Standard deviation(synonyms: standard deviation, standard deviation, square deviation; related terms: standard deviation, standard spread) - in probability theory and statistics, the most common indicator of the dispersion of the values ​​of a random variable relative to its mathematical expectation. For limited arrays of value samples, instead of mathematical expectation the arithmetic mean of the sample population is used.

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  The standard deviation is measured in units of measurement of the random variable itself and is used when calculating the standard error of the arithmetic mean, when constructing confidence intervals, when statistically testing hypotheses, when measuring the linear relationship between random variables. Defined as the square root of the variance of a random variable.

  Standard deviation:

  s = n n − 1 σ 2 = 1 n − 1 ∑ i = 1 n (x i − x ¯) 2 ; (\displaystyle s=(\sqrt ((\frac (n)(n-1))\sigma ^(2)))=(\sqrt ((\frac (1)(n-1))\sum _( i=1)^(n)\left(x_(i)-(\bar (x))\right)^(2)));)
  • Note: Very often there are discrepancies in the names of MSD (Root Mean Square Deviation) and STD (Standard Deviation) with their formulas. For example, in the numPy module of the Python programming language, the std() function is described as "standard deviation", while the formula reflects the standard deviation (division by the root of the sample). In Excel, the STANDARDEVAL() function is different (division by the root of n-1).

  Standard deviation (estimate of the standard deviation of a random variable x relative to its mathematical expectation based on an unbiased estimate of its variance) s (\displaystyle s):

  σ = 1 n ∑ i = 1 n (x i − x ¯) 2 . (\displaystyle \sigma =(\sqrt ((\frac (1)(n))\sum _(i=1)^(n)\left(x_(i)-(\bar (x))\right) ^(2))).)

  Where σ 2 (\displaystyle \sigma ^(2))- dispersion; x i (\displaystyle x_(i)) - i th element of the selection; n (\displaystyle n)- sample size; - arithmetic mean of the sample:

  x ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + … + x n) . (\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\ldots +x_(n)).)

  It should be noted that both estimates are biased. IN general case It is impossible to construct an unbiased estimate. However, the estimate based on the unbiased variance estimate is consistent.

  In accordance with GOST R 8.736-2011, the standard deviation is calculated using the second formula of this section. Please check the results.

  Three sigma rule

  Three sigma rule (3 σ (\displaystyle 3\sigma )) - almost all values ​​of a normally distributed random variable lie in the interval (x ¯ − 3 σ ; x ¯ + 3 σ) (\displaystyle \left((\bar (x))-3\sigma ;(\bar (x))+3\sigma \right)). More strictly - with approximately probability 0.9973, the value of a normally distributed random variable lies in the specified interval (provided that the value x ¯ (\displaystyle (\bar (x))) true, and not obtained as a result of sample processing).

  If the true value x ¯ (\displaystyle (\bar (x))) is unknown, then you should not use σ (\displaystyle \sigma ), A s. Thus, rule of three sigma is converted to the rule of three s .

  Interpretation of the standard deviation value

  A larger standard deviation value shows a greater spread of values ​​in the presented set with the average value of the set; a smaller value, accordingly, shows that the values ​​in the set are grouped around the average 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 most great importance standard deviation - 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. identified with portfolio risk.

  Climate

  Suppose there are two cities with the same average maximum daily temperature, but one is located on the coast and the other on the plain. 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 maximum daily temperatures for a coastal city will be less than for a second city, despite the fact that their average value is the same, which in practice means that the probability that Maximum temperature air of each specific day of the year will differ more strongly from the average value, higher for a city located inside the continent.

  Sport

  Let's assume that there are several football teams that are evaluated according to 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 best values according to 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, the team with great value standard deviation is difficult to predict the result, which in turn is explained by the imbalance, for example, strong defense, but with 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 weak sides commands, and therefore the chosen methods of struggle.

  Material from Wikipedia - the free encyclopedia

  Standard deviation(synonyms: standard deviation, standard deviation, square deviation; related terms: standard deviation, standard spread) - in probability theory and statistics the most common indicator of the dispersion of the values ​​of a random variable relative to its mathematical expectation. With limited arrays of samples of values, instead of the mathematical expectation, the arithmetic mean of the set of samples is used.

  Basic information

  The standard deviation is measured in units of the random variable itself and is used when calculating the standard error of the arithmetic mean, when constructing confidence intervals, when statistically testing hypotheses, when measuring the linear relationship between random variables. Defined as the square root of the variance of a random variable.

