What is skew in statistics. We analyze the formulas for standard deviation and dispersion in Excel
To calculate the simple geometric mean, the formula is used:
Geometric weighted
To determine the weighted geometric mean, the formula is used:
The average diameters of wheels, pipes, and the average sides of squares are determined using the mean square.
Root-mean-square values are used to calculate some indicators, for example, the coefficient of variation, which characterizes the rhythm of production. Here the standard deviation from the planned production output for a certain period is determined using the following formula:
These values accurately characterize the change in economic indicators compared to their base value, taken in its average value.
Quadratic simple
The root mean square is calculated using the formula:
Quadratic weighted
The weighted mean square is equal to:
22. Absolute indicators of variation include:
range of variation
average linear deviation
dispersion
standard deviation
Range of variation (r)
Range of variation- is the difference between the maximum and minimum values of the attribute
It shows the limits within which the value of a characteristic changes in the population being studied.
The work experience of the five applicants in previous work is: 2,3,4,7 and 9 years. Solution: range of variation = 9 - 2 = 7 years.
For a generalized description of differences in attribute values, average variation indicators are calculated based on taking into account deviations from the arithmetic mean. The difference is taken as a deviation from the average.
In this case, in order to avoid the sum of deviations of variants of a characteristic from the average turning to zero (zero property of the average), one must either ignore the signs of the deviation, that is, take this sum modulo , or square the deviation values
Average linear and square deviation
Average linear deviation is the arithmetic average of the absolute deviations of individual values of a characteristic from the average.
The average linear deviation is simple:
The work experience of the five applicants in previous work is: 2,3,4,7 and 9 years.
In our example: years;
Answer: 2.4 years.
Average linear deviation weighted applies to grouped data:
Due to its convention, the average linear deviation is used in practice relatively rarely (in particular, to characterize the fulfillment of contractual obligations regarding uniformity of delivery; in the analysis of product quality, taking into account the technological features of production).
Standard deviation
The most perfect characteristic of variation is the mean square deviation, which is called the standard (or standard deviation). Standard deviation() is equal to the square root of the average square deviation of the individual values of the arithmetic average attribute:
The standard deviation is simple:
Weighted standard deviation is applied to grouped data:
Between the root mean square and mean linear deviations under normal distribution conditions the following ratio takes place: ~ 1.25.
The standard deviation, being the main absolute measure of variation, is used in determining the ordinate values of a normal distribution curve, in calculations related to the organization of sample observation and establishing the accuracy of sample characteristics, as well as in assessing the limits of variation of a characteristic in a homogeneous population.
18. Standard deviation, calculation method, value.
An approximate method for assessing the variability of a variation series is to determine the limit and amplitude, but the values of the variant within the series are not taken into account. The main generally accepted measure of the variability of a quantitative characteristic within a variation series is standard deviation (σ - sigma). The larger the standard deviation, the higher the degree of fluctuation of this series.
The method for calculating the standard deviation includes the following steps:
1. Find the arithmetic mean (M).
2. Determine the deviations of individual options from the arithmetic mean (d=V-M). In medical statistics, deviations from the average are designated as d (deviate). The sum of all deviations is zero.
3. Square each deviation d 2.
4. Multiply the squares of the deviations by the corresponding frequencies d 2 *p.
5. Find the sum of the products (d 2 *p)
6. Calculate the standard deviation using the formula:
when n is greater than 30, or when n is less than or equal to 30, where n is the number of all options.
Standard deviation value:
1. The standard deviation characterizes the spread of the variant relative to the average value (i.e., the variability of the variation series). The larger the sigma, the higher the degree of diversity of this series.
2. The standard deviation is used for a comparative assessment of the degree of correspondence of the arithmetic mean to the variation series for which it was calculated.
Variations of mass phenomena obey the law of normal distribution. The curve representing this distribution looks like a smooth bell-shaped symmetrical curve (Gaussian curve). According to the theory of probability, in phenomena that obey the law of normal distribution, there is a strict mathematical relationship between the values of the arithmetic mean and the standard deviation. The theoretical distribution of a variant in a homogeneous variation series obeys the three-sigma rule.
If in a system of rectangular coordinates the values of a quantitative characteristic (variants) are plotted on the abscissa axis, and the frequency of occurrence of a variant in a variation series is plotted on the ordinate axis, then variants with larger and smaller values are evenly located on the sides of the arithmetic mean.
It has been established that with a normal distribution of the trait:
68.3% of the values of the option are within M1
95.5% of the values of the option are within M2
99.7% of the values of the option are within M3
3. The standard deviation allows you to establish normal values for clinical and biological parameters. In medicine, the interval M1 is usually taken as the normal range for the phenomenon being studied. The deviation of the estimated value from the arithmetic mean by more than 1 indicates a deviation of the studied parameter from the norm.
