Since a population is generally very large, a survey of the entire population takes lots of time and money. Hence we are trying to estimate characteristics of the population using a set of samples. Estimation of population characteristics using samples is called an inferential statistics. However, there may be some difference between the characteristic of the population and the characteristic of the sample. In order to reduce the difference, several methods of sampling have been studied. The most commonly used one is a simple random sampling which collects samples with the same probability of all elements of the population being selected.

In case of the simple random sampling, there are two possible ways to collect samples. One is to include an element selected once again in the population (with replacement), and the other is not to include the selected element back into the population (without replacement). However, in practice, almost all sampling is made without replacement.

Some tools may be needed to ensure that each element of the population is selected equally. We usually use a random number table which is a table of numbers from 0 to 9 without special regularity or partiality. Recently, a random number generator by using a computer which uses the uniform distribution on [0, 1] is widely used to produce a random number. 『eStatU』 provide random number generators for Uniform, Normal and two dimensional Normal distributions.

Simple random sampling is the basic sampling method for all sample surveys on a population. When the characteristics of a population are estimated using a sample, a difference occurs between the estimated value using the sample and the characteristic value of the population, which is called a sampling error. Many sampling methods have been studied to reduce this sampling error.

Each element of the population that is the subject of analysis is called an elementary unit or an observational unit because it is the object that is actually observed in the survey. Depending on the sampling method, these elementary units can be selected individually (e.g. simple random sampling method), or a set of elementary units can be selected for convenience of sampling. Therefore, what is selected to construct a sample is called a sampling unit, distinguishing it from an observational unit. In order to select samples by the sampling unit, a table of the sampling units is required, which is called a sampling frame. Describing how to select samples using the sampling frame is called a sampling design.

In order to conduct a good sample survey, it is necessary to define accurately the above terms in advance. Since a sample survey is a method of scientific research, all steps for a sample survey must be planned in advance and carried out according to the plan.

The following sampling methods are often used in large-scale sample surveys rather than simple random sampling. For detailed information on the estimation and variance of population characteristic values by these sampling methods, please refer to the references as it is beyond the scope of this book.

Stratified sampling method divides the population into an appropriate number of strata and selecting a sample of a fixed size from each stratum using simple random sampling method. For example, if you are interested in the average wage in a population consisting of factory workers, office workerse, and professionals, divide the population into three strata: factory workers, office workers, and professionals. Then, estimate the overall average wage by selecting individuals from each stratum through simple random sampling. Stratification is recommended so that the elements in each stratum are as homogeneous as possible. The variance of the estimated value of stratified sampling is smaller than that of simple random sampling and the cost is reduced. The stratified sampling method has the advantage of being more convenient administratively.

In large-scale sample surveys with a large population, it is difficult to prepare a sampling frame, expensive to survey widely scattered sampling units, and difficult to manage the survey. To solve this problem, for example, when trying to investigate entertainment expenditures by household living in Seoul, if a household (it is called a cluster because there may be many household members) is the sampling unit, all basic regional administrative units in Seoul are listed and then some regional units are selected by simple random sampling. In the selected regional units, some households are selected by simple random sampling and then survey all members in the household. The sampling method from a population using a group of elementary units, that is a cluster, as a sampling unit is called a cluster sampling method.

Suppose you want to get a quick estimate of total sales by examining 2% of a department store's sales slips, approximately 1000 slips each week. 20 sales slips need to be investigated. For this purpose, if the sales slips are filed and stored in sales order, one slip is selected between the first slip and the 50th slip and then continue selections from every 50 slips. For example, if it was the 7th slip, the next is the 57th, 107th, 157th, ... , 957th slip. This is called a systematic sampling, in which one slip is selected out of every 50 slips. Here, 50 is called the sampling interval, and the starting point 7 is called the random starting point, and can be determined randomly within the sampling interval.