  Standard deviation:

  $\backslash sigma=\backslash sqrt(\backslash frac(1)(n)\backslash sum\_(i=1)^n\backslash left(x\_i-\backslash bar(x)\backslash right)^2).$

  Standard deviation(estimate of the standard deviation of a random variable x relative to its mathematical expectation based on an unbiased estimate of its variance) $s$:

  $s=\backslash sqrt(\backslash frac(n)(n-1)\backslash sigma^2)=\backslash sqrt(\backslash frac(1)(n-1)\backslash sum\_(i=1)^n\backslash left(x\_i-\backslash bar\; (x)\backslash right)^2);$

  Three sigma rule

  Three sigma rule ($3\backslash sigma$) - almost all values ​​of a normally distributed random variable lie in the interval $\backslash left(\backslash bar(x)-3\backslash sigma;\backslash bar(x)+3\backslash sigma\backslash right)$. More strictly - with approximately a probability of 0.9973, the value of a normally distributed random variable lies in the specified interval (provided that the value $\backslash bar(x)$ true, and not obtained as a result of sample processing).

  If the true value $\backslash bar(x)$ is unknown, then you should not use $\backslash sigma$, A s. Thus, the rule of three sigma is transformed into the rule of three s .

  Interpretation of the standard deviation value

  A larger standard deviation value shows a greater spread of values ​​in the presented set with the average value of the set; a smaller value, accordingly, shows that the values ​​in the set are grouped around the average 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 estimate how much values ​​from a set may differ from the average value.

  Economics and finance

  Standard deviation of portfolio return $\backslash sigma\; =\backslash sqrt(D[X])$ identified with portfolio risk.

  Climate

  Suppose there are two cities with the same average maximum daily temperature, but one is located on the coast and the other on the plain. 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.

  see also

  Write a review about the article "Root Mean Square Deviation"

  Literature

  • 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..

  Statistical indicators Descriptive statistics Statistical output and examination hypotheses Overall rating Correlation and regression Graphic methods

