### Introduction

### Statistic and Probability

^{2}. If a patient's test result is χ, is this statistically rare against the criteria (e.g., 5 or 1%)? Probability is represented as the surface area in a probability distribution, and the z score that represents either 5 or 1%, near the margins of the distribution, becomes the reference value. The test result χ can be determined to be statistically rare compared to the reference probability if it lies in a more marginal area than the z score, that is, if the value of χ is located in the marginal ends of the distribution (Fig. 1).

### Sampling Distribution (Sample Mean Distribution)

^{2}), the sampling distribution shows normal distribution with mean of µ and variance of σ

^{2}/

*n*. The number of samples affects the shape of the sampling distribution. That is, the shape of the distribution curve becomes a narrower bell curve with a smaller variance as the number of samples increases, because the variance of sampling distribution is σ

^{2}/

*n*. The formation of a sampling distribution is well explained in Lee et al. [2] in a form of a figure.

### T Distribution

^{2}/

*n*, but in reality we do not know σ

^{2}, the variance of the population. Therefore, we use the sample variance instead of the population variance to determine the sampling distribution of the mean. The sample variance is defined as follows:

### Independent T test

^{2}) and found the difference in the means. If this process is repeated 1,000 times, the sampling distribution exhibits the shape illustrated in Fig. 4. When the distribution is displayed in terms of a histogram and a density line, it is almost identical to the theoretical sampling distribution: N(0, 2 × 5

_{2}/6) (Fig. 4).

_{1}- µ

_{2}) was assumed to be 0; thus:

*n*

_{1}and

*n*

_{2}are sufficiently large, the t statistic resembles a normal distribution (Fig. 3).

### Paired T test

_{1}

^{2}is the variance of variable A, σ

_{2}

^{2}is the variance of variable B, and ρ is the correlation coefficient for the two variables. In an independent t test, the correlation coefficient is 0 because the two groups are independent. Thus, it is logical to show the variance of the difference between the two variables simply as the sum of the two variances. However, for paired variables, the correlation coefficient may not equal 0. Thus, the t statistic for two dependent samples must be different, meaning the following t statistic,

*n*

_{1}=

*n*

_{2}=

*n*, and their variance can be represented as

*s*

_{1}

^{2}+

*s*

_{2}

^{2}- 2ρ

*s*

_{1}

*s*

_{2}considering the correlation coefficient. Therefore, the t statistic for a paired t test is as follows: