Does the measurement error follow a normal distribution? Why

Measurement errors are mainly divided into systematic errors and accidental errors.

The systematic error is distributed in a regular manner, and there is an obvious tendency, such as the error of the instrument and the person, which does not obey the normal distribution.

The accidental error is normally distributed, that is, the very large absolute error and the very small absolute error are relatively small, and the middle part of the error is relatively large.

Accidental error four-point characteristic:

1>.Range (boundedness) Under certain observation conditions, the absolute value of the accidental error is not greater than a limit value.

2>.Errors with small absolute values (minimaturities) occur more frequently, and errors with large absolute values occur less frequently.

3>.Positive and negative errors with equal absolute values of signs (equality) occur approximately equally frequently.

4>.Cumulative destructiveness When the number of observations increases infinitely, the arithmetic mean of the accidental error tends to be close to zero.

The area of the rectangular bar on each error interval represents the frequency within which the error occurs – the frequency histogram.

Under certain observation conditions, corresponding to a certain error distribution, when n tends to infinity and dδ tends to 0, the polyline at the top of each rectangle gradually becomes a smooth curve - the error distribution curve.

The frequency distribution of the accidental error increases with n, and the normal distribution is the limit.

Yes, it is necessary to obey the normal distribution, if it does not obey the normal distribution, then it means that there is a gross error in it, and the gross error should be calculated statistically and then removed.

The standard deviation of the normal distribution is normally distributed n ( 2 ), the variance of the socks size d(x) = δ2, e(x) =

Obeying the standard normal distribution, the probability value of the original normal division can be directly calculated by looking at the standard normal distribution table. When dimensional random vectors have similar probability laws, the random vectors follow a multidimensional normal distribution.

Multivariate normal distribution has good properties, for example, the edge distribution of the multivariate normal distribution is still normal, the random vector obtained by any linear transformation is still a multidimensional normal distribution, and especially its linear combination is a univariate normal distribution.

The formula for normalization of the normal distribution: y=(x-)n(0,1).

The standard normal distribution is a probability distribution that is very important in the fields of mathematics, physics, and engineering, and has a significant impact on many aspects of statistics. The expected value = 0, that is, the normal distribution under the condition that the symmetry axis of the curve image is the y-axis and the standard deviation = 1 is denoted as n(0,1).

Definition of normal distribution.

The standard normal distribution, also known as the u distribution, is a normal distribution with 0 as the mean and 1 as the standard deviation, which is denoted as the positive n (0, 1).

The distribution law of the area under the standard normal distribution curve is: the area under the range curve is equal to that, and the area under the range curve is . The statistician also developed a statistical table (when the degrees of freedom are , which can be used to estimate the area under the curve in certain special ranges of U1 and U2 values.

Symmetry: Positive and negative errors occur equally when absolute errors occur.

Unimodality: Errors with small absolute values occur more often than errors with large absolute values.

Bounded: Under certain measurement conditions, the absolute value of random error will not exceed a certain boundary.

Compensatory: As the number of measurements increases, the arithmetic mean of the random error tends to zero.

1. The normal distribution has a unimodality, that is, the curve has a maximum value at the expected value. 2. Symmetry, normal distribution has an axis of symmetry. 3. When the horizontal axis tends to infinity, the probability distribution curve takes the horizontal axis as the asymptotic line.

4. The probability distribution curve has an inflection point on each side at an equal distance from the mean.

It is characterized by unimodality, symmetry, and boundedness.

Standard deviation and expectation are a dimensional that reflects fluctuations in expected values. (u-z δ,u+z δ)z is the quantile of a certain probability, then this set is the range of values of the dependent variable under this probability. In addition, there is no upper and lower bounds for normal distribution.

Let's start with a few concepts:

1. Standard deviation of the sample ≠ standard deviation of the population ≠ statistical standard deviation.

2. On the premise that the population conforms to the normal distribution: the standard deviation of the population = statistical standard deviation.

3. When the sample is representative: the standard deviation of the sample The standard deviation of the population. That is, the standard deviation of the population can be estimated by the standard deviation of the sample.

Then it is necessary to distinguish between statistics in the practical sense and statistics in the mathematical sense

To perform mathematical statistical processing of the actual situation, the premise is that it conforms to the normal distribution function, and under this premise, a series of formulas derived from the normal distribution function can be applied, including the standard deviation formula.

To put it more bluntly: for the actual statistical object, the degree of dispersion of each individual relative to the mean can be expressed by the calculated value of s=((xsample-xaverage) 2 n). For normally distributed functions, the value can represent the half-height width of the function image.

The two were not originally connected in any way. These two are only equal if the distribution of the actual statistical object conforms to a normal function.

Next, let's talk about the problem:

The formula for standard deviation is the result of the derivation of the normal distribution function, but there are conditions for application.