  An excerpt characterizing Standard Deviation

  And, quickly opening the door, he stepped out onto the balcony with decisive steps. The conversation suddenly stopped, hats and caps were taken off, and all eyes rose to the count who had come out.
  - Hello guys! - the count said quickly and loudly. - Thank you for coming. I’ll come out to you now, but first of all we need to deal with the villain. We need to punish the villain who killed Moscow. Wait for me! “And the count just as quickly returned to his chambers, slamming the door firmly.
  A murmur of pleasure ran through the crowd. “That means he will control all the villains! And you say French... he’ll give you the whole distance!” - people said, as if reproaching each other for their lack of faith.
  A few minutes later an officer hurriedly came out of the front doors, ordered something, and the dragoons stood up. The crowd from the balcony eagerly moved towards the porch. Walking out onto the porch with angry, quick steps, Rostopchin hurriedly looked around him, as if looking for someone.
  - Where is he? - said the count, and at the same minute as he said this, he saw from around the corner of the house two dragoons coming out between young man with a long thin neck, with a half-shaved and overgrown head. This young man was dressed in what had once been a dandyish, blue cloth-covered, shabby fox sheepskin coat and dirty prisoner's harem trousers, stuffed into uncleaned, worn-out thin boots. Shackles hung heavily on his thin, weak legs, making it difficult for the young man to walk indecisively.
  - A! - said Rastopchin, hastily turning his gaze away from the young man in the fox sheepskin coat and pointing to the bottom step of the porch. - Put it here! - The young man, clanking his shackles, stepped heavily onto the indicated step, holding the collar of his sheepskin coat with his finger, turned it twice long neck and, sighing, folded his thin, idle hands in front of his stomach with a submissive gesture.
  Silence continued for several seconds while the young man positioned himself on the step. Only in the back rows of people squeezing into one place were groans, groans, tremors and the tramp of moving feet heard.
  Rastopchin, waiting for him to stop at the indicated place, frowned and rubbed his face with his hand.
  - Guys! - said Rastopchin in a metallic ringing voice, - this man, Vereshchagin, is the same scoundrel from whom Moscow perished.
  A young man in a fox sheepskin coat stood in a submissive pose, clasping his hands together in front of his stomach and bending slightly. Emaciated, with a hopeless expression, disfigured by a shaved head young face it was lowered down. At the first words of the count, he slowly raised his head and looked down at the count, as if wanting to tell him something or at least meet his gaze. But Rastopchin did not look at him. On the young man’s long thin neck, like a rope, the vein behind the ear became tense and turned blue, and suddenly his face turned red.
  All eyes were fixed on him. He looked at the crowd, and, as if encouraged by the expression that he read on the faces of the people, he smiled sadly and timidly and, again lowering his head, adjusted his feet on the step.
  “He betrayed his tsar and his fatherland, he handed himself over to Bonaparte, he alone of all Russians disgraced the name of the Russian, and Moscow is perishing from him,” said Rastopchin in an even, sharp voice; but suddenly he quickly looked down at Vereshchagin, who continued to stand in the same submissive pose. As if this look had exploded him, he, raising his hand, almost shouted, turning to the people: “Deal with him with your judgment!” I'm giving it to you!
  The people were silent and only pressed each other closer and closer. Holding each other, breathing in this infected stuffiness, not having the strength to move and waiting for something unknown, incomprehensible and terrible became unbearable. The people standing in the front rows, who saw and heard everything that was happening in front of them, all with fearfully wide-open eyes and open mouths, straining all their strength, held back the pressure of those behind them on their backs.
  - Beat him!.. Let the traitor die and not disgrace the name of the Russian! - shouted Rastopchin. - Ruby! I order! - Hearing not words, but the angry sounds of Rastopchin’s voice, the crowd groaned and moved forward, but stopped again.
  “Count!..” said Vereshchagin’s timid and at the same time theatrical voice amidst the momentary silence that ensued again. “Count, one god is above us...” said Vereshchagin, raising his head, and again the thick vein on his thin neck filled with blood, and the color quickly appeared and ran away from his face. He didn't finish what he wanted to say.
  - Chop him! I order!.. - shouted Rastopchin, suddenly turning pale just like Vereshchagin.
  - Sabers out! - the officer shouted to the dragoons, drawing his saber himself.
  Another even stronger wave swept through the people, and, reaching the front rows, this wave moved the front rows, staggering, and brought them to the very steps of the porch. A tall fellow, with a petrified expression on his face and a stopped raised hand, stood next to Vereshchagin.
  - Ruby! - Almost an officer whispered to the dragoons, and one of the soldiers suddenly, with his face distorted with anger, hit Vereshchagin on the head with a blunt broadsword.
  "A!" - Vereshchagin cried out briefly and in surprise, looking around in fear and as if not understanding why this was done to him. The same groan of surprise and horror ran through the crowd.
  "Oh my God!" – someone’s sad exclamation was heard.
  But following the exclamation of surprise that escaped Vereshchagin, he cried out pitifully in pain, and this cry destroyed him. That barrier stretched to the highest degree human feeling, which was still holding the crowd, broke through instantly. The crime had been started, it was necessary to complete it. The pitiful groan of reproach was drowned out by the menacing and angry roar of the crowd. Like the last seventh wave, breaking ships, this last unstoppable wave rose from the rear ranks, reached the front ones, knocked them down and swallowed everything. The dragoon who struck wanted to repeat his blow. Vereshchagin, with a cry of horror, shielding himself with his hands, rushed towards the people. The tall fellow he bumped into grabbed Vereshchagin’s thin neck with his hands and, with a wild cry, he and he fell under the feet of the crowd of roaring people.
  Some beat and tore Vereshchagin, others were tall and small. And the cries of the crushed people and those who tried to save the tall fellow only aroused the rage of the crowd. For a long time the dragoons could not free the bloodied, beaten half to death factory worker. And for a long time, despite all the feverish haste with which the crowd tried to complete the work once begun, those people who beat, strangled and tore Vereshchagin could not kill him; but the crowd pressed them from all sides, with them in the middle, like one mass, swaying from side to side and did not give them the opportunity to either finish him off or throw him.

  Instructions

  Let there be several numbers characterizing homogeneous quantities. For example, the results of measurements, weighing, statistical observations and so on. All quantities presented must be measured using the same measurement. To find the standard deviation, do the following:

  Determine the arithmetic mean of all numbers: add all the numbers and divide the sum by total numbers.

  Determine the dispersion (scatter) of numbers: add the squares of the previously found deviations and divide the resulting sum by the number of numbers.

  There are seven patients in the ward with temperatures of 34, 35, 36, 37, 38, 39 and 40 degrees Celsius.

  It is required to determine the average deviation from the mean.
  Solution:
  “in the ward”: (34+35+36+37+38+39+40)/7=37 ºС;

  Temperature deviations from the average (in this case, the normal value): 34-37, 35-37, 36-37, 37-37, 38-37, 39-37, 40-37, resulting in: -3, -2, -1 , 0, 1, 2, 3 (ºС);

  Divide the sum of numbers obtained earlier by their number. For accurate calculations, it is better to use a calculator. The result of division is the arithmetic mean of the numbers added.

