Engineering Data Summary and Presentation

Before moving into the theory and practice of statistical process control (SPC), it is important that engineers get involved with exchanging ideas, and at each step of the way colleagues in engineering will be sharing information and data with one another, and engineers will be making presentations to decision makers who have no background in engineering. In those situations, the quality and clarity of displayed data will play a large role in statistical analysis and process control. Well‐constructed graphics and data summaries are essential to good statistical thinking because they focus the engineer on important features of the data. They help the engineer make sense of the data and can provide insight about potential problem‐solving approaches or the type of model should be used.

The computer has become an important tool in the presentation and analysis of data. Although many statistical techniques require only a handheld calculator, this approach can be time‐consuming. Computers can perform the tasks more efficiently. Most statistical analysis is done using a prewritten library of statistical programs. The user enters the data and then selects the types of analysis and output displays that are of interest. Statistical software packages are available for both mainframe machines and personal computers. Among the most popular and widely used packages are Statistical Analysis System for both servers and personal computers (PCs) and Minitab for the PC. We can describe data feature numerically. For example, we can characterize the location or central tendency in the data by ordinary arithmetic average or mean. Because we almost always think of our data as a sample, we will refer to the arithmetic mean as the sample mean (Montgomery et al. 2011).

Sample Mean

If the n observations in a sample are denoted by x₁, x₂, …, x_n, the sample mean is

(8.1)

The sample mean is the average value of all the observations in a given data set. Usually, these data are a sample of observations that have been selected from some large population of observations. We could also think of calculating the average value of all the observations in a population. This average is called the population mean, and it is denoted by the Greek letter μ (mu). When there is a finite number of observation (say, N) in the population, the population mean is

(8.2) equation

Although the sample mean is useful, it does not convey all of the information about a sample of data. The variability or scatter in the data may be described by the sample variance or the sample standard deviation. If the n observations in a sample are denoted by x₁, x₂, …, x_n, then the sample variance is

(8.3) equation

Analogous to the sample variance S², there is a measure of variability in the population called population variance. We will use the Greek letter σ² (sigma squared) to denote the population variance. The positive square root of σ², or σ, will denote population standard deviation. When the population is finite and consists of N values, we may define the population variance as

(8.4) equation

EXAMPLE 8.1

An important quality characteristic of water is the concentration of suspended solid particle in mg/l. Twelve measurements on suspended solids from Wisconsin River by an industrial site are as follows: 42.4, 65.7, 29.8, 58.7, 52.1, 55.8, 57.0, 68.7, 67.3, 67.3, 54.3, and 54.0. Calculate the sample average and sample standard deviation.

SOLUTION

Sample average: images

Sample standard deviation: images

EXAMPLE 8.2

The following data are direct solar intensity measurements (W/m²) on different days at a location in the State of Rajasthan, India: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768, 870, 918, 940, 946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939, 955, 960, 498, 653, 730, and 753.

Calculate the sample mean and sample standard deviation. Prepare a dot diagram of these data. Indicate where the sample mean falls on this diagram. Provide a practical interpretation of the sample mean.