Given a sample of size n, from a normal distribution with theoretical unknown mean μ and known variance σ², it implies that the sample mean

is also normally distributed with mean μ and known variance σ²/n. Given a confident level 100(1 ? α) percentage, the confident interval [1] [2] of theoretical unknown mean μ is:

where Z_α/2 which is the z-value at significant level α is the upper 100α/2 percentage point of standard normal distribution. Let E is the absolute deviation between the sample mean and the theoretical mean μ, we have:

The value E is also called estimated error, which is always less than or equal to.

There is a requirement that how to estimate the sample size n so as to the deviation is less than or equal to the pre-defined error E at given a 100(1 ? α) % confident level. This is the choice of sample size. Following formula [1] indicates that does not exceed the error E if the sample size n is:

(Readers can refer to [1] with regard to pages 252 - 253, 293 - 297, 304 - 305, 309 - 310, 312 - 314, 331 - 333, 344 - 345, 359, 364 - 365 for more details about choice of sample size). There is an issued problem that how to define the error E. Normally, E is defined based on the previous experiment or other study or it can be determined subjectively by specialist. Therefore, this research proposes an objective method to calculate the error E given an available sample. This is an iterative method which is described in next section.

The formula of choice of sample size is re-written:

The z-value Z_α/2 is totally determined and so what we need to do is to calculate the variance σ² and the error

. We will use a novel method when n is considered as a variable. The formula of sample size above

is reduced as below:

where is sample mean and h(i) is proportional to sample size n and the notation denotes the proportion.

Fixing variance σ(i)² and mean μ(i), we have:

Suppose there is an available and given m iteration times, for example, m = 100 and a new sample containing n elements is sampled randomly from at the i^th iteration. This is a form of bootstrap sampling with replacement.

Note that Y_ij (s) are taken randomly from with replacement.

Let M(i) is the sample mean of, we have:

where,

We assume that the sample mean M(i) is approximated to the sample mean and is random variable with theoretical mean μ. We have:

Note that is approximated by M(i).

Summing accumulatively through m iterations corresponding to m sample (s), we have:

Dividing both sides of formula above by, we have:

Let

It is easy to infer that Δ² is sample variance of the set of sample means M(i) (s).

Therefore, the formula for calculating variable h with fixed variance σ² is:

Because the theoretical variance σ² is not defined, it is approximated by sample variance s² of sample.

where is sample mean.

Substituting s² into the formula for calculating variable h, we have:

Finally, the sample size n is calculated by following formula:

It is necessary to have an example for illustrating the proposed formula to calculate sample size without pre-defined error. Given 10-element sample = {X₁ = 8.05, X₂ = 9.60, X₃ = 2.98, X₄ = ?20.26, X₅ = ?6.52, X₆ = ?10.85, X₇ = 8.14, X₈ = 26.48, X₉ = 10.57, X₁₀ = 2.26}, we will estimate the optimal size of the next sample based on. Suppose there are 10 iteration times (m = 10), we have 10 new sample (s) is sampled randomly from with replacement. Table 1 shows such 10 new sample (s) and their means M(i) (s).

The sample variance Δ² of sample means M(i) (s) is:

The mean of sample is:

The sample variance s² of sample is:

Given the confident level 95% (α = 0.05), it is easy to calculate the optimal sample size as follows:

According to results from many experiments, if the origin sample (previous sample) conforms normal distribution, the optimal sample size is 4 - 5 times larger than the size of such origin sample so that it is possible to gain better experiment (testing, analysis, estimation, etc.) because the origin sample that conforms normality is itself good sample. We can implement the proposed formula of sample size choice by R language [3] so that it is convenient to do many experiments on such formula. Following is R language code for implementing the proposed formula.

I invent this method when discussing with the co-author Dr. Hang Ho about choice of sample size. At that time, I make the simile that the ideology of this method is similar to the problem “hen and egg”. Regardless that hen exists before or egg exists before, you feed hen to lay new egg and incubate such egg to hatch new hen. Therefore, given an available random sample is used to estimate the sample size and such sample size is applied to collect new random sample; after that new sample size is estimated based on the new random sample and so on. Now, we analyze the formula for estimating sample size:

The variance s² in numerator expresses the coherent variation of data and the value Δ² in denominator specifies the variation of disturbed data (data is disturbed for many times). It means that Δ² specifies the variation of change (or variation of variation). The smaller the value Δ² is, the more precise the variance s² is and so the sample size is much proportional to s². In other words, the small Δ² makes an increase in sample size. Ratio

approaches 1 when m approaches +∞ and so, the larger the number of iterations is, the more precise the sample size is. If m is small, the sample size tendentiously increases, but the balance is established because Δ² will increase if m is small, and as known the large Δ² makes decrease in sample size. But why the small m makes an increase in Δ² and otherwise? As known the number of iterations m specifies the variation of disturbed data. The larger the number m is, the much more the data is disturbed and so it is easier for the tendency that data is reverted in equilibrium, which causes the decrease in Δ². In other words, the small m makes increase in Δ².

LocNguyen,HangHo, (2015) A Proposed Method for Choice of Sample Size without Pre-Defining Error. Journal of Data Analysis and Information Processing,03,163-167. doi: 10.4236/jdaip.2015.34016