (or be-yakim іnshim) law rozpodіlu. This process is important both theoretically and practically. Virіvnyuvannya empirіchnіh dannyh razkryvaє zakonіrnіnіst іt rozpodіlі, yak buti veiled vypadkovym form svojsya manifest. Installed in such a way, the law can be beaten to a high level of practical tasks.

With rozpodіlom, close to normal, doslidnik zustrichaєtsya in various spheres of science and the galleys of the practical activity of people. In the economy of this kind, roses are used more or less, let's say, in technіtsi chi biology. Mindful by the very nature of social and economic phenomena, they are characterized by a great folding of mutual influences and mutual factors, as well as the presence of low minds that surround a wide group of vipadkivs. But the economist is guilty of working up to a normal level, analyzing the future empirical changes like up to a certain standard. This way, the alignment allows you to reveal the nature of the inner minds, as if they signify the given figure of the rose.

Penetration of the sphere of statistical data in the area of ​​social and economic phenomena made it possible to reveal the basis of a large number of different types of crooked roses. However, it should not be taken into account that the theoretical concept of the curve of the normal distribution of the subdivision is not very suitable for the statistical and mathematical analysis of this type of phenomena. It may not always be acceptable in the analysis of a specific statistical distribution, but in the gallery of the theory and practice of the vibrating method of follow-up may be of paramount importance.

Let's name the main aspects of susceptibility of the normal distribution to the statistical and mathematical analysis.

1. For the purpose of denoting the meaning of a specific sign. It is necessary when reversing the hypotheses if the evidence of that chih empirical rose to be normal.

2. Estimating low parameters, for example, average ones, by the method of maximum likelihood. The essence of yoga is determined by such a law, according to which order is ordered. That assessment is given, as it gives the maximum value. The best approximation to the parameters of the general marriage is given by:

y = - 2 = e 2

3. For the determination of the quality of the selected averages of the general averages.

4. When a predetermined interval has been set, it is necessary to know the approximate value of indications in the general marriage.

Normal rose ( normal distribution) - play an important role in the analysis of data.

Other replacement for the term normal rozpodіl get used to the term gaussian rozpodil in honor of K. Gauss (older terms that practically do not get used to the present hour: the law of Gauss, Gauss-Laplace rose up).

Univariate normal rozpodil

Normal rozpodіl maє shіlnist:

This formula has fixed parameters, - middle, - standard mindfulness.

Width graphs for various parameters are drawn.

The characteristic function of a normal rosacea can be seen:

Differential characteristic function and important t = 0, Otrimuёmo moments and be-any order.

The width curve of a normal distribution is symmetrical, although it may or may not have a single maximum, equal

The standard ventilation parameter changes from 0 to ∞.

Middle change at the boundary of the view -∞ to +∞.

When the parameter is increased, the curve expands to the axis X, when the value is 0, it shrinks around the average value (the parameter characterizes the rise, the rise).

When changing the curve collapses on the axis X(Div. graphics).

Variable parameters and we take into account different models of fluctuation values, which are in telephony.

Typical zastosuvannya normal law in the analysis, for example telecommunication data - signal modeling, description of noise, reshkod, pardons, traffic.

Graphs of one-dimensional normal distribution

Little one

Malyunok 2. Graph of the width of the standard normal distribution with areas that cover 68% and 95% of all warnings

Malyunok 3. Graphs of normal roses with zero mean and varying tolerances (=0.5, =1, =2)

Malyunok 4 Graphs of two normal divisions N(-2,2) and N(3,2).

Please note that the center of the subdivision collapsed when the parameter was changed.

Respect

In the program STATISTICS Under the values ​​of N(3,2), the normal Gaussian law is understood with the parameters: average = 3 and standard deviation =2.

Literature sometimes has another parameter interpreted as dispersion, then. square standard intake.

Calculation of the percentage points of the normal distribution for the help of a movable calculator STATISTICS

For the help of an easy-to-use calculator STATISTICS you can calculate the different characteristics of roses without going into cumbersome tables, like in old books.

Krok 1. Let's start Analysis / Imovirnіsny calculator / Rozpodila.

At the rozdіlі rozpodіlu oberemo normal.

Figure 5. Launching the calculator for immovable roses

Krok 2 Vkazuєmo parametr, scho us tsіkavlyat.

For example, we want to calculate the 95% quantile of the normal distribution from the average 0 and the standard deviation 1.

Specify the number of parameters in the fields of the calculator (div. fields of the average and standard allowance calculator).

We introduce the parameter p = 0.95.

