# 统计代写|抽样理论作业代写sampling theory代考|STAT506

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## 统计代写|抽样理论作业代写sampling theory代考|Random and Systematic Errors

With the exception of accidental errors, such as Increment Preparation Errors (IPE), which affect the integrity of a sample, all other sampling errors are random variables characterized by a given average, nil or not, and a given variance that is never nil. It is by exaggeration, often because it is convenient, that we speak of random error with average nil and variance different from zero, or of systematic errors with variance nil and average different from zero. Actually, all errors such as the Fundamental Sampling Error (FSE), the Increment Delimitation Error (IDE), The Increment Extraction Error (IEE), etc., have two components:

• A random component characterized by the variance only
• A nonrandom component characterized by the average only.
In fact, the variance and the average of an error are physically complementary, even if they are very different properties. Therefore, when several random variables such as $F S E, I D E, I E E$, and so on are independent in probability (it should be clear we are not talking about independent quantities, but independent differences between quantities, which is not the same thing), they are cumulative, and it is perfectly justified to write the following relationships:
If these errors occur separately:
$$T S E=F S E+I D E+I E E+\ldots$$
where TSE is the Total Sampling Error.
For the averages of these errors we can write:
$$m(T S E)=m(F S E)+m(I D E)+m(I E E)+\ldots$$
For the variances of these errors we can write:
$$s^2(T S E)=s^2(F S E)+s^2(I D E)+s^2(I E E)+\ldots$$
The above property of additivity is often well appreciated by those involved in sampling practices.

## 统计代写|抽样理论作业代写sampling theory代考|A Logical Introduction to the Components

Before introducing the various components of the Overall Estimation Error $O E E$, it is beneficial to proceed with an introduction to the fundamental notion of heterogeneity. To find out how it is defined, and to clearly differentiate this notion from the notion of homogeneity. It does not involve a great deal of research to find out that homogeneity is often a relative concept. If we look at a pile of fine sand from a distance, the pile may appear homogeneous; however, we know that as we come closer toward the pile and finally look at it under a magnifying glass, we realize that the homogeneity was only an illusion.

The only reality is a state of great heterogeneity when each individual grain is examined showing differences in sizes, colors, compositions, shapes, densities, opacities, porosities, and so on. It is not long before one wonders whether homogeneity is only a limit case rarely encountered. We indeed live inside a heterogeneous world. And if we try to measure this heterogeneity, we intuitively find that the zero of heterogeneity is homogeneity, just a limit case. Even liquids that appear to be homogeneous are indeed heterogeneous if we consider particles, atoms, ions, and molecules. Thus, there are two ways to look at homogeneity. It can be compared to a mathematical limit never encountered in our universe, and one may say that is pushing the concept too far. This leads to a more practical way, which defines homogeneity as a relative state where all the constituents of a lot are apparently identical (e.g., a lot of calibrated marbles). Even in the second case, homogeneity remains a limit case, and we immediately face a difficulty that consists in clearly defining this limit. As we move from major to minor constituents, or constituents at the trace level, we will have to push down this limit more and then return to the point where we have to face reality, which is essentially heterogenous.

As far as sampling is concerned, we may as well forget about homogeneity and remove this word from our vocabulary, because it leads to misconceptions and dangerous optimistic and wishful thinking assumptions. To state that a material is homogeneous is indeed almost always wishful thinking. Because the Theory of Sampling is a preventive “tool” we choose the safe hypothesis that we are only dealing with heterogeneity and we intend to measure this heterogeneity, the zero of which is homogeneity.
In our attempt to measure heterogeneity we will have to clearly differentiate two categories of heterogeneity:

• the Constitution Heterogeneity $\mathrm{CH}$
• the Distribution Heterogeneity DH.

# 抽样理论代考

## 统计代写|抽样理论作业代写采样理论代考|随机和系统误差

$$T S E=F S E+I D E+I E E+\ldots$$

$$m(T S E)=m(F S E)+m(I D E)+m(I E E)+\ldots$$

$$s^2(T S E)=s^2(F S E)+s^2(I D E)+s^2(I E E)+\ldots$$

## 统计代写|抽样理论作业代写采样理论代考|组件逻辑介绍

. c

• the Constitution Heterogeneity $\mathrm{CH}$
• the Distribution Heterogeneity DH.

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