1-3 Measure of Location Just as graphics can enhance the display of data, numerical descriptions are also of value. In this section and the next, we present several important numerical measures for describing the characteristics of data. Suppose that the data are x1, x2…xn, where each xi is a number. One important characteristics of a set of numbers is its location, or central tendency. This section presents methods for describing the location or centre of a data set. Section 1-4 will present methods for describing the variability in data.

The Mean The most common measure of location or centre of data is the ordinary arithmetic average or mean. Because we usually think of the data as a sample, we will refer to the arithmetic mean as the sample mean. Definition If the observations in a sample of size n are x1, x2…xn, then the sample mean is

Example 1-4 The sample mean tension bond strength for the 10 observations collected from the modified cement mortar from section 1-2.1 is

The value of the sample mean is more precise than the precision associated with an individual observation. Consequently, many times we will report the sample mean to one more digit beyond that used in the individual measurements. A physical interpretation of the sample mean as a measure of location is shown in Fig., which is a dot diagram of the tension bond strength data. Notice that the sample mean = 16.764 can be thought of as a "balance point”. That is, if each observation represents one pound of mass placed at the point on the x-axis, then a fulcrum located at would exactly balance this system of weights.

For the aluminium - lithium alloy strength data in table 1-1 we find that the sum of all 80 observations is so the sample mean is From examination of either the stem-and-leaf plot in Fig. 1-3 or the histogram in Fig. 1-6 it seems that the sample mean 162.7 psi is a "typical" value of comparative strength, since it occurs near the middle of the data where the observations are concentrated. However, this impression can be misleading. Suppose that the histogram looked like Fig 1-12 The mean of these data is still a measure of central tendency but it does not necessarily imply that most of the observations are concentrated around it.

Remember, if we think of the observations as having unit mass, the sample mean is the point at which the histogram will exactly balance. The sample mean represents the average value of all the observations in the sample. We can also think of calculating the average value of all the observations in a population. This is called the population mean, and it is denoted by the Greek Letter m (mu). When there are a finite number of observations (say N) in the population, then the mean is

Example 1-5 The April 22, 1991 issue of Aviation Week and Space Technology reports that during Operation Desert Storm, US Air Force F-117A pilots flew 1270 combat sorties for a total of 6905 hours. Therefore, the mean duration of an F-117 A mission during this operation was

In subsequent chapters, we will discuss models for infinite populations, and we will give a more general definition of m. In many practical applications of statistics to engineering problems the mean is unknown and it is impossible (or at least impractical) to examine all the members of the population, as in example 1-5. In the chapters on statistical inference, we will present methods for making inferences about the population mean that are based on the sample mean. For example, we will use the sample mean as a point estimate of m.

1-4 Measures of Variability Location or central tendency does not necessarily provide enough information to describe data adequately. For example, consider the compressive strengths (in psi) obtained from two samples of Aluminium-Lithium Alloy: Sample 1: 130, 150, 145, 158, 165, 140. Sample 2: 90, 128, 205, 140, 165, 160. The mean of both samples is 148 psi. However, referring to the dot diagram in Fig. We see that the scatter or variability of sample 2 is much greater than that of sample 1.

1-4-1 The Sample Range A very simple measure of variability is the sample range, defined as the difference between the largest and smallest observations. That is the range is r = max (xi) - min (xi) For the two samples of compressive strength above, we find that the range of the first sample is First sample: r1 = 165 - 130 = 35 Second sample: r2 = 205 - 90 = 115 Clearly, the larger the range, the greater the variability in the data. The sample range is easy to calculate, but it ignores all the information in the sample between the smallest and largest observations. For example, the two samples Sample 1: 1, 3, 5, 8, 9 Sample 2: 1, 5, 5, 5, 9 However, in the second sample there is variability only in the two extreme values, while in the first sample the three middle values vary considerably. Sometimes, when the sample size is small, say n £ 10 the information loss associated with the range is not too serious. The range is widely used in statistical applications in quality control where sample size of 4 or 5 are commonly employed. Both have the same range (r=8)

The Sample Variance and Sample Standard Deviation The most important measures of variability are the sample variance and the sample standard deviation. Definition If x1, x2… xn is a sample of observations, then the sample variance is The sample standard deviation, S, is the positive square root of the sample variance. The units of measurements for the sample variance are the square of the original units of the variable. Thus, if x is measured in pounds per square inch (psi), the units for the sample variance are (psi)2. The standard deviation has the desirable property of measuring variability in the original units of the variable of interest x.

Example 1-8 Table 1-3 displays the quantities needed for calculating the sample variance and sample standard deviation for sample 2 of aluminium-lithium alloy compressive strengths. These are plotted in Fig. 1-15. The numerator of S2 is Calculation of the sample variance and sample standard deviation

Computation of S2 The computation of S2 requires calculation of , n subtractions, and n squaring and adding operations. If the original observations or the deviation is are not integers, the deviations may be tedious to work with, and several decimals may have to be carried to ensure numerical accuracy. A more efficient computational formula for the sample variance is obtained as follows:

Note

Table 1-9 Table 1-4 displays the calculations required for computing the sample variance and standard deviation using the shortcut method, equation 1-6 Table 1-4 calculations of the sample variance and sample standard deviation These results agree exactly with those obtained previously.

Population Variance Analogous to the sample variance s2, there is a measure of variability in the population called the population variance. We will use the Greek Letter s2 (sigma squared) to denote the population variance. The positive square root of s2, or s, will denote the population standard deviation. When the population is finite and consists of N values, we may define the population variance as More general definitions of the variance s2 will be given in subsequent chapters. We observed previously that the sample mean could be used to make inferences about the population mean. Similarly, the sample variance can be used to make inferences about the population variance.