Check mark "Zvorotna f.r." to show up automatically. Check the "Graph" box.

Press the button "Calculate" at the top right corner.

Baby 6. Adjustment of parameters

Krok 3 For the Z field, the result is expected: the value of the quantile is more than 1.64 (div. coming soon).

Figure 7. Revisiting the result of the robotic calculator

Malyunok 8. Graphs of the width and function of the rozpodіlu. Straight line x=1.644485

9. Graphs of the normal distribution function. Vertical dotted lines-x=-1.5, x=-1, x=-0.5, x=0

10. Graphs of the normal distribution function. Vertical dotted lines - x = 0.5, x = 1, x = 1.5, x = 2

Evaluation of the parameters of the normal distribution

The value of the normal distribution can be calculated for help interactive calculator.

Double normal rose

A one-dimensional normal rose is naturally focused on two-world normal rose.

For example, if you see the signal at least one point, then you have enough of a one-dimensional subdivision, two points - a two-dimensional one, three points - a trivial one.

The general formula for a two-world normal rozpodіlu may look:

De - parna correlation between x1і x2;

x1 obviously;

The average and standard ventilation of the change x2 obviously.

What a vipadkovі size X 1і X 2 independent, then the correlation is more equal 0 = 0, obviously the middle term in the exponent vanishes, and we can:

f(x 1 ,x 2) = f(x 1)*f(x 2)

For non-independent values, the two-world width is separated in the case of two one-world numbers.

Graphs of the width of the two-world normal distribution

Figure 11. Graph of the width of a two-world normal distribution (zero vector of means, single matrix of covariance)

figure 12

Malyunok 13. Graph of the width of a two-world normal subdivision (zero vector of mat. score, matrix of variance z 1 on the head diagonal and 0.5 on the side diagonal)

Figure 14. Cross-section of the width plot of a two-world normal subdivision (zero vector of mat. score, matrix of variance z 1 on the head diagonal and 0.5 on the side diagonal) with a plane z = 0.05

Malyunok 15. Graph of the width of a two-world normal subdivision (zero vector of mat. score, variance matrix of 1 on the head diagonal and -0.5 on the side diagonal)

Malyunok 16. Peretin graph of the width of a two-world normal rozpodіl (zero vector mat. scoring, matrix of variance z 1 on the head diagonal and -0.5 on the side) with a plane z=0.05

Malyunok 17. The cross-section of the graphs of the thickets of the two-world normal rose under the plane z=0.05

For a short understanding of the two-world normal rose, try the same task.

Manager. Look at the graph of the two-world normal distribution. Think about how you can visualize it as a wrapper for a graphic of a one-world normal rozpodіlu? When is it necessary to fix the deformation?

In the article, it is reportedly shown that such a normal law has been subdivided into a vipadkovoї value and, as it is, it is practically a task for the hour of perfection.

Normal rozpodіl in statistics

The history of the law is 300 years old. Abraham de Moivre became the first leader, who foresaw an approximation as early as 1733. Through many rocks, Carl Friedrich Gauss (1809) and Pier-Simon Laplace (1812) introduced mathematical functions.

Laplace also revealed the miraculous law and formulated central boundary theorem (CPT), zgіdno z kakoyu sum of the great number of small and independent values ​​may be normal rozpodіl.

The normal law does not fix the equal fallows of one change in the other. The nature of the fallow land is fixed less. The specific shape of the rosette is determined by special parameters. For example, y = ax + b- The price is straight. However, de same won’t pass and under some kind of impudence, it is determined by the parameters aі b. Same with normal roses. It was understood that this function, as it describes the trend of high concentration of the value of the center, that exact form is set by special parameters.

The curve of a normal rozpodіlu Gaus may look like this.

The schedule of the normal rozpodіlu guessing zvіn, so you can guess the name ring-like curve. The graph may have a "hump" in the middle and a sharp decrease in thickness along the edges. In whom I mean the essence of a normal rozpodіlu. Imovirnist that the vipadical value appears to the center richly, lower those that are strongly inspired in the middle.

On a small scale, there are images of two wooden houses under the Gauss curve: blue and green. Give it to me. intervals, in both plots equal. Ale pomitno viznyayutsya heights. The blue lot is far away from the center, and may be lower in height, lower is green, as it is located in the very center of the rose. From the same, the areas are also considered, so that the immovability of falling into the designated intervals.

The formula for a normal rozpodіlu (shіlnostі) is as follows.