The purpose of statistical experiments or surveys is to find out some information about a population. Information about the population usually refers to a characteristic value of the population, such as the mean and variance which are called parameters. Since it is so difficult or costly to investigate the population parameters, they are usually estimated by using characteristic values of a set of samples such as the sample mean and sample variance. <Figure 6.2.1> is a simulation to show the relationship between population data of size 10,000 and sample data (approximately 10%) using 『eStatU』 which shows characteristic values of a population are similar to characteristic values of a set of samples.

Characteristic values of a set of samples are called sample statistic and the distribution of the sample statistic is called a sampling distribution. The sampling distribution identifies the relationship between the sample statistic and the population parameter and it makes possible to estimate and test a population parameter. In this section, let's first look at the sampling distribution of sample means and find out how to estimate the population mean.

The following example is to find out the sampling distribution of sample means.

The relationship between the population mean and all possible sample means in [Example 6.2.1] is observed even if the population has a different shape of distribution. If the population is very large, it is not possible to find all possible samples as shown in [Example 6.2.1] and to find a distribution of sample means. Therefore, the following theoretical research has been developed.

If a population is normally distributed with \( N(μ, σ^2 ) \), the distribution of all possible sample means is exactly a normal distribution such as \( N(μ, \frac {σ^2 }{n} ) \). If the population is an infinite population with the mean μ and variance \( σ^2 \), then the distribution of all possible sample means is approximately a normal distribution such as \( N(μ, \frac {σ^2 }{n} ) \) if the sample size is large enough. This is referred to as the Central Limit Theorem, which is a key theory in statistics, specifically summarized as follows.

The central limit theorem is very important as a theory underlying modern statistics. <Figure 6.2.4> shows a simulation using 『eStatU』 that, when a population is a normal distribution, the distribution of sample means is also normal, but variances become smaller as the sample size increases. .

When a sample survey is conducted, only one set of samples is selected from the population to estimate a characteristic value of the population, such as the population mean. In general, we consider the sample mean of selected samples as an estimate of the population mean. Do you think this sample mean can estimate the population mean well even if it is only one set of samples?

This is a basic question on the estimation that everyone can think about at least once. The sampling distribution of all possible sample means which we studied in the previous section is the answer to this question. That is, whatever a population distribution is, if the sample size is large enough, all possible sample means are clustered around the population mean in the form of a normal distribution. Therefore, the sample mean obtained from one set of samples is usually close to the population mean. Even in the worst case, the difference between the population mean and sample mean, known as an error, is not so significant, and it is possible to estimate the population mean by using the sample mean. The larger the sample size, the more sample means are concentrated around the population mean based on the central limit theorem and hence we can reduce the error of the estimation.

A value of one observed sample mean is called a point estimate of the population mean.

In general, the sample statistic used to estimate a population parameter must have good characteristics, so that the estimate can be accurate. If the average of all possible sample statistics is equal to the population parameter, then the sample statistic has a good chance to estimate the population parameter and it is called an unbiased estimator. In the previous section we found that a sample mean is an unbiased estimator of the population mean.

If the value of a sample statistic becomes closer and closer to the population parameter when the sample size grows, the sample statistic is called a consistent estimator. The variance of all possible sample means is closer to zero as the sample size increases by the central limit theorem studied in the previous section, so the sample mean is closer to the population mean. Therefore, the sample mean is a consistent estimator of the population mean.

If a sample statistic has the least variance when there are several unbiased estimators for the population parameter, it is called an efficient estimator. The sample mean is also an efficient estimator. Consequently, a sample mean has all good characteristics necessary to estimate the population mean.

