Instruction

1

The first is to understand how this formed the sample average. For example, given some set of numeric values that consists of n units. All these units form the so-called sample. The sum of all these numbers is a formula to be expressed as ΣXi (Xi is any of the values of this sample, where i = 1,2,3...i-1,i. That is, i is the number of values in the sample). Then, in order to find the sample mean, you add together all the values from the sample and divided by their number n.

2

On top of all the recorded data can be expressed by only one formula, which is listed above. The sample mean is the simplest of the characteristics that reveal the nature of the sampling. It is widely used in mathematical statistics, probability theory, and also in many other fields of knowledge.

3

In the school curriculum is not given any formulas for finding average, although it becomes immediately clear that when children in math class in 5th grade are asked to find the average value of any numbers the children already know in order to find the average of these numbers, they will need to fold them all and then divided by their number. In fact, they also find the sample mean.

Note

When you look at the formula for the sample mean, then I am immediately reminded of course of school mathematics in which it was necessary to find an arithmetic mean. Indeed, the formula is almost identical, but from a mathematical point of view while the sample mean is not considered some kind of preset already set (as given in the book of problems in mathematics), and the space in which there are many random values. Exploring this space is some kind of sampling frame, from which subsequently is the sample mean.

# Advice 2: How to find the coefficient of variation

Mathematical statistics is impossible without studying

**variation**and, in particular, the calculation**of the coefficient****of variation**. He has received the greatest application in practice due to simple calculation and clarity of results.You will need

- - variation of several numeric values;
- calculator.

Instruction

1

First, find the sample mean. To do this, add up all the values of the variational series and divide them by the number of study items. For example, if you want to find the coefficient

**of variation of**three indicators 85, 88 and 90 to calculate the sample mean we need to add these values and divide by 3: x(CP)=(85+88+90)/3=87,67.2

Then, calculate the margin of error the sample mean (standard deviation). For this purpose from each sample subtract the average value found in first step. Lift all the difference in a square and fold the received results among themselves. You got the numerator of the fraction. In the example the calculation would look like this: (85-87,67)^2+(88-87,67)^2+(90-87,67)^2=(-2,67)^2+0,33^2+2,33^2=7,13+0,11+5,43=12,67.

3

To get the denominator, multiply the number of sample points n (n-1). In the example this would look like 3x(3-1)=3x2=6.

4

Divide the numerator by the denominator and from the resulting numbers to Express a fraction to get the margin of error SX. You get 12,67/6=2 and 11. The root of 2.11 is equal to 1.45.

5

Proceed to the main point: find the coefficient

**of variation**. To do this, divide the margin of error for the sample mean found in the first step. In the example of 2.11/87,67=0,024. To get the result in percent, multiply the resulting number by 100% (0,024х100%=2,4%). You have found the coefficient**of variation**and is equal to 2,4%.6

Please note, the estimated coefficient

**of variation**is quite small, so the variation of the sign is considered weak and the target population can be considered homogeneous. If the ratio exceeded the 0.33 (33%), the average value cannot be considered typical, and study through it the totality of would be wrong.Useful advice

You can check the result by eye, to ensure his fidelity. Rate about the sample elements, if they are almost identical, you should get a small percentage. The greater the scatter of the index, the greater the coefficient of variation.

# Advice 3: How to find confidence interval

The objective of any statistical calculations is to build a probabilistic model of a random event. This allows you to collect and analyze data on specific observations or experiments. A confidence

**interval**is used when a small sample that allows to determine with a high degree of reliability.You will need

- - table of the function values of the Laplace.

Instruction

1

The confidence

**interval**in probability theory is used to estimate the mathematical expectation. In relation to the parameter analyzed by statistical methods, this is**the interval**that covers the value of this quantity with a given accuracy (the degree or level of reliability).2

Let the random variable x is normally distributed and known standard deviation. Then the condence

**interval**is equal to: m(x) – t·σ/√n < M(x) < m(x) + t·σ/√n, where m(x) – sample mean of sample x, t is the argument of the function Laplace, σ – standard deviation, n – sample size, M(x) is the mathematical expectation. The expression standing on the left and right of M(x) are called the confidence limits.3

The Laplace function is used in the formula to determine the probability of a parameter value in this

**interval**. Generally, when solving such problems requires either to calculate the function via argument, or Vice versa. The formula for finding features is a rather cumbersome integral, therefore for simplification of work with probabilistic models, use a ready-made table of values.4

Example:Find confidence

**interval**confidence level of 0.9 for the estimated sign of some General sample x, if it is known that the standard deviation σ is equal to 5, the sample mean m(x) = 20, size n = 100.5

Solution:Identify the values involved in the formula are unknown to you. In this case, the mathematical expectation and the Laplace argument.

6

The statement says that the value of the function equal to 0.9, therefore, determine t from the table:Φ(t) = 0,9 → t = 1,65.

7

Substitute all known numbers into the formula and calculate the confidence limits:20 – 1,65·5/10 < M(x) < 20 + 1,65·5/1019,175 < M(x) < 20,825.

# Advice 4: How to calculate standard deviation

The standard

**deviation of**the term probability theory and mathematical statistics, measure of spread of values of a random variable around its expected value.Instruction

1

The standard

**deviation**calculated when carrying out statistical tests of various hypotheses, and to detect relationships between random variables, build confidence intervals, etc. This statistic is the most common type of variance used in the calculations, it is particularly convenient when tabular calculations.2

Together with the concept of standard deviation is useful to consider other statistical concept of a sample. This term is used to indicate the sampling results of homogeneous observations. Mathematically, sampling is a sequence X whose elements are random variables x1, x2, ..., xn taken at random from a finite set of observations.

3

There are several formulas to calculate the standard deviation: the classic, the formula using the value of the average value without it. Respectively:σ = √(∑(x_i – HSR)2/(n - 1));σ = √((∑x_i2 – n·х_ср2)/(n - 1));σ = √((∑x_i2 – ((∑x_i)2/n)/(n - 1)).

4

Depending on the task, you can use a particular formula example: let the given histogram table of the distribution of the random variable consisting of the column values themselves are the values and column percentage frequencies of each value, which we denote by p_i. Find the standard

**deviation**by the formula using the average value.5

Solution.To solve the problem, you must determine the average value of a random variable:HSR = ∑p_i·x_i/∑p_i,

6

For convenience, complete the table multiple columns, this will facilitate the solution of the problem. In the third column write down the works of p_i·x_i, i.e. the values of the first and second columns. The fourth column of the complete works of p_i·x_i2. Now append the row with the sum of values of columns 2-4. It is convenient to do in a computer program, such as Microsoft Excel.

7

We can now calculate the standard

**deviation**by the formula, substituting the appropriate values from the table.:σ = √(∑p_i·x_i2 - ((∑p_i·x_i)2/∑p_i)/∑p_i).