4. In medicine, the three-sigma rule is used in pediatrics for individual assessment of the level of physical development of children (sigma deviation method), for the development of standards for children's clothing
5. The standard deviation is necessary to characterize the degree of diversity of the characteristic being studied and to calculate the error of the arithmetic mean.
The value of the standard deviation is usually used to compare the variability of series of the same type. If two series with different characteristics are compared (height and weight, average duration of hospital treatment and hospital mortality, etc.), then a direct comparison of sigma sizes is impossible , because standard deviation is a named value expressed in absolute numbers. In these cases, use coefficient of variation (Cv), which is a relative value: the percentage ratio of the standard deviation to the arithmetic mean.
The coefficient of variation is calculated using the formula:
The higher the coefficient of variation , the greater the variability of this series. It is believed that a coefficient of variation of more than 30% indicates the qualitative heterogeneity of the population.
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X i - random (current) variables;
X̅– the average value of random variables for the sample is calculated using the formula:
So, variance is the average square of deviations . That is, the average value is first calculated, then taken the difference between each original and average value is squared , is added and then divided by the number of values in the population.
The difference between an individual value and the average reflects the measure of deviation. It is squared so that all deviations become exclusively positive numbers and to avoid mutual destruction of positive and negative deviations when summing them up. Then, given the squared deviations, we simply calculate the arithmetic mean.
The answer to the magic word “dispersion” lies in just these three words: average - square - deviations.
Standard deviation (MSD)
Taking the square root of the variance, we obtain the so-called “ standard deviation". There are names "standard deviation" or "sigma" (from the name of the Greek letter σ .). The formula for the standard deviation is:
So, dispersion is sigma squared, or is the standard deviation squared.
The standard deviation, obviously, also characterizes the measure of data dispersion, but now (unlike dispersion) it can be compared with the original data, since they have the same units of measurement (this is clear from the calculation formula). The range of variation is the difference between extreme values. Standard deviation, as a measure of uncertainty, is also involved in many statistical calculations. With its help, the degree of accuracy of various estimates and forecasts is determined. If the variation is very large, then the standard deviation will also be large, and therefore the forecast will be inaccurate, which will be expressed, for example, in very wide confidence intervals.
Therefore, in methods of statistical data processing in real estate assessments, depending on the required accuracy of the task, the two or three sigma rule is used.
To compare the two-sigma rule and the three-sigma rule, we use Laplace’s formula:
F - F ,
where Ф(x) is the Laplace function;
Minimum value
β = maximum value
s = sigma value (standard deviation)
a = average
In this case, a particular form of Laplace’s formula is used when the boundaries α and β of the values of the random variable X are equally spaced from the center of the distribution a = M(X) by a certain value d: a = a-d, b = a+d.  Or   (1) Formula (1) determines the probability of a given deviation d of a random variable X with a normal distribution law from its mathematical expectation M(X) = a. If in formula (1) we take sequentially d = 2s and d = 3s, we obtain: (2), (3). |
Two sigma rule
It can be almost reliably (with a confidence probability of 0.954) that all values of a random variable X with a normal distribution law deviate from its mathematical expectation M(X) = a by an amount not greater than 2s (two standard deviations). Confidence probability (Pd) is the probability of events that are conventionally accepted as reliable (their probability is close to 1).
Let's illustrate the two-sigma rule geometrically. In Fig. Figure 6 shows a Gaussian curve with the distribution center a. The area limited by the entire curve and the Ox axis is equal to 1 (100%), and the area of the curvilinear trapezoid between the abscissas a–2s and a+2s, according to the two-sigma rule, is equal to 0.954 (95.4% of the total area). The area of the shaded areas is 1-0.954 = 0.046 (»5% of the total area). These areas are called the critical region of the random variable. Values of a random variable falling into the critical region are unlikely and in practice are conventionally accepted as impossible.
The probability of conditionally impossible values is called the significance level of a random variable. The significance level is related to the confidence probability by the formula:
where q is the significance level expressed as a percentage.
Three sigma rule
When solving issues that require greater reliability, when the confidence probability (Pd) is taken equal to 0.997 (more precisely, 0.9973), instead of the two-sigma rule, according to formula (3), the rule is used three sigma
According to three sigma rule with a confidence probability of 0.9973, the critical area will be the area of attribute values outside the interval (a-3s, a+3s). The significance level is 0.27%.