The formula is composed of two mathematical constants:

π - Number of pi 3.142;

e- the basis of the natural logarithm 2.718;

two changing parameters that set the shape of a particular curve:

m- mathematical refinement (for example, other dzherelakh can have other meanings, for example, µ or a);

σ2– dispersion;

well i change myself x, on the basis of which the amount of imovirnost is calculated.

The specific form of the normal rose under the deposit in the form of 2 parameters: ( m) and ( σ2). Briefly N(m, σ 2) or N(m, σ). Parameter m(matochuvannya) determines the center of the rozpodіlu, which shows the maximum height of the graph. Dispersion σ2 characterize the range of variations, that is, the “spreading” of data.

The parameter of mathematical refinement shifts the center of the spread to the right and to the left, without interfering with the shape of the crookedness.

And the dispersion axis defines the hospitality of the curve. If there is a small rozkid given, then the whole mass is concentrated towards the center. If the Danes have a great rozkid, then the stench spreads over a wide range.

There is no direct practical zastosuvannya. For the development of possibilities, it is necessary to integrate the function of space.

Imovirnist of the fact that the vipad value appears less for the deak value x, signify function of the normal distribution:

Vikoristovuyuchi mathematical authority, whether it be some kind of uninterrupted rozpodіlu, clumsily rozrahuvat whether it be іnshі іmovіrnostі, oskіlki

P(a ≤ X< b) = Ф(b) – Ф(a)

Standard normal rose

The normal rozpodіl lie down in the parameters of the average dispersion, through which it is nasty to see the power. Good mother, the singing star rose, which is to fall on the scale of the tribute. I won't. called standard normal rose. In fact, the normal normal distribution, only with the parameters of mathematical scoring 0, and the variance - 1, is shortly written N(0, 1).

Whether it’s a normal rose, it’s easy to transform into a standard way of normalization:

de z- new change, like a victorious deputy x;
m- mathematical ochіkuvannya;
σ - standard care.

For vibratory data, estimates are taken:

Arithmetic mean of the variance of the new change z now it's also ok 0 and 1 is clear. It is easy for him to perekonatisya with additional elementary algebraic transformations.

Literature has a name z-score. Tse vono same - normalized data. Z-score can be directly compared with theoretical possibilities, tk. її scale zbіgaєtsya іz sharp.

Now, marveling at how it looks like the width of the standard normal rosette (for z-scores). Guess what the Gaussian function might look like:

Let's pretend to be a deputy (x-m)/σ letter z, and deputy σ - alone, taken away the function of the width of the standard normal distribution:

Gustina schedule:

The center, as it was cleared, is located at point 0. At this point, the Gauss function reaches its maximum, which in turn confirms the acceptance of the average value of its average value (tobto. x-m=0). The strength of this point is 0.3989, e 0 \u003d 1 and it is necessary to loosen up less spіvіdnoshennia 1 on the root of 2 pі.

In this order, behind the graph, it is clearly visible that the meanings that may be small in the middle, fall more often for others, and those, which are very far away from the center, grow significantly more slowly. The scale of the abscissa axis is changed in standard views, which allows you to change from one to the other and to take into account the universal structure of the normal distribution. The curve of Gauss for the rationing of data miraculously demonstrates the greater power of the normal distribution. For example, it is symmetrical about the axis of ordinates. At the boundaries of ±1σ, the arithmetic mean concentrates the majority of all values ​​(it is estimated for the time being by an eye). At the boundaries of ±2σ, there are more data. At the boundaries of ±3σ, all data are rebuffed. Remaining power widely in the house under the name three sigma rule for a normal rose.

The function of the normal standard rozpodіl allows you to rozrakhovuvat ymovіrnostі.

I understood, I don’t care about anyone by hand. Everything is underpinned and placed in special tables, as if it were a kind of assistant to statistics.

Table of normal distribution

Tables of normal distribution are divided into two types:

- Table strength;

- Table functions(Integral in width).

table strength vikoristovuєtsya rarely. Tim is not smaller, more like you look out there. It is permissible, it is necessary to take a margin for z = 1, then. schіlnіst znachennya, vіddalénogo vіd expectation by 1 sigma. Below is a table layout.

Fallow in the organization of these data, it is necessary to take the name of the column of that row. Our butt has a row 1,0 i stovpets 0 , because there are no hundred of them. Shukane value is 0.2420 (0 to 2420 omissions).

The Gaussian function is symmetric along the ordinate axis. Tom φ(z)= φ(-z), then. space for 1 the same power for -1 which is clearly visible on the little one.

Sob not marnuvati papers, tables are used only for positive values.

In practice, more often than not, the values functions standard normal rozpodіlu, tobto ymovіrnostі for rіznih z.