In contrast to the point estimate for the population mean, estimating population mean by using an interval is called an interval estimation. If the population is a normal distribution with the mean μ and variance \(σ^2 \), the distribution of all possible sample means is a normal distribution with the mean μ and variance \(\frac {σ^2}{n} \), so the probability that one sample mean will be included in the interval \( μ ± 1.96 \times \frac {σ}{\sqrt{n}} \) is 95% as follows. $$\small P(\mu - 1.96 \times \frac {σ}{\sqrt{n}} < \overline X < \mu + 1.96 \times \frac {σ}{\sqrt{n}} ) = 0.95 $$ We can rewrite this formula as follows. $$\small P(\overline X - 1.96 \times \frac {σ}{\sqrt{n}} < \mu < \overline X + 1.96 \times \frac {σ}{\sqrt{n}} ) = 0.95 $$ Assuming σ is known, the meaning of the above formula is that 95% of intervals obtained by applying the formula \(\small [ \overline {X} - 1.96 \times \frac {σ}{\sqrt{n}}, \overline {X} + 1.96 \times \frac {σ}{\sqrt{n}} ] \) for all possible sample means include the population mean. The formula of this interval is referred to as the 95% confidence interval of the population mean. $$\small \left[ \overline {X} - 1.96 \times \frac {σ}{\sqrt{n}}, \overline {X} + 1.96 \times \frac {σ}{\sqrt{n}} \right] $$

Generally, since \(\small \overline {X} \sim N(μ, \frac {σ^2}{n} ) \), the standardized random variable of \(\small \overline X \), \(\small Z = \frac {\overline X - \mu}{\frac {\sigma}{\sqrt n}} \), follows the standard normal distribution \(N(0,1)\). Therefore, the following probability of the standard normal random variable \(Z\) is \(1-α\) . $$\small P \left( -z_{\alpha/2} < \frac {\overline{X} - \mu } {\sigma/\sqrt{n}} < z_{\alpha/2} \right) = P ( -z_{\alpha/2} < Z < z_{\alpha/2} ) = 1 - \alpha $$ This formula can be written as follows: $$\small P \left( \mu - z_{\alpha/2} \frac {\sigma} {\sqrt{n}} < \overline X < \mu + z_{\alpha/2} \frac {\sigma} {\sqrt{n}} \right) = 1 - \alpha $$ This formula can also be written as follows: $$\small P \left( \overline X - z_{\alpha/2} \frac {\sigma} {\sqrt{n}} < \mu < \overline X + z_{\alpha/2} \frac {\sigma} {\sqrt{n}} \right) = 1 - \alpha $$ The confidence interval for the population mean μ is as follows if the population is normally distributed and the population variance \( \sigma^2 \) is known.

100(1-α)% here is called a confidence level, which refers to the probability of intervals that will include the population mean among all possible intervals calculated by the confidence interval formula. Usually, we use 0.01 or 0.05 for α. \( z_{α} \) is the upper α percentile of the standard normal distribution. In other words, if \(Z\) is the random variable which follows the standard normal distribution, the probability that \(Z\) is greater than \( z_{α} \) is α, i.e., $$ P(Z > z_{α} ) = α $$ For example, \( z_{0.025;} \) = 1.96, \( z_{0.95} \) = -1.645, \( z_{0.005} \) = 2.575.

<Figure 6.2.6> shows a simulation of the 95% confidence interval for the population mean by extracting 100 sets of samples with the sample size \(n\) = 20 from a population of 10,000 numbers which follow the standard normal distribution N(0,1). In this case, 96 of the 100 confidence intervals contain the population mean 0. This result might be different on your computer, because the program use a random number generator which depends on computer. Whenever we repeat these experiments, the result may also vary slightly.

One problem in estimating the unknown population mean by using the formula of the confidence interval in the previous section is that either the population variance may not be known or the population is not normally distributed. If the sample size is large enough, a confidence interval of the population mean can be obtained approximately using the sample variance instead of the population variance on the previous formula. However, if the sample size is small and the sample variance is used, a confidence interval based on the t distribution should be used.

The \(t\) distribution was studied by a statistician W. S. Gosset, who worked for a brewer in Ireland, and published his study result in 1907 under the alias Student. So \(t\) distribution is often referred to as Student's \(t\) distribution. The \(t\) distribution is not just a single distribution, but it is a family of distributions under a parameter called a degree of freedom, 1,2, ... , 30, ... and denoted as \(t_1 ,t_2 , ... , t_{30} , ... \)

The shape of the \(t\) distribution is symmetrical about zero (y axis), similar to the standard normal distribution, but it has a tail that is flat and longer than the standard normal distribution. <Figure 6.2.7> shows the standard normal distribution N(0,1) and \(t\) distribution with 3 degrees of freedom at the same time by using the \(t\) distribution module of 『eStatU』.