In other words, the probability that the absolute value of the deviation will exceed three times the standard deviation is very small, namely 0.0027 = 1-0.9973. This means that only 0.27% of cases will this happen. Such events, based on the principle of the impossibility of unlikely events, can be considered practically impossible. Those. sampling is highly accurate.
This is the essence of the three sigma rule:
If a random variable is distributed normally, then the absolute value of its deviation from the mathematical expectation does not exceed three times the standard deviation (MSD).
In practice, the three-sigma rule is applied as follows: if the distribution of the random variable being studied is unknown, but the condition specified in the above rule is met, then there is reason to assume that the variable being studied is normally distributed; otherwise it is not normally distributed.
The level of significance is taken depending on the permitted degree of risk and the task at hand. For real estate valuation, a less precise sample is usually adopted, following the two-sigma rule.
The root mean square or standard deviation is a statistical indicator that evaluates the amount of fluctuation of a numerical sample around its average value. Almost always, the majority of values are distributed within plus or minus one standard deviation from the mean.
Definition
The standard deviation is the square root of the arithmetic mean of the sum of squared deviations from the mean. Strict and mathematical, but absolutely incomprehensible. This is a verbal description of the formula for calculating standard deviation, but to understand the meaning of this statistical term, let's understand everything in order.
Imagine a shooting range, a target and an arrow. The sniper shoots at a standard target, where hitting the center gives 10 points, depending on the distance from the center the number of points decreases, and hitting the extreme areas gives only 1 point. Each shooter's shot is a random integer value between 1 and 10. A target riddled with bullets is a perfect illustration of the distribution of a random variable.
Expected value
Our novice shooter practiced shooting for a long time and noticed that he hit different values with a certain probability. Let's say, based on a large number of shots, he found out that he hits 10 with a 15% probability. The remaining values received their probabilities:
- 9 - 25 %;
- 8 - 20 %;
- 7 - 15 %;
- 6 - 15 %;
- 5 - 5 %;
- 4 - 5 %.
Now he is preparing to take another shot. Which value is he most likely to hit? The mathematical expectation will help us answer this question. Knowing all these probabilities, we can determine the most likely outcome of the shot. The formula for calculating the mathematical expectation is quite simple. Let's denote the shot value as C and the probability as p. The mathematical expectation will be equal to the sum of the product of the corresponding values and their probabilities:
Let's define the expectation for our example:
- M = 10 × 0.15 + 9 × 0.25 + 8 × 0.2 + 7 × 0.15 + 6 × 0.15 + 5 × 0.05 + 4 × 0.05
- M = 7.75
So, it is most likely that the shooter will hit the 7 point zone. This area will be the most heavily shot, which is an excellent result of the most frequent hits. For any random variable, the expected value means the most common value or the center of all values.
Dispersion
Dispersion is another statistical indicator that illustrates the spread of a value. Our target is densely riddled with bullets, and the dispersion allows us to express this parameter numerically. If the mathematical expectation shows the center of the shots, then the dispersion is their spread. In essence, dispersion means the mathematical expectation of deviations of values from the expected value, that is, the average square of deviations. Each value is squared so that the deviations are only positive and do not cancel each other in the case of identical numbers with opposite signs.
D[X] = M − (M[X]) 2
Let's calculate the spread of shots for our case:
- M = 10 2 × 0.15 + 9 2 × 0.25 + 8 2 × 0.2 + 7 2 × 0.15 + 6 2 × 0.15 + 5 2 × 0.05 + 4 2 × 0.05
- M = 62.85
- D[X] = M − (M[X]) 2 = 62.85 − (7.75) 2 = 2.78
So our deviation is 2.78. This means that from the area on the target with a value of 7.75, the bullet holes are spread out by 2.78 points. However, in its pure form, the variance value is not used - the result is the square of the value, in our example this is a square point, but in other cases it could be square kilograms or square dollars. Dispersion as a square value is not informative, so it represents an intermediate indicator for determining the standard deviation - the hero of our article.
Standard deviation
To convert variance into meaningful points, kilograms, or dollars, we use standard deviation, which is the square root of the variance. Let's calculate it for our example:
S = sqrt(D) = sqrt(2.78) = 1.667
We received the points and can now use them to connect with the mathematical expectation. The most likely result of the shot in this case would be expressed as 7.75 plus or minus 1.667. This is enough to answer, but we can also say that it is almost certain that the shooter will hit the target area between 6.08 and 9.41.
Standard deviation or sigma is an informative indicator that illustrates the spread of a value relative to its center. The larger the sigma, the greater the spread the sample shows. This is a well-studied coefficient and the interesting three-sigma rule is known for the normal distribution. It has been established that 99.7% of the values of a normally distributed quantity lie in the region of plus or minus three sigma from the arithmetic mean.