Such tables have less positive meaning. Therefore, for understanding, that significance whatever needs of knowledge next to the nobility power of the standard normal rose.

Function Ф(z) it is symmetrical with its value of 0.5 (and not the axis of ordinates, like Gustina). Looks fair equanimity:

This fact of the testimony in the image:

Function value Ф(-z)і Ф(z) divide the chart into 3 parts. Moreover, the upper and lower parts are equal (marked with checkmarks). In order to add imovirnist Ф(z) up to 1, enough to add an insufficient value Ф(-z). Viyde rivnist, vzad troch more.

It is necessary to know the possibility of consumption in the interval (0;z) so that the imovirnіst vіdkhilennya vіd zero in a positive bіk to deykoї kіlkostі standard intakes sufficient for the value of the function of the standard normal range of 0.5:

For accuracy, you can marvel at the little ones.

On the curve of Gauss, this situation looks like a square from the center to the right to z.

Dosit often analytics to call for the possibility of repentance in the offending side of zero. Oskilki function is symmetrical to the center, the front formula needs to be multiplied by 2:

Little one below.

Below the Gauss curve is the central part, surrounded by selected values -z zliva that z right-handed

The assigned characteristics should be taken to respect, because The tabular values ​​rarely correspond to the cycle interval.

For ease of tasking with assistants, call publish tables for the function of the form:

If it is necessary to be able to look at the offending side of zero, then, as we have changed, the table value for the given function is simply multiplied by 2.

Now let's look at specific examples. Below is a table of a standard normal distribution. We know the table values ​​for three z: 1.64, 1.96 and 3.

How to understand the meaning of these numbers? Pochnemo s z=1.64, for which tabular value to become 0,4495 . Easier to explain the sens baby.

Tobto the ability of what is standardized normally rozpodіlena vipadkovy value to spend in the interval vіd 0 before 1,64 , dear 0,4495 . When the order is dismissed, it is necessary to sound out the resilience of resentment in the offending side, to this we multiply the value 0,4495 for 2 ta we take approximately 0.9. The occupied area under the Gauss curve is shown below.

In this way, 90% of the normal distribution values ​​are reduced to the interval ±1.64σ as the arithmetic mean. I'm not vipadkovo vibrating the meaning z=1.64, because the surrounding area is about the arithmetic mean, which occupies 90% of the entire area, sometimes victorious for rozrahunku dovirchih іnvalіv. If the value is not taken into the designated area, it will come on slightly (10% in total).

Prote for re-verification of hypotheses often vikoristovuєєєєєєєєєєєєєєєєєєєє єє є scho Ҕ 95% sіх sluzhen. Half of the ymovirnosti vіd 0,95 – ce 0,4750 (Div. other seen in the table of values).

For the price of imovirnosti z = 1.96. Tobto. in the midst of mayzha ±2σ in average є 95% value. Less than 5% fall out of the middle.

One more cicava that often vikoristovuetsya tabular meaning vіdpovіdaє z=3, there is one behind our table 0,4986 . Multiply by 2 and subtract 0,997 . Otzhe, at the border ±3σ in the arithmetic mean, all values ​​are put in.

This is how the 3 sigma rule looks for a normal distribution on the diagram.

For the help of statistical tables, you can take away whether it is possible. However, the method is more common, non-handy, and very old. Today everyone works on the computer. Let's move on to the practice of rozrachunkiv in Excel.

Normal spread in Excel

Excel has a lot of functions for the fluff and the return value of the normal rozpodіlu.

NORM.S.DIST function

Function NORM.ST.DIST recognized for rozrahunku schіlnostі ϕ(z) what is the immovability Φ(z) for normative data ( z).

\u003d NORM.ST.DIS(z; integrand)

z- the value of the standardized change

integral- if 0, then the insurance coverage isϕ(z) , where 1 is the value of the function Ф(z), then. flexibility P(Z

Razrahuyemo schіlnіst і znachnіnі ї ї ї for raznyh z: -3, -2, -1, 0, 1, 2, 3(їх it is possible to put it in the middle A2).

For the rozrahunka of Gustini, you need the formula =NORM.ST.DIS(A2;0). On the diagram below, there is a red dot.

For the distribution of the value of the function =NORM.ST.DIST(A2;1). On the diagram, the area under the normal curve is filled.

In fact, more often it is necessary to explain the stability of the fact that the vipad value does not appear beyond the average between average values ​​(in mean-square deviations, which z), then. P(|Z| .