The \(t\) distribution is closer to the standard normal distribution as degrees of freedom increase above 30. This is why a confidence interval can be obtained approximately by using the standard normal distribution if the sample size is greater than 30. Denote \(t_{n:α}\) as the 100(1-α)% percentile of the \(t\) distribution with \(n\) degrees of freedom. For example, \(t_{7:0.05}\) is the 100(1-0.05)=95% percentile of the \(t\) distribution and its value is 1.895 as <Figure 6.2.8>. In the standard normal distribution, this value was 1.645. Since the \(t\) distribution is symmetrical, \(t_{n:α} = -t_{n:1-α}\).

In order to find a percentile value of the \(t_{7}\) distribution using 『eStatU』, click on '\(t\) distribution' in the main menu of 『eStatU』 and then set the degree of freedom (df) to 7, and set the probability value in the fifth option below the \(t\) distribution graph to 0.95, then \(t_{7:0.05}\) = 1.895 will appear as in <Figure 6.2.8>.

Consider the interval estimation of the population mean when you do not know the population variance, but assume that the population is a normal distribution. If \( X_1 , X_2 , ... , X_n \) is the random sample of size \(n\) from the normal population, then it can be shown that the distribution of \( \frac {\overline X -\mu}{S/\sqrt{n}} \), where σ is replaced with S, is the \(t\) distribution with \( n-1 \) degrees of freedom. $$\small \frac {\overline X -\mu}{\frac{S}{\sqrt{n}}} \sim t_{n-1} $$ Hence the probability of the (1 - α)% interval is as follows. $$\small P \left( -t_{n-1;\alpha/2} < \frac {\overline{X} - \mu } {S\sqrt{n}} < t_{n-1:\alpha/2} \right) = 1 - \alpha $$ The left hand side of the above formula can be summarized as the confidence interval for the population mean when the population variance is unknown as follows:

If we know the relationship between the population variance and the sample variance, it is possible to estimate the population variance. In this section, the distribution of all possible sample variances is discussed in section 6.3.1 and the estimation of the population variance using the sample variance is discussed in section 6.3.2.

Consider the following example to understand the sampling distribution of sample variances.

As observed in the above example, the sampling distribution of the sample variances is an asymmetric distribution with many small sample variances and a few large sample variances. In general, when the population is normally distributed and the population variance is \(σ^2\), all possible sample variances scaled by a constant follow the chi-square \(χ^2\) distribution. More accurately, the sample statistic $$ \frac {(n-1)S^2}{\sigma^2} $$ follows a chi-square distribution with \(n-1\) degrees of freedom.

This chi-square distribution is a family of distributions depending on the degree of freedom, such as \(χ^2_{1} , χ^2_{2} , ... , χ^2_{27} , ...\) etc. The chi-square distribution is an asymmetrical distribution as <Figure 6.3.2>. If the degree of freedom is small, the shape of the chi-square distribution is much skewed to the right.

A cumulated probability and a percentile of the chi-square distribution can be calculated by using 『eStatU』 as in <Figure 6.3.3>.

[ ]

The sampling distribution of all sample variances is summarized as follows:

Examples of estimating the population variance are as follows.

In order to estimate the population variance, the sampling distribution of all possible sample variances should be used. As discussed in [Example 6.3.1], for an infinite population, the mean of all possible sample variances is the population variance. That is, the sample variance \(S^2\) is an unbiased estimator of the population variance \(σ^2\). Therefore, the sample variance is used to estimate the population variance. In addition, estimation of the population standard deviation \(σ\) uses the sample standard deviation \(S\), but it should be noted that the sample standard deviation is not an unbiased estimator of the population standard deviation. However, as the sample size increases, there is no significant error in using \(S\) as an estimator of \(σ\).

