Calculate the standard deviation. What is standard deviation - using standard deviation function to calculate standard deviation in excel

standard deviation(synonyms: standard deviation, standard deviation, standard 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.

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The standard deviation is measured in units of measurement of the random variable itself and is used in calculating the standard error of the arithmetic mean, in constructing confidence intervals, in statistical verification of hypotheses, in measuring the linear relationship between random variables. It is 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 RMS (Standard Deviation) and SRT (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 (divide by the root of the sample). In Excel, the STDEV() function is different (dividing by the square root of n-1).

Standard deviation(estimation 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 sample element; 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 the general case, it is impossible to construct an unbiased estimate. However, an estimate based on an unbiased variance estimate is consistent.

In accordance with GOST R 8.736-2011, the standard deviation is calculated according to the second formula of this section. Please check your 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 - approximately with a probability of 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 processing the sample).

If the true value x ¯ (\displaystyle (\bar (x))) unknown, then you should use σ (\displaystyle \sigma ), a s. Thus, the rule of three sigma is transformed into the rule of three s .

Interpretation of the value of the standard deviation

A larger value of the standard deviation indicates a greater spread of values in the presented set with the mean of the set; a smaller value, respectively, indicates 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 of 7 and standard deviations of 7, 5, and 1, respectively. The last set has a small standard deviation because the values in the set are clustered around the mean; the first set has the largest value of the standard deviation - the values within the set strongly diverge from the average value.

In a general sense, the standard deviation can be considered a measure of uncertainty. For example, in physics, the 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 mean value of the measurements is very different from the values predicted by the theory (large standard deviation), then the obtained values or the method of obtaining them should be rechecked. is 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. Coastal cities are known to have many different daily maximum temperatures less than inland cities. Therefore, the standard deviation of the maximum daily temperatures in the coastal city will be less than in the second city, despite the fact that they have the same average value of this value, which in practice means that the probability that the maximum air temperature of each particular day of the year will be stronger differ from the average value, higher for a city located inside the continent.

Sport

Let's assume that there are several football teams that are ranked 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 the best values in 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 has a hard time predicting the result, which in turn is explained by an imbalance, for example, a strong defense but a weak attack.

The use of the standard deviation of the parameters of the team allows one to predict the result of the match between two teams to some extent, evaluating the strengths and weaknesses of the teams, and hence the chosen methods of struggle.

Answers to examination questions on public health and health care.
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81. Standard deviation, calculation method, application.

An approximate method for assessing the fluctuation of a variational series is the determination of the limit and amplitude, however, the values of the variant within the series are not taken into account. The main generally accepted measure of the fluctuation of a quantitative trait within the range of variations 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 mean are denoted as d (deviate). The sum of all deviations is equal to zero.

3. Square each deviation d 2 .

4. Multiply the squared deviations by the corresponding frequencies d 2 *p.

5. Find the sum of the products  (d 2 * p)

6. Calculate the standard deviation by 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.

The value of the standard deviation:

1. The standard deviation characterizes the spread of the variant relative to the average value (i.e., the fluctuation 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 compliance of the arithmetic mean with the variation series for which it was calculated.

Variations of mass phenomena obey the law of normal distribution. The curve representing this distribution has the form of 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 the system of rectangular coordinates on the abscissa axis the values of the quantitative trait (options) are plotted, and on the ordinate axis - the frequency of occurrence of the variant in the variation series, 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 variant values are within М1

95.5% of the variant values are within M2

99.7% of the variant values are within M3

3. The standard deviation allows you to set the normal values for clinical and biological parameters. In medicine, the M1 interval is usually taken outside the normal range for the phenomenon under study. The deviation of the estimated value from the arithmetic mean by more than 1 indicates the 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 (method of sigma deviations), for the development of standards for children's clothing

5. The standard deviation is necessary to characterize the degree of diversity of the trait under study and calculate the error of the arithmetic mean.