Significantly, why is it more comfortable to fall in the fall of the value at the boundary ±1z, ±2z and ±3z zero. Required formula 2Ф(z)-1, Excel =2*NORM.ST.DIST(A2,1)-1.

On the diagram, you can clearly see the main basic power of the normal distribution, including the rule of three sigmas. Function NORM.ST.DIST– this is an automatic table of the value of the function of the normal subdivision of Excel.

You can stand and zavorotne zavdannya: for imovirnistyu P(Z know the standardized value z that is the quantile of the standard normal distribution.

NORM.S.OBR function

NORM.ST.INV rozrakhovu zvorotne znachennya znachennya funktsii standard normal rozpodіlu. The syntax consists of one parameter:

=NORM.ST.OBR(immovirnist)

imovirnist- Tseimovirnist.

This formula is victorious so very often, like in the past, and even behind the same tables, not only fluctuations, but quantiles are brought to light.

For example, in case of rozrahunka dovіrchih _intervalіv vznachaєєєєєєєєєєєєєєєє імовірнієєє, for what it is necessary to razrahuvat value z.

Vrakhovuychi those, scho dovirchy _interval folded from the upper and lower boundaries, and those, scho normal rose symmetrically to zero, enough to take the upper boundary (positive recovery). The lower boundary is taken as a negative sign. Significantly trust the imovirnist yak γ (gamma), then the upper boundary of the trusted interval is covered by such a formula.

Razrahuemo in Excel value z(which shows the difference in the average in sigmas) for the number of emotions, including t, as you know, remind me of any statistics: 90%, 95% and 99%. The middle B2 has a significant formula: =NORM.ST.OBR((1+A2)/2). Changing the value of the change (the change in the middle of A2) is taken away by the difference between the intervals.

Confidence interval for 95% confidence is 1.96, so it may be 2 r.m.s. It is easy to put the stars in the mind to estimate the possible rise of normal fluctuations. Confidence intervals of 90%, 95% and 99% are generally consistent with confidence intervals of ±1.64, ±1.96 and ±2.58 σ.

In general, none of the functions NORM.ST.RASP and NORM.ST.OBR allow the growth of any kind of rozrahunok, but with a normal rozpodilom. But, to make it easier to change the number of functions, Excel has a few other functions. For example, for rozrahunku dovirchih іnintervalіv srednої it is possible to beat TRUST.NORM. To recheck the arithmetic mean, the formula Z.TEST.

Let's look at some more brown formulas from butts.

NORM.DIST function

Function NORM.DIST wind up in NORM.ST.DIST deprive us of what її vikoristovuyut for the processing of data, be it on some scale, and not less rationing. The parameters of the normal subdivision are specified in the syntax.

=NORM.DIST(x,average,standard_display,integral)

middle- mathematical refinement, as it is victorious as the first parameter of the normal distribution model

standard_v_dkl– root-mean-square displacement – ​​another parameter of the model

integral- if 0, then the security is secured, if 1 - the value of the function, then. P(X

For example, the value for the value 15, which is similar to the normal selections for the queens 10, the standard allowance 3, is developed as follows:

If the remaining parameter is set to 1, it is assumed that the value of normal deviation will appear less than 15 when the parameters are set to a subdivision. In such a rite, money can be invested without intermediary for weekend tributes.

NORM.INV function

Ce quantile of the normal distribution, tobto. the value of the pivotal function. Advance syntax.

=NORM.OBR(improvement, average, standard_output)

imovirnist- Imovirnist

middle- uterus

standard_v_dkl- root-mean-square ventilation

Assigned to those who NORM.ST.INV, only the function works on a given scale.

Butt testimonies from the video like statistics.

Modeling a normal rose

For some tasks, the generation of normal vipad numbers is necessary. There are no ready functions. However, Excel has two functions to rotate reversed numbers: VIPADMIZHі RAND. The first one is vipadkovі rіvnomіrno razpodіleni tsіlі numbers at the designations of the boundaries. Another function generates evenly distributed subdivision numbers between 0 and 1. RAND.

Let's say, for the experiment, it is necessary to take a selection from a normally distributed general marriage with 10 queens and a standard birth rate of 3. For one vipadical value, we write an Excel formula.

NORM.INV(RAND();10;3)

Let's go for the necessary amount of the middle one and the normal selection is ready.

To model the standardized data, follow the rules NORM.ST.OBR.

The process of converting equal numbers to normal numbers can be shown on the next diagram. In the case of equal dynamics, which are generated by the RAND formula, a horizontal line is drawn to the graph of the function of the normal distribution. Then let's see the point of the crossover of imitations from the graph lowered the projections on the horizontal axis.