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Standard Deviation, a mathematical term, is the square of the difference from the mean
Arithmetic mean
(i.e., variance) of
Arithmetic square root
, use
sigma
Indicates. The standard deviation is also known as the standard deviation, or experimental standard deviation, in
Probability statistics
In the most commonly used as
Statistical distribution
Degree of measurement basis.
The standard deviation is the variance
Arithmetic square root
. The standard deviation reflects one
Data set
the
Degree of dispersion
.
Mean number
The same two sets of data may not have the same standard deviation.
- Chinese name
- Standard deviation
- Foreign name
- Standard Deviation
- Field of application
- Statistics
Relation to variance: Variance = standard deviation squared.
In the experiment, a single measurement will inevitably produce errors, so we often measure multiple times and then use the measured value
Mean value
Represents the amount measured and characterizes the distribution of the data with an error bar, where the height of the error bar is ±
Standard error
. This is the standard deviation.
Coefficient of variation:
Among them,
The average of the index data.
Standard deviation
(
Standard Deviation
), in
Probability statistics
In the most commonly used as
Statistical distribution
Degree (statistical dispersion) is measured. Standard deviation is defined as the units of the population
Standard value
Rather than
Mean number
From the
Difference squared
the
Arithmetic mean
the
Square root
. It reflects inter-individual relationships within a group
Degree of dispersion
. The results measured to the degree of distribution, in principle, have two properties:
Is a non-negative value and has the same unit as the measurement data. A standard deviation of a total or one
Random variable
There is a difference between the standard deviation of sigma and the standard deviation of the number of samples in a subset.
In simple terms, a standard deviation is a set of data
Mean value
A measure of dispersion. A large standard deviation represents a large difference between most values and their mean;
For example, the two sets of numbers {0,5,9,14} and {5,6,8,9} both have an average of 7, but the second set has a smaller standard deviation.
Standard deviation can be used as a measure of uncertainty. In the physical sciences, for example, do
repeatability
When measuring, the standard deviation of the set of measured values represents the values of those measurements
precision
. When deciding whether the measured values match
Predicted value
The standard deviation of the measured value plays a decisive role: if the measured mean deviates too far from the predicted value (and is compared with the standard deviation value), the measured value and the predicted value are considered to contradict each other. This is easy to understand, because if the measured values all fall outside a certain range of values, it is reasonable to infer whether the predicted values are correct.
Standard deviation is applied to investments as a measure of the stability of returns. A higher standard deviation means that returns are far from the past
Mean number
The more volatile the return, the higher the risk. In contrast, a smaller standard deviation means a more stable return and less risk.
For example, six students from each group A and B took the same language test, and the scores of group A were 95, 85, 75, 65, 55, 45, and the scores of Group B were 73, 72, 71, 69, 68, 67. The mean for both groups is 70, but the standard deviation for Group A is about 17.08 points and for Group B is about 2.16 points, indicating that the gap between group A students is much larger than the gap between group B students.
For the population (that is, to estimate the population variance), divide the square root by n (corresponding
excel
Function: STDEV.P);
If sampling (i.e., estimating the sample variance), divide the square root by (n-1) (corresponding to excel function: STDEV.S);
Because we're dealing with a lot of samples, it's widely used
Sign of the square root
Inside divide by n minus 1.
Subtract the sum of the squares of all the numbers from their mean, divide the result by the number of numbers in the set (or the number minus one, i.e. the variance), and then take the square root of the resulting value.
The dark blue area is the distance
Mean value
A range of values within one standard deviation. in
Normal distribution
In, this range is occupied
ratio
Is 68.2% of the total value (i.e. 1). For a normal distribution, the ratio within two standard deviations (dark blue, blue) adds up to 95.4%. For a normal distribution, the ratio within plus or minus three standard deviations (dark blue, blue, light blue) adds up to 99.6%.
And because of this property of standard deviation, we get
The three Sigma rule
(three-sigma guideline).
Normal distribution diagram
Standard deviation reflects a set of data
Degree of dispersion
One of the most common forms of quantification is representation
precision
Important indicators. When we talk about standard deviation, we need to understand why it's there. We use methods to detect it, but the detection method always has errors, so the detection value is not its true value. The gap between the detected value and the true value is the most decisive indicator to evaluate the detection method. But what the true value is, we don't know. So how do you quantify the detection method
accuracy
Becomes a problem. This is also the purpose of clinical work quality control: to ensure the accuracy and reliability of each batch of experimental results.