The value of the standard deviation is usually used to compare the fluctuations of the same type of series. If two rows with different characteristics are compared (height and weight, average duration of hospital stay and hospital mortality, etc.), then a direct comparison of sigma sizes is impossible. , because standard deviation - a named value, expressed in absolute numbers. In these cases, apply the coefficient of variation (CV) , which is a relative value: the percentage of the standard deviation to the arithmetic mean.

The coefficient of variation is calculated by the formula:

The higher the coefficient of variation , the greater the variability of this series. It is believed that the coefficient of variation over 30% indicates the qualitative heterogeneity of the population.

From Wikipedia, the free encyclopedia

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 a linear relationship between random variables. Defined as the square root of the variance of a random variable.

Standard deviation:

$\sigma=\sqrt(\frac(1)(n)\sum_(i=1)^n\left(x_i-\bar(x)\right)^2).$

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

$s=\sqrt(\frac(n)(n-1)\sigma^2)=\sqrt(\frac(1)(n-1)\sum_(i=1)^n\left(x_i-\bar (x)\right)^2);$

three sigma rule

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

If the true value $\bar(x)$ unknown, then you should use $\sigma$ , a s. Thus, the rule of three sigma is transformed into the rule of three s .

Interpretation of the value of the standard deviation

Practical use

In practice, the standard deviation allows you to estimate how much values from a set can differ from the average value.

Economics and finance

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

Climate

Sport

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Literature

Borovikov V. STATISTICS. The art of computer data analysis: For professionals / V. Borovikov. - St. Petersburg. : Peter, 2003. - 688 p. - ISBN 5-272-00078-1..

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An excerpt characterizing the standard deviation

And, quickly opening the door, he stepped out with resolute steps onto the balcony. The conversation suddenly ceased, hats and caps were removed, and all eyes went up to the count who came out.
- Hello guys! said the count 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 the chambers, slamming the door hard.
A murmur of approval ran through the crowd. “He, then, will control the useh of the villains! And you say a Frenchman ... he will untie the whole distance for you! people said, as if reproaching each other for their lack of faith.
A few minutes later an officer hurried out of the front door, ordered something, and the dragoons stretched out. The crowd moved greedily from the balcony to the porch. Coming out on the porch with angry quick steps, Rostopchin hastily looked around him, as if looking for someone.
- Where is he? - said the count, and at the same moment as he said this, he saw from around the corner of the house coming out between two dragoons a young man with a long, thin neck, with his head half-shaven and overgrown. This young man was dressed in what used to be a dapper, blue-clothed, shabby fox sheepskin coat and in dirty, first-hand prisoner's trousers, stuffed into uncleaned, worn-out thin boots. Shackles hung heavily on thin, weak legs, making it difficult for the young man's hesitant gait.
- BUT! - said Rostopchin, hastily turning his eyes away from the young man in the fox coat and pointing to the bottom step of the porch. - Put it here! The young man, rattling his shackles, stepped heavily onto the indicated step, holding the pressing collar of the sheepskin coat with his finger, turned his long neck twice and, sighing, folded his thin, non-working hands in front of his stomach with a submissive gesture.
There was silence for a few seconds as the young man settled himself on the step. Only in the back rows of people squeezing to one place, groaning, groans, jolts and the clatter of rearranged legs were heard.
Rostopchin, waiting for him to stop at the indicated place, frowningly rubbed his face with his hand.