In the previous section, when a population was normally distributed, we found that the sample variance multiplied by a constant, \( \frac {(n-1)S^2}{\sigma^2} \), follows the chi-square distribution with \(n-1\) degrees of freedom. Using this, we can find the confidence interval of the population variance \(σ^2\) as follows:

If we know the relationship between the population proportion and sample proportion, it is possible to estimate the population proportion. In this section, the distribution of all possible sample proportions is discussed in section 6.4.1 and the estimation of the population proportion using the sample proportion is discussed in section 6.4.2.

Consider the sampling distribution of all possible sample proportions by using the following example.

When the sample size is large, the sampling distribution of all possible sample means in general is as follows:

If the population size is \(N\) which is finite and the sampling is without replacement, the variance of \(\hat p\) becomes \(\frac{p(1-p)}{n} \frac{N-n}{N-1}\). The term \(\frac{N-n}{N-1}\) is called the finite population correction factor.

Some practical examples to estimate the population proportion are as follows.

Assume that the proportion of a population is \(p\). As discussed in Section 6.1, since the sample proportion, \(\hat p\), meets all the criteria of a good estimator when estimating the population proportion \(p\), the sample proportion is used to estimate the population proportion. The sampling distribution of all possible sample proportions is approximately a normal distribution with the mean \(p\) and variance \(\frac{p(1-p)}{n}\) and the standard error of the sample proportions is \(\sqrt{\frac{p(1-p)}{n}}\). But, since the population proportion \(p\) is unknown, we use \(\sqrt{\frac{\hat p(1- \hat p)}{n}}\) as an estimate of the standard error of the sample proportions.

From the fact that the distribution of the sample proportion \(\hat p\) is approximated to a normal distribution, interval estimation of the population proportion \(p\) can be done as follows.

Until now, we have studied the estimation of a population parameter using a given sample. However, it is often necessary to first determine how large a sample should be before obtaining such a sample. This problem is closely related to the precision of the estimate. We could see in the previous section that the width of the confidence interval was reduced if the sample size is larger. However, as sampling is costly, it usually determines the minimum sample size required to achieve this precision that would satisfy the researcher.

The 100(1-α)% confidence interval of a population mean μ is as follows as explained in section 6.2. $$\small \left[ \overline X - z_{\alpha/2} \frac {\sigma} {\sqrt{n}} ,\; \overline X + z_{\alpha/2} \frac {\sigma} {\sqrt{n}} \right] $$ The term \( z_{\alpha/2} \frac {\sigma} {\sqrt{n}}\) which is the half size of the confidence interval is called the maximum allowable error bound to estimate the population mean and is denoted as \(d\).

If the maximum allowable error bound and the confidence level 1-α are given, the sample size can be determined by solving the following equation. $$ z_{\alpha/2} \frac {\sigma} {\sqrt{n}} = d $$ The solution of \(n\) in the above equation is as follows. $$ n = {\left( \frac { z_{\alpha/2} {\sigma} } {d} \right)}^2 $$

Since the population standard deviation σ in the above equation is usually unknown, the estimated value from past experience or from data obtained by a preliminary survey is used. The estimation of the population standard deviation σ through a preliminary survey can be done by using the range as follows: $$ \hat \sigma = \frac {range}{4} = \frac {max - min}{4} $$

As explained in section 6.4, the 100(1-α)% confidence interval of a population proportion \(p\) is as follows. $$ \left[ \hat p - z_{α/2} \sqrt{\frac{\hat p (1-\hat p)}{n} }, \hat p + z_{α/2} \sqrt{\frac{\hat p (1-\hat p)}{n} } \right] $$

The term \(z_{α/2} \sqrt{\frac{\hat p (1-\hat p)}{n} }\) which is the half size of the confidence interval is called the maximum allowable error bound to estimate the population proportion and is denoted as \(d\).

Therefore, if the maximum allowable error bound \(d\) and the confidence level 1-α are given, the sample size can be determined by solving the following equation. $$ z_{α/2} \sqrt{\frac{\hat p (1-\hat p)}{n} } = d $$ The solution of the above equation is as follows. $$ n = \hat p (1-\hat p) ( {\frac{z_{α/2}}{d}} )^2 $$

In the above equation, the value of the sample proportion \(\hat p\) is usually used by the value of past experience or preliminary investigation. If there is no information about the population proportion, the value of 0.5 is usually used.