Although it is impossible to know the true value of a sample, each sample will always have a true value, no matter what it is. It can be imagined that a good detection method, the detection value should be very tightly dispersed around the true value. If it is not tight, the distance from the true value will be large, and the accuracy is of course not good, and it is impossible to imagine a method with a large degree of dispersion that will measure accurate results. Therefore, the dispersion is the most important and basic index to evaluate the quality of the method.
There are many ways to evaluate and quantify the dispersion of a set of data:
The most direct and simplest way is
Maximum value
-
Minimum value
(i.e. range) to evaluate the dispersion of a set of data. This method is most common in daily life, such as the elimination of the highest and lowest scores in the competition is a very poor specific application.
Because of the uncontrollability of errors, it is unscientific to judge a set of data by only two data points. So people don't use ranges in more demanding fields. In fact, the dispersion is the data deviation
Mean value
The extent of. Therefore, the difference between the data and the mean (we call it the dispersion) can be added to reflect an accurate degree of dispersion. The greater the sum, the greater the dispersion.
But because
Accidental error
A surname
Normal distribution
Yes,
Deviation from mean
There are positive and negative, and for large samples the algebraic sum of deviations from the mean is zero. In order to avoid the plus-minus problem, there are two ways in mathematics: one is to take
Absolute value
That is, the sum of the absolute values of the deviation from the mean is often said. In order to avoid the sign problem, the most commonly used method in mathematics is another method - squared, so that it is all
Nonnegative number
. Therefore, the sum of squares of the mean deviation is an index to evaluate the dispersion.
Due to the sum of the squares of the deviation from the mean
Sample number
It can only reflect the dispersion of the same sample, and it is difficult to compare the same sample in actual work, so in order to eliminate the influence of the number of samples, increase
comparability
Average the sum of squares of the deviation from the mean, which is what we call the variance and is a good indicator of the dispersion.
Sample size
The bigger it is, the more it reflects the real situation, and
Arithmetic mean
But completely ignored this problem, this has long been considered in statistics, in statistics, the average difference of the sample is mostly divided by
Degree of freedom
(n-1), which means the degree to which an instinct is free to choose. When there's only one left, it can't be free anymore, so it's n-1 degrees of freedom.
Because the variance is the square of the data, it is too different from the detection value itself, and it is difficult for people to measure intuitively, so the variance is commonly used
Sign of the square root
And that's the standard deviation we're talking about.
In statistics, the average difference of a sample is usually divided by the degree of freedom (n-1), which means the degree to which the sample is free to choose. When there's only one left, it can't be free anymore, so it's n-1 degrees of freedom.
Standard deviation can objectively and accurately reflect the degree of dispersion of a set of data, but for different items or different samples of the same item, standard deviation is lack of comparability, so for
methodology
Evaluation is introduced again
Coefficient of variation
CV.
The mean and standard deviation of a set of data are often used as the basis for reference. Intuitively, if the center of the value is considered as the mean, then the standard deviation is
Statistical distribution
One of the "natural" measurements.
from
geometry
From the point of view of sigma, the standard deviation can be understood as a sigma from
N-dimensional space
As a function of the distance from a point to a line. As a simple example, there are three values in a set of data
. They can determine a point in three dimensions
. Imagine a straight line through the origin. If all 3 values in this set of data are equal, then point P is a point on the line L, and the distance from P to L is 0, so the standard deviation is also 0. If the three values are not all equal, the point P is perpendicular to L as PR, and PR crosses L at R, then the coordinates of R are those of the three values
Mean number
:
Using some algebra, it is not difficult to find that the distance between the point P and R (that is, the distance between the point P and the line L) is |PR|. In n-dimensional space, the same rule applies, just replace 3 with n.
Standard deviation and
Standard error
All are
Mathematical statistics
The content of the two is not only relatively similar in literal terms, but also both indicate distance from a certain one
Standard value
or
Median value
The degree of dispersion, that is, both expressed
Degree of variation
But there is a big difference between the two.