Let's look at an example
Currency pair volatility
It is known that the methods of mathematical statistics are widely used in the foreign exchange market. Many trading terminals have built-in tools for calculating the volatility of an asset, which demonstrates a measure of the volatility of the price of a currency pair. Of course, financial markets have their own specifics for calculating volatility, such as the opening and closing prices of stock exchanges, but as an example, we can calculate the sigma for the last seven daily candles and roughly estimate the weekly volatility.
The pound/yen currency pair is rightfully considered the most volatile asset in the Forex market. Suppose that theoretically during the week the closing price of the Tokyo Stock Exchange took the following values:
145, 147, 146, 150, 152, 149, 148.
Let's enter this data into the calculator and calculate the sigma equal to 2.23. This means that on average the Japanese yen changed by 2.23 yen every day. If everything was so wonderful, traders would make millions from such movements.
Conclusion
Standard deviation is used in statistical analysis of numerical samples. This is a useful coefficient for assessing the spread of data, since two sets with seemingly the same mean value can be completely different in the spread of values. Use our calculator to find small sample sigmas.
In statistical testing of hypotheses, when measuring a linear relationship between random variables.
Standard deviation:
Standard deviation(estimate of the standard deviation of the random variable Floor, the walls around us and the ceiling, x relative to its mathematical expectation based on an unbiased estimate of its variance):
where is the dispersion; - The floor, the walls around us and the ceiling, i th element of the selection; - sample size; - arithmetic mean of the sample:
It should be noted that both estimates are biased. In the general case, it is impossible to construct an unbiased estimate. However, the estimate based on the unbiased variance estimate is consistent.
Three sigma rule
Three sigma rule() - almost all values of a normally distributed random variable lie in the interval. More strictly - with no less than 99.7% confidence, the value of a normally distributed random variable lies in the specified interval (provided that the value is true and not obtained as a result of sample processing).
If the true value is unknown, then we should use not, but the Floor, the walls around us and the ceiling, s. Thus, the rule of three sigma is transformed into the rule of three Floor, walls around us and the ceiling, s .
Interpretation of the standard deviation value
A large value of the standard deviation shows a large spread of values in the presented set with the average value of the set; a small value, accordingly, shows that the values in the set are grouped around the middle value.
For example, we have three number sets: (0, 0, 14, 14), (0, 6, 8, 14) and (6, 6, 8, 8). All three sets have mean values equal to 7, and standard deviations, respectively, equal to 7, 5 and 1. The last set has a small standard deviation, since the values in the set are grouped around the mean value; the first set has the largest standard deviation value - the values within the set diverge greatly from the average value.
In a general sense, standard deviation can be considered a measure of uncertainty. For example, in physics, standard deviation is used to determine the error of a series of successive measurements of some quantity. This value is very important for determining the plausibility of the phenomenon under study in comparison with the value predicted by the theory: if the average value of the measurements differs greatly from the values predicted by the theory (large standard deviation), then the obtained values or the method of obtaining them should be rechecked.
Practical use
In practice, standard deviation allows you to determine how much the values in a set may differ from the average value.
Climate
Suppose there are two cities with the same average maximum daily temperature, but one is located on the coast and the other is inland. It is known that cities located on the coast have many different maximum daytime temperatures that are lower than cities located inland. Therefore, the standard deviation of the maximum daily temperatures for a coastal city will be less than for the second city, despite the fact that the average value of this value is the same, which in practice means that the probability that the maximum air temperature on any given day of the year will be higher differ from the average value, higher for a city located inland.
Sport
Let's assume that there are several football teams that are rated on some set of parameters, for example, the number of goals scored and conceded, scoring chances, etc. It is most likely that the best team in this group will have better values on more parameters. The smaller the team’s standard deviation for each of the presented parameters, the more predictable the team’s result is; such teams are balanced. On the other hand, a team with a large standard deviation is difficult to predict the result, which in turn is explained by an imbalance, for example, a strong defense but a weak attack.
Using the standard deviation of team parameters makes it possible, to one degree or another, to predict the result of a match between two teams, assessing the strengths and weaknesses of the teams, and therefore the chosen methods of fighting.
Technical analysis
see also
Literature
| This article is proposed for deletion.
An explanation of the reasons and the corresponding discussion can be found on the page Wikipedia: To be deleted/December 17, 2012. |
* Borovikov, V. STATISTICS. The art of data analysis on a computer: For professionals / V. Borovikov. - St. Petersburg. : Peter, 2003. - 688 p. - ISBN 5-272-00078-1.
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