For the whole, that is, n infinity. In this case, the formula divided by n is used to calculate the formula, which meets the conditions for the application of the formula.

For samples, n is a finite value that does not meet the applicable conditions, so the formula divided by n cannot be directly applied.

In order to be able to estimate the standard deviation of an infinite population from a limited sample, approximate calculations must be used. As for how to approximate the calculation, there can be many kinds of theoretically, and the formula of dividing by n-1 has been shown to be able to obtain a result of comparing the standard deviation of the population at any time, which is called unbiased estimation. In mathematical terms, it is:

The error between this estimate and the positive value is convergent. In layman's terms, this estimate is more reliable.

Mathematically, the larger n is, the closer this estimate is to the true value. The practical implication is that the larger the sample size, the more representative it is of the population.

As for the specific derivation and proof process of these formulas, I have actually forgotten. Because it is basically not used in actual use, it is enough to remember the result and understand the meaning.

The standard deviation of a probability distribution or random variable is the positive square root value of the variance, denoted by the symbol. The standard deviation reflects how discrete a data set is, and the smaller the standard deviation, the less these values deviate from the mean, and vice versa. The probability distribution reflects the full picture of the random variable, and the standard deviation indicates the dispersion of the measured values.

Application of standard deviation to probability distributions.

Basic concepts. Probability distribution is one of the basic concepts of probability theory, which is used to express the probability law of the value of random variables. For the convenience of use, the probability distribution takes different manifestations according to the different types of random variables.

Standard deviation is also known as standard deviation, standard deviation describes the average of the distance from the mean (mean deviation) of each data, which is the square root after the square sum of the deviations, and is denoted by . The standard deviation is the arithmetic square root of the variance. The standard deviation reflects how discrete a data set is, and the smaller the standard deviation, the less these values deviate from the mean, and vice versa.

The magnitude of the standard deviation can be measured by the magnification of the standard deviation versus the mean value. The standard deviation of two datasets with the same mean may not be the same.

The probability distribution reflects the whole picture of the random variable, but in practical application, it is more related to several numerical eigenquantities that represent the probability distribution. These characteristic quantities mainly include expectation, variance, and standard deviation.

The formula for normalization of the normal distribution: y=(x-)n(0,1).

Standard normal distribution.

It is a probability distribution that is very important in the fields of mathematics, physics, and engineering, and has a significant impact on many aspects of statistics. Expectations.

0, that is, the axis of symmetry of the curve image is the y-axis, the standard deviation.

Normal distribution under 1 condition, denoted as n(0,1).

Definition of normal distribution.

The standard normal distribution, also known as the u distribution, is a normal distribution with 0 as the mean and 1 as the standard deviation, and is denoted as n(0,1).

The distribution of the area under the standard normal distribution curve is: the area under the range curve is equal to, and the area under the range curve is . Statisticians have also developed a table for statistical purposes (degrees of freedom.

, with the help of this table, it is possible to estimate the number of branches under the curve in some special range of U1 and U2 values.

Most random errors obey a normal distribution.

Random errors that obey a normal distribution are characterized by symmetry, where the absolute error is equal and the positive and negative errors occur equally times. Unimodal, absolute.

Small errors occur more often than large errors in absolute values. Bounded: Under certain measurement conditions, the absolute value of random error will not exceed a certain boundary. Compensatory, the arithmetic average of random errors as the number of measurements increases.

Tend to zero. <>

Random error, also known as accidental error and uncertain error, is due to a series of mutually compensating errors formed by small random fluctuations in the measurement process. The reason for this is the influence of various unstable random factors in the analysis process, such as room temperature and relative humidity.

and instability of environmental conditions such as air pressure, small differences in analyst operation, and instability of the instrument. The magnitude and positive and negative of the random error are not fixed, but after many measurements, it will be found that the probability of the occurrence of positive and negative random errors with the same absolute value is roughly equal, so they can often cancel each other out, so the random error can be reduced by increasing the number of parallel measurements and averaging them.

Obeys a normal distribution.

The characteristics of random error are:

1. The normal distribution has a unimodality, that is, the curve has a maximum value at the expected value.

2. Symmetry, normal distribution has an axis of symmetry.

3. When the horizontal axis tends to infinity, the probability distribution curve takes the horizontal axis as the asymptotic line.

4. The probability distribution curve has an inflection point on each side at an equal distance from the mean.

Random error has the following rules:

1) Size: Absolute.

A small error is more likely to occur than an error with a large absolute value.

2) Symmetry: Positive and negative errors with equal absolute values have an equal probability of occurring.

3) Boundedness: The probability of an error with a large absolute value is close to zero. The absolute value of the error does not exceed a certain boundary.

4) Compensatory: the arithmetic mean of the error of the measured value under certain measurement conditions.

It tends to zero as the number of measurements increases.

Therefore, I do not choose "boundedness" and am satisfied.