Statistics 简明教程

Statistics - Simple random sampling

简单随机样本被定义为总体中每个元素都有相等和独立的机会被选择的样本。对于具有 N 个单位的总体，以所有可能的 NCn 样本组合来选择 n 个样本单位的概率由 1/NCn给出，例如，如果我们有一个由五个元素（A、B、C、D、E）组成的总体，即 N 为 5，而我们想要大小为 n = 3 的样本，那么有 5C3 = 10 个可能的样本，并且任何单个单位成为样本成员的概率由 1/10 给出。

A simple random sample is defined as one in which each element of the population has an equal and independent chance of being selected. In case of a population with N units, the probability of choosing n sample units, with all possible combinations of NCn samples is given by 1/NCn e.g. If we have a population of five elements (A, B, C, D, E) i.e. N 5, and we want a sample of size n = 3, then there are 5C3 = 10 possible samples and the probability of any single unit being a member of the sample is given by 1/10.

简单随机抽样可以用两种不同的方式进行，即“有放回”或“无放回”。当在选择下一个之前依次将单位选入样本后放回已选单位时，它是有放回的简单随机样本。如果所选单位在下次抽取前未被放回，并且连续单位的抽取仅从总体中剩余的单位中进行，则将其称为无放回的简单随机样本。因此，在前一种方法中，一次选定的单位可能会重复，而在后一种方法中，一次选定的单位不会重复。由于无放回的简单随机样本具有更高的统计效率，因此它是首选的方法。

Simple random sampling can be done in two different ways i.e. 'with replacement' or 'without replacement'. When the units are selected into a sample successively after replacing the selected unit before the next draw, it is a simple random sample with replacement. If the units selected are not replaced before the next draw and drawing of successive units are made only from the remaining units of the population, then it is termed as simple random sample without replacement. Thus in the former method a unit once selected may be repeated, whereas in the latter a unit once selected is not repeated. Due to more statistical efficiency associated with a simple random sample without replacement it is the preferred method.

可以通过抽签法或随机数表中的任一方法抽取简单的随机样本。

A simple random sample can be drawn through either of the two procedures i.e. through lottery method or through random number tables.

  1. Lottery Method - Under this method units are selected on the basis of random draws. Firstly each member or element of the population is assigned a unique number. In the next step these numbers are written on separate cards which are physically similar in shape, size, color etc. Then they are placed in a basket and thoroughly mixed. In the last step the slips are taken out randomly without looking at them. The number of slips drawn is equal to the sample size required. Lottery method suffers from few drawbacks. The process of writing N number of slips is cumbersome and shuffling a large number of slips, where population size is very large, is difficult. Also human bias may enter while choosing the slips. Hence the other alternative i.e. random numbers can be used.

  2. Random Number Tables Method - These consist of columns of numbers which have been randomly prepared. Number of random tables are available e.g. Fisher and Yates Tables, Tippets random number etc. Listed below is a sequence of two digited random numbers from Fisher & Yates table: 61, 44, 65, 22, 01, 67, 76, 23, 57, 58, 54, 11, 33, 86, 07, 26, 75, 76, 64, 22, 19, 35, 74, 49, 86, 58, 69, 52, 27, 34, 91, 25, 34, 67, 76, 73, 27, 16, 53, 18, 19, 69, 32, 52, 38, 72, 38, 64, 81, 79 and 38. The first step involves assigning a unique number to each member of the population e.g. if the population comprises of 20 people then all individuals are numbered from 01 to 20. If we are to collect a sample of 5 units then referring to the random number tables 5 double digit numbers are chosen. E.g. using the above table the units having the following five numbers will form a sample: 01, 11, 07, 19 and 16. If the sampling is without replacement and a particular random number repeats itself then it will not be taken again and the next number that fits our criteria will be chosen.

因此，可以使用这两种程序中的任何一种来抽取简单的随机样本。然而，在实践中，人们发现简单随机样本需要大量的时间和精力，并且不切实际。

Thus a simple random sample can be drawn using either of the two procedures. However in practice, it has been seen that simple random sample involves lots of time and effort and is impractical.