First of all, from
Statistical sampling
Speaking of the aspects. Real life or
Investigation and research
We are often unable to investigate a certain kind of desire
Target group
All members are tested, and only some members can be selected out of all members (that is, samples) for investigation, and then statistical principles and methods are used to analyze the data obtained, and the data results are the results of the sample, and then the sample results are inferred about the overall situation. A population can take more than one sample, the more samples taken, its
Sample mean
The closer it is to the average of the population.
legend
Standard deviation
It means
Sample data
The degree of dispersion. The standard deviation is
Sample mean
The square root of the variance, standard deviation is usually relative to the mean of the sample data, usually expressed in M±SD, indicating how far a sample data observation is from the mean. And you can see here that the standard deviation is affected by
Extreme value
The impact of... The smaller the standard deviation, the more aggregated the data. The larger the standard deviation, the more discrete the data. The size of the standard deviation depends on the test, if a test is an academic test, the standard deviation is large, indicating the student's score
Degree of dispersion
Large, more able to measure the academic level of students; If a test measures a certain psychological quality, a small standard deviation indicates that the questions written are homogeneous, and the smaller standard deviation is better. The standard deviation is closely related to the normal distribution: in the normal distribution, 1 standard deviation is equal to 68.26% of the area of the curve under the normal distribution, and 1.96 standard deviation is equal to 95% of the area. This is...
Test score
equivalent
Plays an important role on.
Standard error
It's the sampling error. Because numerous samples can be extracted from a population, the data of each sample is an estimate of the data of the population. The standard error represents the current sample's estimate of the population, and the standard error represents
Sample mean
And the population mean
Relative error
. The standard error is divided by the standard deviation of the sample
Sample size
To calculate the square root of theta. As you can see here, the standard error is more affected by the sample size. The larger the sample size, the smaller the standard error, so
Sampling error
The smaller the sample, the better representative it is of the population.
A normally distributed population, taking n samples, can get the sample mean, estimated by the sample mean
Population mean
You need to consider the variance or standard deviation of the sample mean.
[1]
Excel
STDEV.S, STDEV.P,
STDEVA
,
STDEVPA
Four functions, respectively
Sample standard deviation
,
Population standard deviation
, contain
Logical value
The sample standard deviation of the operation, the population standard deviation of the operation that includes the logical value (excel uses"
Standard deviation
").
The difference in calculation method is: sample standard deviation ^2=
Sample variance
* (Number of data -1); Population standard deviation ^2=
Population variance
* Number of data.
excel decomposition of the function:
stdev(A1:E10)=sqrt(DEVSQ(A1:E10)/(COUNT(A1:E10)-1))
(2) stdevp function can be decomposed into (assuming the overall data is a matrix such as A1:E10) :
stdevp(A1:E10)=sqrt(DEVSQ(A1 :E10) /(COUNT(A1 :E10))
Standard deviation is a statistical measure of the difference between a value in a set of values and its mean value. Standard deviation is used
Evaluation price
Possible degree of change or volatility. The bigger the standard deviation,
Price fluctuation
The wider the range, stocks, etc
Financial instrument
The more the performance fluctuates.
Call the function "STDEV.S" in excel to estimate the standard deviation of the sample. Standard deviation reflects relative to the mean
Degree of dispersion
.
Fund algorithm
Standard deviation is a tool to measure the volatility of a fund. Standard deviation is the degree to which a fund is likely to change. The bigger the standard deviation, the more likely the fund's future net worth is to change,
stability
The smaller it is, the higher the risk.
For example, a fund with a one-year standard deviation of 30% means that its net worth could rise by 30% in a year, but it could also fall by 30%. So if there are two
yield
For the same fund, the investor should choose the fund with the lower standard deviation (bear the same risk and get the same return), and if there are two funds with the same standard deviation, the investor should choose the fund with the higher return (bear the same risk but have a higher return). It is recommended that investors take both the benefits and risks into account to judge the fund. For example, Fund A has a two-year yield of 36% and a standard deviation of 18%; The two-year yield of fund B is 24%, and the standard deviation is 8%. From the data point of view, the return of fund A is higher than that of fund B, but the risk is also greater than that of fund B. "Per unit of A fund
Risk-return rate
"For 2
Fund B is 3
. Therefore, the original evaluation of fund A is better only based on income, but after standard deviation
Risk factor
After adjustment, the B fund is even better.
In addition, standard deviation can also be used to judge fund attributes. According to Morningstar,
Equity fund
The average standard deviation of the active fund is 5.04, and the average standard deviation of the active fund is 5.14. The mean standard deviation of conservatively allocated funds was 4.86; Normal
Bond fund
The mean standard deviation was 2.91;
Money fund
The mean standard deviation is 0.19; It can be seen that the more active the fund, the greater the standard deviation; And if the investor holds a fund with a standard deviation higher than
Mean value
The risk is higher, investors may wish to watch
Olympic Games
At the same time, also inspect the fund.