When producing a product in general, there is a variation in the quality characteristic values even if it is produced under the same working conditions. The causes of the variation can be divided into chance causes that can always occur in the production process and assignable causes that occur due to such as worker's carelessness, use of defective materials, abnormalities in production equipment, etc. If the quality characteristic value varies up to a certain limit around a specific value, it is regarded as a change due to chance causes and is tolerated. If it exceeds the limit, it is regarded as an explainable cause, and the cause is investigated and an action is taken to prevent the occurrence of defective products. If the quality level is monitored by controlling the process to prevent defectives as above, it is called a statistical quality control. One of the statistical tools widely used for this purpose is the control chart which was first introduced by W.A Shewart in 1924.

A control chart is a diagram to manage changes in quality characteristic values. As shown in <Figure 6.6.1>, there is an upper control limit (UCL) at the top, a center line (CL) in between, and a lower control limit (LCL) at the bottom. The characteristic value of an inspected sample in each time are plotted as a line graph. When the quality characteristic values do not exceed the control limit lines and are not related to each other, the process is said to be under control. If the quality characteristic value is outside the control limits or, if the characteristic values are related each other even they are within the control limits, the process is said to be out of control.

Control charts are divided into a control chart by variable and control chart by attribute.

A control chart by variable refers to a control chart when the quality characteristic value is a quantitative variable such as temperature, pressure, strength, weight, etc. The quantitative variable has the advantage of being able to obtain a lot of information about quality characteristics, but it has the disadvantage of requiring a lot of purchase and maintenance costs for measuring instruments, and requiring a lot of manpower and time for measurement. There are many types of control charts for variable, but the sample mean control chart (\( \overline X \) chart) that uses the sampling distribution of sample means studied in this chapter and the sample range control chart (\(R\) chart) that manage dispersion are commonly used in production process. These control charts requires the assumption that the quality characteristic values follow a normal distribution \(N(\mu, \sigma^2 )\). If the quality characteristic value is normally distributed, the distribution is completely determined by the mean \(\mu\) and variance \(\sigma^2\). Therefore, if the sample mean and sample range are managed simultaneously, the distribution of the quality characteristic is managed.

The theoretical basis of the sample mean control chart (\( \overline X \) chart) is based on the sampling distribution theory of sample means studied in Section 6.2. If a population distribution of quality characteristic values is a normal distribution with a mean of \(\mu\) and variance of \(\sigma^2\), that is \(N(\mu , \sigma^2 ) \) , the sampling distribution of all possible sample means of size \(n\) follows a normal distribution with a mean of \(\mu\) and variance of \(\frac {\sigma^2}{n}\), that is \(N(\mu , \frac{\sigma^2}{n}) \). Therefore, according to the characteristics of the normal distribution, the interval $$ [\mu - 3 \frac{\sigma}{\sqrt{n}},\quad \mu + 3 \frac{\sigma}{\sqrt{n}}] $$ contains 99.74% of all possible sample means. The right limit of this interval is called the upper control limit(\(UCL_{\overline X}\)) of the \(\overline X\) chart, the left limit is called the lower control limit (\(LCL_{\overline X}\)), and \(\mu\) is called the center line (\(CL_{\overline X}\)) as following. $$ \begin{align} UCL_{\overline X} &= \mu + 3 \frac{\sigma}{\sqrt{n}} \\ CL_{\overline X} &= \mu \\ LCL_{\overline X} &= \mu - 3 \frac{\sigma}{\sqrt{n}} \end{align} $$ Since the mean \(\mu\) and standard deviation \(\sigma\) of the quality characteristic values are not known, they are estimated using samples. In a production process that produces a large number of products, several samples of size \(n\) are selected, and the average of all sample means is used to estimate \(\mu\) and the sample variance or sample range is used to estimate \(\sigma\). Estimation of \(\sigma\) using the sample ranges are commonly used in quality control sites because the sample range is an easily understandable statistics to workers.

