The probability distribution is the exhaustive characteristic of a random variable, however in some cases it happens to specify enough the numerical characteristics expressing some essential features of distribution of probability of a random variable. As such characteristics the moments of distribution of various orders are considered. The moment mk[x] of order k (about of origin) about discrete random variable x name the sum:
, (1)
Where xi - all possible values of a random variable, pi probability of random variable, n - volume of sample. The first moment m1[x] about origin refers to as a expection (mean) of random variable x and is designated M[x] or mx.
The population mean is the characteristic of position of a random variable on a numerical axis. At great volume of sample arithmetic mean, considered under the formula (2), will converge to its population mean(pin-1).
(2)
Sometimes with a view of reception of the best conformity of the probability distribution of empirical lines to some theoretical circuits transformation of initial set is carried out. For example, for transformation of some x1,x2,...xn in a line lgx1, lg x2 ,... lg xn arithmetic mean mlgx is equal
(3)
Then size G represents geometrical mean.
, x>0 (4)
For transformation of some x1,x2,...xn in a line 
with arithmetic mean m1/x.
(5)
H harmonious mean.
(6)
Except for a expectation (mean) of a random variable other characteristics of position, in particular a mode and a median are sometimes used also. A median - the characteristic of the center of grouping which is equal to value of a member ranged on decrease or increase of the lines occupying average position. For lines in volume 2m+1 - Me=xm+1, for lines with volume 2m - Me=0.5 (xm+xm+1). A mode - the most probable (most frequently meeting) size in the given statistical line, representing the greatest ordinate of a curve of distribution in case of one-topmost distribution. At presence of several tops the curve of distribution will have accordingly and some fashions.
Central moment mk[x] of k-order about random variable x name the moment of k-order the order random variable x*=x - mx:
(7)
For a discrete random variable:
, (8)
(in the elementary case is accepted, as well as the formula (2), p=n-1)
Between the initial and central moments there are following ratio:
(9)
(10)
(11)
The central moment of 2-order refers to as a variance and is designated D[x] or Dx. The variance of a random variable serves as a measure of its variance. It characterizes disorder of values of a random variable concerning its population mean. The variance has dimension of a square of size, therefore in some cases it is more convenient to use other characteristic of variance - standard deviation s.
(12)
Comparison of lines to different absolute values of random variables is convenient for carrying out with the help of coefficient of variation Cv.
(13)
The considered measures of position and variance not completely describe properties of statistical lines. As the characteristic of asymmetry of probability distributions the coefficient of asymmetry Cs is used.
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(14)
For the characteristic of a degree of a smoothness considered distribution in comparison with normal the coefficient of excess is used. Evasion of this parameter from zero testifies to available differences.
(15)
For system of random variables, as well as for one random variable, as numerical characteristics the moment about origin and central moments of probability distribution are used. The second mixed moment of system of two random variables, determined under the formula (16), refers to as the covariance.
(16)
For independent random variables cov(x,y)=m1[x]m1[h] =0. The size rxy refers to as correlation coefficient of random variables x and h.
(17)
For independent random variables rxy =0. This necessary condition of independence, but not sufficient. On calculations of correlation coefficient it is convenient to use the formula (18).
(18)
When it is required to estimate connection inside lines, for calculation of autocorrelation coefficient the formula (19) is used.
, (19)
where 
, 
.
Listed in the this work of statistics are dot estimations (i.e. are defined by one number) and of a solvency, unbiased and efficiency should meet the requirements: 1) solvent the estimation aspiring on probability to the estimated parameter with increase of volume of sample refers to; 2) unbiased the estimation which population mean is equal to the estimated parameter at any volume of sample refers to; 3) effective the estimation possessing the minimal variance at fixed number of supervision refers to.
In practice use formulas:
(20)
, (21)
Where Ds and Css - the displaced selective estimations.
For estimating of statistics use the following methods: 1) a method of the moments; 2) a graphic method; 3) a method of the greatest plausibility.
It is given: materials of observation (tab. 1, 2); number-1 designates misses in observation.
The task:
1) Define a mean and a variance;
2) Write the formula for calculation (unbiased) variance, a standard deviation, coefficient of a variation, coefficient of asymmetry and coefficient of an excess as, similar to expression (19; by example: where is ..);
3) Fill in available misses as arithmetic mean of two nearest real values;
4) Calculate a method of the moments for considered lines of the characteristic in view of amendments on biasing arithmetic mean, geometrical mean, harmonious mean, a variance, a standard deviation, coefficient of a variation, coefficient of asymmetry, coefficient of an excess, coefficient of autocorrelation (for one year).