Stock price
The fluctuation of is
Stock market risk
The performance, therefore
Stock market
Risk analysis
Exactly.
Stock market price
Fluctuations are analyzed.
volatility
Represents the value of the future price
uncertainty
This uncertainty is generally described by variance or standard deviation (Markowitz,1952). Below is a list of China and US stocks for some periods
Statistical index
, of which
Chinese securities market
The data is composed of"
Chanlong
"Software download,
American stock market
Data taken from
ECI
"WorldStockExchangeDataDisk".
|
A given year
|
performance
|
volatility
|
||
|
Shanghai Composite index
|
Standard & Poor's index
|
|||
|
1996
|
110.93
|
16.46
|
0.2376
|
0.0573
|
|
1997
|
0.13
|
31.01
|
0.1188
|
0.0836
|
|
1998
|
8.94
|
26.67
|
0.0565
|
0.0676
|
|
1999
|
17.24
|
19.53
|
0.1512
|
0.0433
|
|
2000
|
43.86
|
10.14
|
0.097
|
0.0421
|
|
2001
|
15.34
|
13.04
|
0.0902
|
0.0732
|
|
2002
|
20.82
|
23.37
|
0.0582
|
0.1091
|
Through calculation, we can get:
Shanghai Composite Index performance
Expected value
Material (110.93 to 0.13 + 8.94 + 17.24 + 43.86-15.34-20.82) / 7 = 20.6685714
Standard & Poor's
Performance expectation ≈6.731429
S&p expected volatility ≈0.068029
The performance standard deviation of Shanghai Composite Index is ≈45.2489073
The standard deviation of SSE volatility is ≈0.063167
Standard & Poor's index
Performance standard deviation ≈21.70647
Standard deviation of S&P volatility ≈0.023647
Because the standard deviation is
Absolute value
Can not directly compare China and the US by standard deviation, and
Coefficient of variation
You can make a direct comparison. It can be obtained by calculation: coefficient of variation C·V= standard deviation SD÷ average MN×100%
Sse performance variation coefficient ≈2.18926148
The volatility coefficient of Shanghai Stock Exchange is ≈0.5462
Standard & Poor's coefficient of performance variation ≈3.2247
S&p volatility coefficient of variation ≈0.3476
Through comparison, it can be seen that the variation coefficient of SSE volatility is greater than that of S&P volatility, indicating that in the long run
Chinese stock market
Relatively poor stability, or a less mature
Stock market
.
Analysis chart
Suppose an enterprise's funds come through
Issue bonds
And stock, and both belong to
risk
Assets. Where the yield on the bond is
Risk by standard deviation
To measure; The yield of the stock is
, the risk is
; Stocks and bonds
Correlation coefficient
for
,
covariance
for
; The proportion of bonds is
, the proportion of stocks is (
*
). On the basis of
Portfolio theory
The external investors of the enterprise should
Enterprise investment
acquired
Expected rate of return
for
, the variance is
- 1.
Corporate debt funds and equity funds are complete Positive correlation That is, the correlation coefficient
- 2.
Corporate debt funds and equity funds are completely negatively correlated, that is, their correlation coefficient is -1. The expected rate of return received by the investor and its variance are respectively. According to portfolio theory, only if the proportion of investment is greater than
- 3.
The correlation coefficient between debt funds and equity funds is greater than -1 and less than 1. In theory, there are two kinds of a business Financing method There is a high degree of correlation between the two financing methods on the one hand System risk On the other hand, they also bear the same corporate risk. Therefore, from the perspective of practice, the correlation degree between different financing methods of enterprises can not be complete Positive correlation And negative correlation. For an enterprise, debt funds have fixed claims on the enterprise, and equity funds have only residual claims on the enterprise, so the fluctuation of debt funds is unlikely to be as large as that of equity funds. At the same time, the risk of enterprises will affect the debt funds and equity funds of enterprises at the same time, so the correlation coefficient of debt funds and equity funds of enterprises cannot be negative. The correlation coefficient between different financing methods of enterprises is generally between 0 and 1.
So in what proportion will the value of the enterprise reach the maximum? On the basis of
Portfolio theory
when
, and
When, can appear
Superior to
. See, decide
Enterprise capital structure
The immediate factors are mainly different
Financing method
The rate of return and risk and between them
Correlation coefficient
.