- Guys! - said Rostopchin in a metallic voice, - this man, Vereshchagin, is the same scoundrel from whom Moscow died.
The young man in the fox coat stood in a submissive pose, with his hands clasped together in front of his stomach and slightly bent over. Emaciated, with a hopeless expression, disfigured by a shaved head, his young face was lowered down. At the first words of the count, he slowly raised his head and looked down at the count, as if he wanted to say something to him or at least meet his gaze. But Rostopchin did not look at him. On the long, thin neck of the young man, like a rope, a vein behind the ear tensed and turned blue, and suddenly his face turned red.
All eyes were fixed on him. He looked at the crowd, and, as if reassured by the expression which he read on the faces of the people, he smiled sadly and timidly, and lowering his head again, straightened his feet on the step.
“He betrayed his tsar and fatherland, he handed himself over to Bonaparte, he alone of all Russians has dishonored the name of a Russian, and Moscow is dying from him,” Rastopchin said in an even, sharp voice; but suddenly he quickly glanced down at Vereshchagin, who continued to stand in the same submissive pose. As if this look blew him up, he, raising his hand, almost shouted, turning to the people: - Deal with him with your judgment! I give it to you!
The people were silent and only pressed harder and harder on each other. Holding each other, breathing in this infected closeness, 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 frightened wide-open eyes and gaping mouths, straining with all their strength, kept the pressure of the rear ones on their backs.
- Beat him! .. Let the traitor die and not shame the name of the Russian! shouted Rastopchin. - Ruby! I order! - Hearing not words, but the angry sounds of Rostopchin's voice, the crowd groaned and moved forward, but again stopped.
- Count! .. - Vereshchagin's timid and at the same time theatrical voice said in the midst of a momentary silence. “Count, one god is above us…” said Vereshchagin, raising his head, and again the thick vein on his thin neck became filled with blood, and the color quickly came out and fled from his face. He didn't finish what he wanted to say.
- Cut him! I order! .. - shouted Rostopchin, suddenly turning as pale as Vereshchagin.
- Sabers out! shouted the officer to the dragoons, drawing his saber himself.
Another even stronger wave soared through the people, and, having reached the front rows, this wave moved the front ones, staggering, brought them to the very steps of the porch. A tall fellow, with a petrified expression on his face and with a stopped raised hand, stood next to Vereshchagin.
- Ruby! almost whispered an officer to the dragoons, and one of the soldiers suddenly, with a distorted face of anger, hit Vereshchagin on the head with a blunt broadsword.
"BUT!" - Vereshchagin cried out shortly and in surprise, looking around in fright 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 from Vereshchagin, he cried out plaintively in pain, and this cry ruined him. That barrier of human feeling, stretched to the highest degree, which still held the crowd, broke through instantly. The crime was begun, it was necessary to complete it. The plaintive groan of reproach was drowned out by the formidable and angry roar of the crowd. Like the last seventh wave breaking ships, this last unstoppable wave soared up from the back rows, reached the front ones, knocked them down and swallowed everything. The dragoon who had struck wanted to repeat his blow. Vereshchagin with a cry of horror, shielding himself with his hands, rushed to the people. The tall fellow, whom he stumbled upon, seized Vereshchagin's thin neck with his hands, and with a wild cry, together with him, fell under the feet of the roaring people who had piled on.
Some beat and tore at Vereshchagin, others were tall fellows. 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 bloody, beaten 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 crushed 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 leave him.

