An Overview of Basic Statistics

Editor's Notes

• #10: Solution: See Working Problem slides
• #13: Solution: See Working Problem slides
• #14: In describing a measure, we may have one of the following four scenarios:   Inaccurate and imprecise measure Accurate but imprecise measure Precise but inaccurate measure Precise and accurate measure
• #15: Solution: See Working Problem slides
• #16: Process and system In general, the term ‘process’ can be used to imply one or more of five basic elements: machine, material, methodology, people, and environment. The term ‘system’ is typically used when we are prepared to specify inputs and outputs.
• #17: Process and system Process Example: We may describe education as a process in which material implies course material (papers, books, etc.), machine implies media projector or computers, methodology implies the method of teaching (face-to-face, or distance learning), people implies teachers and students, and environment implies a class room or a laboratory.
• #18: System Example: a weaving machine (a loom) is a system in which the input is the yarns, and the output is the fabric
• #19: How to describe data The starting point in any statistical analysis is to read data using the language of statistics. As in any language where a sentence will not be understood or communicated without an organized sentence structure, the language of statistics uses the so called ‘descriptive statistics’ to establish an organized and meaningful display of data with the goal being to reduce the flood of data presented down to few statistics that can fully describe the data. Another important reason for performing descriptive statistics is to detect any abnormality in the data that may result from typographical errors, data acquisition errors, or mixing of data sets. For this reason, it is always a good practice to begin any statistical analysis by performing an analysis of descriptive statistics. The following example illustrates these points. Give the student time to describe each data set before moving to the next slide Solution: Quiz 1: all grades are identical; each student made 90 in the first quiz. Is this possible? Quiz 2: most grades range from 77 to 88, but two grades seem exceptional, 30 and 99. Quiz 3: grades seem to be split into two groups: a group in the high 60s and low to mid 70s and another group in the high 80s or above Quiz 4: grades seem to reflect a typical quiz grades with few students making relatively low grades, 76 and 78; few students making high grades, 95 and 96; and the majority are in the intermediate range (82, 84, 85, 86, 87, 88).
• #22: This example is an over simplification of a frequency diagram and should be considered only for illustration of the concept of graphing frequency. In the next slide, the example will illustrate a typical Histogram.
• #23: Discuss this with students without detailed description of how you construct a histogram as this will be covered in Chapter 3. The frequency distribution here simply illustrates two key points: The bulk of the data concentrating from 130 to 170 Outliers or extreme values
• #27: We will learn more about Probability in Chapters 4, 5, and 6
• #29: The most common sampling techniques are: Random sampling Stratified sampling Cluster sampling Random sampling- In random sampling, every element in the population will have the same opportunity of being drawn in the sample. For example, in a class room of 40 students, we can draw a representative sample of 10 students by writing each student name or identification number on a post card, shuffling the 40 cards, placing them in a box, and blindly withdraw 10 cards representing the sample. This is the physical approach to perform random sampling. In Chapter 7 we will discuss how to perform random sampling using Excel® data analysis in a matter of seconds. Stratified sampling- In some situations, complete random sampling may not be possible due to the difficulty of accessing some portions of the population. In this case, we may use the so-called ‘stratified sampling’. In this approach, we simply divide the population into pre-specified categories, or strata, and draw samples from each category in random fashion and with size proportional to the category size in the population. This approach ensures that each category of the population is represented in the sample. For example, suppose we have a small city population of 100,000 people; 80,000 white, 15,000 African American, and 5,000 Hispanic. If these ethnic groups are intermingled in the city, we should then use stratified sampling to ensure that each group is represented in the sample and in proportional-weight allocated fashion. Accordingly, if the sample size is 100 people, 80 people should be selected randomly from the white people group, 15 from the African American group, and 5 people from the Hispanic community group. Figure 1.6 illustrates the differences between random sampling and stratified sampling. Cluster Sampling- This is used when the population itself consists of natural or pre-designed clusters, with each cluster having its own unique features. For example, a company may have three or four factories in different areas producing the same product. In this case, a cluster sample should be a sample consisting of components from each factory or cluster.