Let \( {\overline X}_{1}, {\overline X}_{2}, ... , {\overline X}_{k} \) be the sample means obtained by selecting \(k\) samples of size \(n\), and let \( \overline {\overline X} \) be the average of these sample means. Let \( {R}_{1}, {R}_{2}, ... , {R}_{k} \) be the sample range (maximum - minimum) of each sample, and let \(\overline R\) be the average of these ranges as following: $$ \overline {\overline X} = \frac{\sum_{i=1}^{k} {\overline X}_{i} }{k} , \quad {\overline R} = \frac{\sum_{i=1}^{k} {R}_{i} }{k} $$ The mean \(\mu\) is estimated by using \( \overline {\overline X} \). Since the sample standard deviation \(s\) is not an unbiased estimator of the population standard deviation \(\sigma\), \(\sigma\) is estimated by using the mean of the ranges \(\overline R\) and the coefficient \(d_2\) that depends on the number of samples, \(n\), as follows. $$ \hat \sigma = \frac {\overline R}{d_2} $$ \(\hat \sigma\) is an unbiased estimator of the population standard deviation \(\sigma\). Here, \(d_2\) is a constant that depends on the number of samples, \(n\), and is shown in Table 6.6.1.

Table 6.6.1 Constants which are used in control charts

sample size \(n\)	\(A_2\)	\(D_3\)	\(D_4\)	\(d_2\)
2	1.880	0	3.267	1.128
3	1.023	0	2.574	1.693
4	0.729	0	2.282	2.059
5	0.577	0	2.114	2.326
6	0.483	0	2.004	2.534
7	0.419	0.076	1.924	2.704
8	0.373	0.136	1.864	2.847
9	0.337	0.184	1.816	2.97
10	0.308	0.223	1.777	3.078
11	0.285	0.256	1.774	3.173
12	0.266	0.284	1.716	3.258
13	0.249	0.308	1.692	3.336
14	0.235	0.329	1.671	3.407
15	0.223	0.348	1.652	3.472
16	0.212	0.364	1.636	3.532
17	0.203	0.379	1.621	3.588
18	0.194	0.392	1.608	3.64
19	0.187	0.404	1.596	3.689
20	0.18	0.414	1.586	3.735
21	0.173	0.425	1.575	3.778
22	0.167	0.434	1.566	3.819
23	0.162	0.443	1.557	3.858
24	0.157	0.452	1.548	3.895
25	0.153	0.459	1.541	3.931

Therefore, the center line (\(CL_{\overline X}\)) and control limit lines of the \(\overline X\) chart using the estimated values of \(\mu\) and \(\sigma\) are as follows. $$ \begin{align} UCL_{\overline X} &= \overline {\overline x} + 3 \frac{\overline R / d_2}{\sqrt{n}} \\ CL_{\overline X} &= \overline {\overline x} \\ LCL_{\overline X} &= \overline {\overline x} - 3 \frac{\overline R / d_2}{\sqrt{n}} \end{align} $$ Here, by setting \( \frac{ 3 / d_2 }{\sqrt{n}} \) as a new constant \(A_2\), the control limit lines can be written as follows. The constant \(A_2\) which depends on \(n\) is also summarized in Table 6.6.1. $$ \begin{align} UCL_{\overline X} &= \overline {\overline x} + A_2 \overline R \\ CL_{\overline X} &= \overline {\overline x} \\ LCL_{\overline X} &= \overline {\overline x} - A_2 \overline R \end{align} $$