  Pay attention to all stages of the calculation, since an error in even one of the calculations will lead to an incorrect final indicator. Check your calculations at every stage. The arithmetic average has the same meter as the summed numbers, that is, if you determine the average attendance, then all your indicators will be “person”.

  This method calculations are used only in mathematical and statistical calculations. For example, the arithmetic mean in computer science has a different calculation algorithm. The arithmetic mean is a very relative indicator. It shows the probability of an event, provided that it has only one factor or indicator. For the most in-depth analysis, many factors must be taken into account. For this purpose, the calculation of more general quantities is used.

  The arithmetic mean is one of the measures of central tendency, widely used in mathematics and statistical calculations. Finding the arithmetic average for several values ​​is very simple, but each task has its own nuances, which are simply necessary to know in order to perform correct calculations.

  Quantitative results of similar experiments.

  How to find the arithmetic mean

  Finding the arithmetic mean for an array of numbers should begin by determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be equal to 184. When writing, the arithmetic mean is denoted by the letter μ (mu) or x (x with a bar). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, so the arithmetic mean will be equal to 184/5 and will be 36.8.

  Features of working with negative numbers

  If the array contains negative numbers, then the arithmetic mean is found using a similar algorithm. The difference only exists when calculating in the programming environment, or if the problem has additional conditions. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps:

  1. Finding the general arithmetic average using the standard method;
  2. Finding the arithmetic mean of negative numbers.
  3. Calculation of the arithmetic mean of positive numbers.

  The responses for each action are written separated by commas.

  Natural and decimal fractions

  If an array of numbers is presented decimals, the solution is carried out using the method of calculating the arithmetic mean of integers, but the result is reduced according to the requirements of the problem for the accuracy of the answer.

  When working with natural fractions they should be reduced to a common denominator, which is multiplied by the number of numbers in the array. The numerator of the answer will be the sum of the given numerators of the original fractional elements.

  Carrying out any statistical analysis unthinkable without calculations. In this article we will look at how to calculate variance, standard deviation, coefficient of variation and other statistical indicators in Excel.

  Maximum and minimum value

  Average linear deviation

  The average linear deviation is the average of the absolute (modulo) deviations from in the analyzed data set. The mathematical formula is:

  a– average linear deviation,

  X– analyzed indicator,

  X̅– average value of the indicator,

  n

  In Excel this function is called SROTCL.

  

  After selecting the SROTCL function, we indicate the data range over which the calculation should occur. Click "OK".

  Dispersion

  (module 111)

  Perhaps not everyone knows what , so I’ll explain, it’s a measure that characterizes the spread of data around the mathematical expectation. However, usually only a sample is available, so the following variance formula is used:

  

  s 2– sample variance calculated from observational data,

  X– individual values,

  X̅– arithmetic mean for the sample,

  n– the number of values ​​in the analyzed data set.

  Corresponding Excel function — DISP.G. When analyzing relatively small samples (up to about 30 observations), you should use , which is calculated using the following formula.

  

  The difference, as you can see, is only in the denominator. Excel has a function for calculating sample unbiased variance DISP.B.

  

  Select the desired option (general or selective), indicate the range, and click the “OK” button. The resulting value may be very large due to the preliminary squaring of the deviations. Dispersion in statistics is a very important indicator, but it is usually used not in its pure form, but for further calculations.

  Standard deviation

  The standard deviation (RMS) is the root of the variance. This indicator is also called standard deviation and is calculated using the formula:

  by general population

  

  by sample

  

  You can simply take the root of the variance, but Excel has ready-made functions for standard deviation: STDEV.G And STDEV.V(for the general and sample populations, respectively).

  

  Standard and standard deviation, I repeat, are synonyms.

  Next, as usual, indicate the desired range and click on “OK”. The standard deviation has the same units of measurement as the analyzed indicator, and therefore is comparable to the original data. More on this below.

  The coefficient of variation

  All indicators discussed above are tied to the scale of the source data and do not allow one to obtain a figurative idea of ​​the variation of the analyzed population. To obtain a relative measure of data dispersion, use the coefficient of variation, which is calculated by dividing standard deviation on average. The formula for the coefficient of variation is simple:

  There is no ready-made function for calculating the coefficient of variation in Excel, which is not a big problem. The calculation can be made by simply dividing the standard deviation by the mean. To do this, write in the formula bar:

  STANDARDDEVIATION.G()/AVERAGE()

  The data range is indicated in parentheses. If necessary, use the sample standard deviation (STDEV.B).