• #30: In general, there are two main sources of variability in a process: (i) inherent variability, and (ii) induced variability. Inherent variability normally results from natural sources or from some form of limitation in any process element (material, machine, people, etc). For example, humans are inherently different by virtue of their creation. This is a positive variability as no one can conceive a world with all human being alike in everything. Even twins are inherently different in so many aspects. Some statistics suggest that there are roughly over 120 million living human twins in the world today, and nearly 10 million monozygotic twins (twins derived from a single fertilized egg zygote, or human identical twins). Similarly, plants, animals, and all types of God-created things are inherently different.   Induced variability is defined by the differences between things, which might otherwise be thought to be alike because they were produced under conditions, which were as nearly alike as it is possible to make them. This type of variability can be observed in products that are made to be identical (coffee makers, refrigerators, cars, etc.), yet they exhibit some differences resulting from human or manufacturing errors. For example, a manufacturer producing ceramic tiles set the machines and the manufacturing operations to produce the same dimensions of ceramic tiles every day, yet these tiles may encounter some variability in thickness, area, and even color in the absence of good control of the manufacturing process.
• #32: A variable represents a phenomenon or a characteristic of a process or system under investigation in which different elements are associated with different values. There are three main types of variables:   Continuous variables Discrete variables Special variables   A continuous variable can take any value in a specified interval falling within its plausible range. For example, the diameter of a fine metal rod may take a value of 40, 40.25, 40.75 or 41 millimeter. These data imply continuity in the variable values. Similarly, variables such as student grades, people income, ages, material strength, thickness, length, temperature, concentration, and weight are all of continuous type. In one sentence, continuous variables are variables whose sample spaces (or all possible outcomes) are intervals.   Discrete variables are variables whose sample spaces are sets of discrete points. They are associated with situations in which a count is made such as the count of pass or fail, stop or go, and good or bad. Educators use discrete variables to measure performance parameters such as the number of failing students, the number of drop-out students, and the number of absences. Engineers and manufacturers use discrete values to describe many parameters such as the number of defects, the number of rejects, and the number of machine failures. A discrete variable can take only integer values (e.g. 0, 1, 2, ...). Specifying the variable as being continuous or discrete is critical for performing a reliable statistical analysis. In Chapters 5 and 6, we will discuss probability distributions for discrete and continuous variables, respectively.   Special variables are those that are not easily described in quantitative manner. This type of variables is typically initiated from nonnumeric sources. For example, when we describe the comfort resulting from wearing some clothing, we may use terms such as smooth, flexible, harsh, rough, soft and stiff. In order to use statistics to describe the extent of comfort, we should first express these characterizations in some numeric format (e.g. establishing a numeric index). Depending on the structure of this format, the special variable can then be treated as a discrete or a continuous variable.   Another way to classify variables is by dividing them into two main categories: ‘quantitative’ and ‘qualitative’ variables. Quantitative variables are those that can be measured numerically. These include income, weight, temperature, grades, etc. Qualitative variables are those that cannot be described numerically but may be converted into some kind of numerical codes depending on their natures. Categorical variables are just another name for qualitative variables. For example, when we categorize students as freshman, sophomore, junior, and senior, we are in essence using qualitative variable. If we have to include these categories in a data set, it would be useful to associate them with some codes such as ‘1’ for freshman, ‘2’ for sophomore, ‘3’ for junior and ‘4’ for senior. This is a key step in dealing with this type of variables in a statistical analysis. Other qualitative variables include: color, taste, style, pain level, gender, race, and job position.
• #34: Data can talk! Or data often try to tell us something!! But, to what extent data can truly tell us what we need to know? The key to answering this question stems from the level of measurement of the data under consideration. In order to describe a variable we have to use associated data and in order to use the data in a meaningful analysis, we must realize the data level of measurement. For example, if we have data representing male and female and we decided to give male, say a code number 1, and female a code number 2, then taking the average of these two numbers, which will be 1.5, would mean nothing and will serve no purpose in our analysis. The reason, this average exhibits no meaning is that the data used is at its lowest level of measurement or at what is generally called the nominal level of measurements; it is only a code with the numbers 1 and 2 being used to distinguish males from females in the data set in case we need to analyze each category separately or compare between the two in terms of other variables such as income, weight, ages, etc.   In general, there are four basic levels of measurements as illustrated by the pyramid of Figure 1.7, where the data resolution and meaningful aspects progressively increase as we move from the base of the pyramid to the top. The lowest level is the nominal measurement data. At this level, variables are typically qualitative, mutually exclusive and exhaustive. They typically have no meaningful magnitude or logical order and we can do very little with this type of data in terms of statistical analysis except for data organization and information management. Examples of nominal measurements include: eye or skin color, gender, people names, and religion.   