When the population distribution of quality characteristic values is a normal distribution with a mean of \(\mu\) and variance of \(\sigma^2\), that is \(N(\mu, \sigma^2 )\), let the sample range of a sample of size \(n\) be the random variable \(R\). The distribution of a random variable \(W = \frac {R}{\sigma}\) which divides the sample range \(R\) by a constant \(\sigma\) can be derived theoretically. Let \(d_3\) be the standard deviation of the random variable \(W\) which depends on \(n\). Since the sample range is \(R = W \sigma \), the standard deviation of \(R\) becomes \(\sigma_R = d_3 \sigma \). Since we do not know \(\sigma\), if we use the estimated value, \(\hat \sigma = \frac {\overline R}{d_2}\), the estimated standard deviation of \(R\) will be \(\sigma_R = d_3 \hat \sigma = d_3 \frac {\overline R}{d_2}\). Therefore, the center line and control limit lines of the \(R\) chart are as follows. $$ \begin{align} UCL_{\overline R} &= {\overline R} + 3 \hat {\sigma_R} = \overline R + 3 d_3 \frac{\overline R}{d_2} \\ CL_{\overline R} &= {\overline R} \\ LCL_{\overline X} &= {\overline R} - 3 \hat {\sigma_R} = \overline R - 3 d_3 \frac{\overline R}{d_2} \end{align} $$ To make calculations easier, if we let \(D_3 = 1 - 3 \frac{d_3}{d_2}\), \(D_4 = 1 + 3 \frac{d_3}{d_2}\), the center line and control limit lines of the \(R\) chart are as follows. The constants \(D_3\) and \(D_4\) are also listed in Table 6.1.1. $$ \begin{align} UCL_{\overline R} &= {\overline R} D_4 \\ CL_{\overline R} &= {\overline R} \\ LCL_{\overline X} &= {\overline R} D_3 \end{align} $$

When managing the quality of a product, quality characteristics such as weight, length, and temperature cannot be measured sometimes and we can only judge whether the product ‘passes’ or ‘fails’. In this case, control charts by attribute are used. A control chart for the defective fraction rate (\(p\) chart) and a control chart for the number of defective items (\(C\) chart) are commonly used. \(p\) chart is explained only In this book. Please refer other references for \(C\) chart. .

Assume that in one production process, the product is produced with the defective rate \(p\) and maintained at a constant level. If we select a sample of size \(n\) from this production process, the number of defective products included in the sample follows a binomial distribution as discussed in Chapter 5. In other words, if the defective rate of products in the process is \(p\), and let the random variable X be ‘the number of defective products in the sample of size n’, the probability density function \(f(x)\) of X is as follows. $$ f(x)= \frac{n!}{x!(n-x)!} p ^{x} (1-p) ^{n-x} \quad x=0,1, ... ,n $$ The mean of X is E(X) = \(np\) and the variance of X is V(X) = \(np(1-p)\). Therefore, if the defective rate of the process is \(p\), the mean and variance of the defective rate of a sample, \(\hat p = \frac {X}{n}\), are E(\(\hat p\)) = \(p\), Var(\(\hat p\)) = \(\frac{p(1-p)}{n}\). The distribution of the defective rate of a sample, \(\hat p\), follows approximately a normal distribution when the sample size, \(n\), is sufficiently large. In general, if the defective rate of the process, \(p\), is not a value very close to 0 or 1 (usually \(np\) > 5, \(n(1-p)\) > 5 is considered satisfactory), \(\hat p\) follows approximately a normal distribution \(N(p, \frac{p(1-p)}{n} )\). Therefore, the following interval includes approximately 99.74% of all possible sample defective rates due to the characteristics of the normal distribution. $$ [p - 3 \sqrt {\frac{p(1-p)}{n}}, p + 3 \sqrt {\frac{p(1-p)}{n}} ] $$ The right limit of this interval is called the upper control limit of the \(p\) chart (\(UCL_p\)), the left limit of this interval is called the lower control limit (\(LCL_p\)), and \(p\) is called the center line (\(CL_p\)). However, since the defective rate of the population, \(p\), is not known, several samples are selected and the average rate of defective products included in all selected samples, \(\overline p\), is used as an estimator of \(p\) as follows. $$ \overline p = \frac {\text {number of defective products in all samples} } {\text {number of products in all samples}} $$ The center line and control limit lines of the \(p\) chart using this estimator \(\overline p\) are as follows. $$ \begin{align} UCL_{p} &= \overline p + 3 \sqrt {{\overline p}(1-{\overline p})/n } \\ CL_{p} &= \overline p \\ LCL_{p} &= \overline p + 3 \sqrt {{\overline p}(1-{\overline p})/n } \end{align} $$