Wise mathematicians and statisticians came up with a more reliable indicator, although for a slightly different purpose - mean linear deviation. This indicator characterizes the measure of the spread of the values of the data set around their average value.

In order to show the measure of the spread of data, you must first determine what this very spread will be considered relative to - usually this is the average value. Next, you need to calculate how far the values of the analyzed data set are far from the average. It is clear that each value corresponds to a certain amount of deviation, but we are also interested in a general estimate covering the entire population. Therefore, the average deviation is calculated using the formula of the usual arithmetic mean. But! But in order to calculate the average of the deviations, they must first be added. And if we add positive and negative numbers, they will cancel each other out and their sum will tend to zero. To avoid this, all deviations are taken modulo, that is, all negative numbers become positive. Now the average deviation will show a generalized measure of the spread of values. As a result, the average linear deviation will be calculated by the formula:

a is the average linear deviation,

x- the analyzed indicator, with a dash on top - the average value of the indicator,

n is the number of values in the analyzed dataset,

the summation operator, I hope, does not scare anyone.

The average linear deviation calculated using the specified formula reflects the average absolute deviation from the average value for this population.

The red line in the picture is the average value. The deviations of each observation from the mean are indicated by small arrows. They are taken modulo and summed up. Then everything is divided by the number of values.

To complete the picture, one more example needs to be given. Let's say there is a company that manufactures cuttings for shovels. Each cutting should be 1.5 meters long, but, more importantly, all should be the same, or at least plus or minus 5 cm. However, negligent workers will cut off 1.2 m, then 1.8 m. . The director of the company decided to conduct a statistical analysis of the length of the cuttings. I selected 10 pieces and measured their length, found the average and calculated the average linear deviation. The average turned out to be just right - 1.5 m. But the average linear deviation turned out to be 0.16 m. So it turns out that each cutting is longer or shorter than necessary by an average of 16 cm. There is something to talk about with workers . In fact, I have not seen the real use of this indicator, so I came up with an example myself. However, there is such an indicator in the statistics.

Dispersion

Like the mean linear deviation, the variance also reflects the extent to which the data spread around the mean.

The formula for calculating the variance looks like this:

(for variation series (weighted variance))

(for ungrouped data (simple variance))

Where: σ 2 - dispersion, Xi– we analyze the sq indicator (feature value), – the average value of the indicator, f i – the number of values in the analyzed data set.

The variance is the mean square of the deviations.

First, the mean is calculated, then the difference between each baseline and mean is taken, squared, multiplied by the frequency of the corresponding feature value, added, and then divided by the number of values in the population.

However, in its pure form, such as, for example, the arithmetic mean, or index, dispersion is not used. It is rather an auxiliary and intermediate indicator that is used for other types of statistical analysis.

Simplified way to calculate variance

standard deviation

To use the variance for data analysis, a square root is taken from it. It turns out the so-called standard deviation.

By the way, the standard deviation is also called sigma - from the Greek letter that denotes it.

The standard deviation obviously also characterizes the measure of data dispersion, but now (unlike dispersion) it can be compared with the original data. As a rule, mean-square indicators in statistics give more accurate results than linear ones. Therefore, the standard deviation is a more accurate measure of data scatter than the mean linear deviation.

One of the main tools of statistical analysis is the calculation of the standard deviation. This indicator allows you to make an estimate of the standard deviation for a sample or for the general population. Let's learn how to use the standard deviation formula in Excel.

Let's immediately define what the standard deviation is and what its formula looks like. This value is the square root of the arithmetic mean of the squares of the difference between all the values of the series and their arithmetic mean. There is an identical name for this indicator - standard deviation. Both names are completely equivalent.

But, of course, in Excel, the user does not have to calculate this, since the program does everything for him. Let's learn how to calculate standard deviation in Excel.

Calculation in Excel

You can calculate the specified value in Excel using two special functions STDEV.V(according to the sample) and STDEV.G(according to the general population). The principle of their operation is absolutely the same, but they can be called in three ways, which we will discuss below.

Method 1: Function Wizard

Method 2: Formulas tab

Method 3: Entering the formula manually

There is also a way where you don't need to call the argument window at all. To do this, enter the formula manually.

As you can see, the mechanism for calculating the standard deviation in Excel is very simple. The user only needs to enter numbers from the population or links to cells that contain them. All calculations are performed by the program itself. It is much more difficult to understand what the calculated indicator is and how the results of the calculation can be applied in practice. But understanding this already belongs more to the realm of statistics than to learning how to work with software.

Portal for the student. Self-training

Calculate the standard deviation. What is standard deviation - using standard deviation function to calculate standard deviation in excel

Encyclopedic YouTube

three sigma rule

Interpretation of the value of the standard deviation

Climate

Sport

81. Standard deviation, calculation method, application.

Basic information

three sigma rule

Interpretation of the value of the standard deviation

Practical use

Economics and finance

Climate

Sport

see also

Write a review on the article "Standard deviation"

Literature

An excerpt characterizing the standard deviation

Dispersion

Calculation in Excel

Method 1: Function Wizard

Method 2: Formulas tab

Method 3: Entering the formula manually

Portal for the student. Self-training

Encyclopedic YouTube

three sigma rule

Interpretation of the value of the standard deviation

Climate

Sport

81. Standard deviation, calculation method, application.

Basic information

three sigma rule

Interpretation of the value of the standard deviation

Practical use

Economics and finance

Climate

Sport

see also

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Literature

An excerpt characterizing the standard deviation

Dispersion

Calculation in Excel

Method 1: Function Wizard

Method 2: Formulas tab

Method 3: Entering the formula manually

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