  The coefficient of variation is usually expressed as a percentage, so you can frame a cell with a formula in a percentage format. The required button is located on the ribbon on the “Home” tab:

  

  You can also change the format by selecting from the context menu after highlighting the desired cell and right-clicking.

  The coefficient of variation, unlike other indicators of the scatter of values, is used as an independent and very informative indicator of data variation. In statistics, it is generally accepted that if the coefficient of variation is less than 33%, then the data set is homogeneous, if more than 33%, then it is heterogeneous. This information can be useful for preliminary characterization of the data and for identifying opportunities for further analysis. In addition, the coefficient of variation, measured as a percentage, allows you to compare the degree of scatter of different data, regardless of their scale and units of measurement. Useful property.

  Oscillation coefficient

  Another indicator of data dispersion today is the oscillation coefficient. This is the ratio of the range of variation (the difference between the maximum and minimum values) to the average. There is no ready-made Excel formula, so you will have to combine three functions: MAX, MIN, AVERAGE.

  

  The coefficient of oscillation shows the extent of the variation relative to the mean, which can also be used to compare different data sets.

  Overall, with using Excel many statistical indicators are calculated very simply. If something is not clear, you can always use the search box in the function insert. Well, Google is here to help.

  According to the sample survey, depositors were grouped according to the size of their deposit in the city’s Sberbank:

  Define:

  1) scope of variation;

  2) average deposit size;

  3) average linear deviation;

  4) dispersion;

  5) standard deviation;

  6) coefficient of variation of contributions.

  Solution:

  This distribution series contains open intervals. In such series, the value of the interval of the first group is conventionally assumed to be equal to the value of the interval of the next one, and the value of the interval last group equal to the value of the previous interval.

  The value of the interval of the second group is equal to 200, therefore, the value of the first group is also equal to 200. The value of the interval of the penultimate group is equal to 200, which means that the last interval will also have a value of 200.

  1) Let us define the range of variation as the difference between the largest and lowest value sign:

  The range of variation in the deposit size is 1000 rubles.

  2) The average size of the contribution will be determined using the weighted arithmetic average formula.

  Let us first determine discrete quantity feature in each interval. To do this, using the simple arithmetic mean formula, we find the midpoints of the intervals.

  The average value of the first interval will be:

  

  the second - 500, etc.

  Let's enter the calculation results in the table:

  Deposit amount, rub.	Number of depositors, f	Middle of the interval, x	xf
  200-400	32	300	9600
  400-600	56	500	28000
  600-800	120	700	84000
  800-1000	104	900	93600
  1000-1200	88	1100	96800
  Total	400	-	312000

  The average deposit in the city's Sberbank will be 780 rubles:

  3) The average linear deviation is the arithmetic mean of the absolute deviations of individual values ​​of a characteristic from the overall average:

  

  The procedure for calculating the average linear deviation in the interval distribution series is as follows:

  1. The weighted arithmetic mean is calculated, as shown in paragraph 2).

  2. Absolute deviations from the average are determined:

  3. The resulting deviations are multiplied by frequencies:

  4. Find the sum of weighted deviations without taking into account the sign:

  5. The sum of weighted deviations is divided by the sum of frequencies:

  It is convenient to use the calculation data table:

  Deposit amount, rub.	Number of depositors, f	Middle of the interval, x
  200-400	32	300	-480	480	15360
  400-600	56	500	-280	280	15680
  600-800	120	700	-80	80	9600
  800-1000	104	900	120	120	12480
  1000-1200	88	1100	320	320	28160
  Total	400	-	-	-	81280

  

  The average linear deviation of the size of the deposit of Sberbank clients is 203.2 rubles.

  4) Dispersion is the arithmetic mean of the squared deviations of each attribute value from the arithmetic mean.

  Calculation of variance in interval rows distribution is made according to the formula:

  

  The procedure for calculating variance in this case is as follows:

  1. Determine the weighted arithmetic mean, as shown in paragraph 2).

  2. Find deviations from the average:

  3. Square the deviation of each option from the average:

  4. Multiply the squares of the deviations by the weights (frequencies):

  

  5. Sum up the resulting products:

  

  6. The resulting amount is divided by the sum of the weights (frequencies):

  

  Let's put the calculations in a table:

  Deposit amount, rub.	Number of depositors, f	Middle of the interval, x
  200-400	32	300	-480	230400	7372800
  400-600	56	500	-280	78400	4390400
  600-800	120	700	-80	6400	768000
  800-1000	104	900	120	14400	1497600
  1000-1200	88	1100	320	102400	9011200
  Total	400	-	-	-	23040000