Statistics in Engineering and Science

Statistics in Engineering and Science

1. Statistics in Experimentation

   • Types of studes
     • Data are the center of Both and are subject to a variety of sources that induce variation in measurements. The variation can occur because of:
       • Fixed differences among machines
       • Random differences due to changes in ambient conditions
       • Measurement error in instrument readings
       • Effects due to many other known or unknown influences.
     • Experimental : The variables of interest often can be controlled and fixed at predetermined values for each test run in the experiment.
     • Observational: Many of the variables of interest cannot be controlled, but they can be recorded and analyzed.
   • Statistics
     • Is the science of problem-solving in the presence of variability
     • Played a role in both the design of the experiment and the formal analysis of the results
     • Scatterplot
       • A two-dimensional graph of individual data values for pairs of quantitative variables, can be made for any pair of variables so long as both are quantitative.
   • Statistical design and analysis of experiments
     • To ensure that relevant sources of variation could be identified and measured.
       • Project planning
         • Statistical considerations
           • What is to be measured?, How large is the likely varation? , What are the influential factors?
       • Experimental design
         • Control known sources of variation. Allowed estimatation of the size of the uncontrolled variation. Permit an investigation of suitable models
           • Uncontrollable variation can arise from
             • Experimental error is introduced whenever test conditions are changed.
             • Measurement errors arise from the inability to obtain exactly the same measurement on two successive test runs when all experimental conditions are unchanged.
       • Data analysis
         • Make inferences on design factors. Guide subsequent designs. Suggest more appropriate models.

2. Population and Samples

   • Population or Process
     • Population: A statistical population consists of all possible items or units possessing one or more common characteristics under specified experimental or observational conditions.
       • Depending on the experimental conditions, many different populations of observed values could be defined.
         • It is important to define the fundamental qualities or quantities of interest in an experiment. We term these qualities or quantities variables.
           • Variable is a property or characteristic on which information is obtained in an experiment.
             • Response Variable: Any outcome or result of an experiment.
             • Factors: Controllable experimental variables that can influence the observed values of response variables
           • Observation: The collection of information in an experiment, or actual values obtained on variables in an experiment.
           • Observed value : Actual measurement or attribute that is the result of an individual observation
     • Process : Is a repeatable series of actions that results in an observable characteristic or measurement.
   • Sample
     • A group of observations taken from a population or a process if it is not feasible to collect information on every item in a population
       • Ways to collect samples
         • Convenience sample
           • Is chosen simply by taking observations that are easily or inexpensively obtained. All other considerations are secondary to the economic or rapid collection of data.
         • Judgmental sampling
           • One’s experience and professional judgment are used to select representative observations from a population of interest.
         • Simple Random Sample.
           • A simple random sample of size n is obtained when items are selected from a fixed population or a process in such a manner that every group of items of size n has an equal chance of being selected as the sample.
             • Systematic random samples
               • Are obtained by sampling every kth (e.g., every 5th, 10th, or 100th) unit in the population.
             • Stratified random samples
               • Are based on subdividing a heterogeneous population into groups, or strata, of similar units and selecting simple random samples from each of the strata.
             • Cluster sampling
               • Based on subdividing the population into groups, or clusters, of units in such a way that it is convenient to randomly sample the clusters and then either randomly sample or completely enumerate all the observations in each of the sampled clusters

3. Parameters and Statistics

   • Parameters
     • Numerical characteristic of a population or a process, often are denoted by Greek letters. They are fixed constants representing an entire population of data values
   • Statistics
     • Numerical characteristic that is computed from a sample of observations. They are “random” variables and their numerical values depend on which particular observations from the population are included in the sample.
     • it has its own probability, or Sampling, Distribution.
       • A sampling distribution is a theoretical MODEL that describes the probability of obtaining the possible values of a sample statistic.
         • Averages from simple random samples of size n also follow a normal probability model.
       • Distribution
         • A tabular, graphical, or theoretical description of the values of a variable using some measure of how frequently they occur in a population, a process, or a sample.
           • Histograms
             • Especially useful when large numbers of data must be processed. Constructed by dividing the range of the data into several intervals (usually of equal length), counting the number of observations in each interval, and constructing a bar chart of the counts.
           • Relative frequency histograms
             • When the sample size is large, it can be advantageous to construct. Histograms and frequency distributions either the proportions (counts/sample size) or the percentages (proportions × 100%) of observations in each interval are calculated and graphed, rather than the frequencies themselves.

4. Mathematical and Statistical Modeling

   • Experiments are conducted to characterize, confirm, or reject models
     • Mathematical Model.
       • A model is termed mathematical if it is derived from theoretical or mechanistic considerations that represent exact, error-free assumed relationships among the variables.
     • Statistical Model.
       • A model is termed statistical if it is derived from data that are subject to various types of specification, observation, experimental, and/or measurement errors