Ordinal measurement level data are those used for sequencing or ranking. For example, you may wish to rate your professor after completing a statistics course according to a particular trait, say as excellent, very good, good, fair, and poor. Numbers assigned for these ratings may be: 0 (poor), 1(fair), 2(good), 3(very good), and 4(excellent). These numbers, though at equal distances, do not imply a rating magnitude; meaning, a rating of 4 is not twice as much a rating of 2. However, the order is meaningful as the higher the number the better the rating. Again, the data here are mutually exclusive and exhaustive.   Interval measurement level data exhibit all the characteristics of ordinal measurement data but in addition, the difference between values is constant. In other words, they exhibit constant increment that has some meaning. Temperature is the classic example of this type of measurements. Twenty degrees is certainly the same anywhere on the Fahrenheit temperature scale, but 60 degrees is not twice as hot as 30 degrees. Interval data measurement has no value-related zero. For example, on a temperature scale, zero Fahrenheit does not mean an absence of temperature or heat; it only implies cold and if we change the units to Celsius, this zero will roughly become -18 degree. Again, interval measurement data are mutually exclusive, exhaustive, and equal differences in the variable are represented by equal differences in the measurements. Other examples of interval measurements include shoe sizes and IQ scores.   Ratio measurement level data represent the most commonly used data in statistical analysis. Virtually all quantitative data are essentially ratio data. They have meaningful magnitude, order, and ratios between numbers. The zero value of ratio measurements means an absence of the characteristic magnitude. These include: weight, length, strength, income, profit, area, etc.
• #36: Solution:   a. Interval b. Ratio c. Nominal d. Ordinal e. Ratio
• #39: A normal data set will yield a distribution called ‘the Gaussian’ or the ‘normal distribution’. In its ideal shape, the normal distribution is characterized by a symmetrical bell-shaped distribution as shown here. It has equal values of mean, mode, and median and they are all located at the center of the distribution. These are the key parameters that describe the center of the data, and they are collectively called ‘measures of central tendency’. The spread of the data is described by the so called ‘measures of data dispersion’. These include the range, and the standard deviation. In Chapter 2, we will discuss how these measures are calculated. The normal distribution is fully described mathematically by two parameters: the mean of the data, and the standard deviation, which is a measure of the deviation of all values from the center. What students need to know at this point is the shape of the normal curve and how this shape relates to the distribution of values of a variable. It will also be useful that students begin to describe data by sketching a rough normal curve illustrating the variable values and their distribution. This point is illustrated in the example in the next slide.
• #40: Example Figure 1.9 shows the distributions of the area of ceramic tiles of two types of tiles produced by two different suppliers. The tiles are advertized at the same price and with the same nominal specifications by the two suppliers. Describe the normal distributions of area for these two types of tiles.   This example is provided without data, only the shapes of the distributions of the ceramic tile area, to assist students in generally describing a normal distribution. As you can see from the two distributions, both type A and type B tiles have the same mean value of area. They also have the same mode (the most frequent value) as indicated by the peak of the distributions. In addition, they both have the same median (This is the value right in the middle of all values). These are all measures of central tendency. The difference between the distributions is in the width of the distribution; this reflects the variability or the dispersion in the data with the wider distribution indicating higher variability or larger range of values. Since it is important to buy tiles that are virtually identical in area (to avoid mismatch of tiles), type A should be the tiles of choice as they exhibit much smaller variability in area than type B.
• #41: The variable in the previous example (ceramic tile area) represents the case of ‘nominal-the-best-variable’; that is a variable, which should exhibit an exact value, not too low and not too high. Other variable types are the ‘larger-the-better’ type, and the ‘smaller-the-better-type’. The Figure here shows expected frequency distributions of these types of variables. Larger the better type variables include: student’s grade, people income, and machine efficiency. Smaller-the better type variables include: number of defects, car accidents, and number of failing students.
• #45: In practice, we often deal with a very large population of a process (machine, people, material, etc.). Typically, our goal is to determine the performance of this population through examining one or more population attributes. For example, a company producing millions of electronic items, say DVD players wishes to examine important attributes such as picture resolution and audio performance. Since testing these attributes for the entire population of DVD players is time consuming and can be very costly, the company draws a random sample of few DVD players from the population, tests the two attributes in the sample, produces sample data, and analyzes the data to determine key statistics such as the mean and the range. These systematic steps produce sample descriptive statistics. In doing so, the company’s interest is not the sample per se but rather the whole population of electronic items from which the sample was drawn. In order to estimate population parameters from sample statistics, we use ‘inferential statistics’ as shown here.