Statistics and permeability engineering reports
- 1. Statistics and Probabilities first lecture 2th class Asst.lect. Ali Khaleel Faraj
- 2. The course syllabus Chapter One: introduction fundamental elements of statistics, types of data, Methods of describing data, measures of central tendency, measures of variation measures of relative standing Chapter Two: Probability discreet of random variable: the probability distribution, the binomial probability distribution, The hypogeometric probability distribution, Poison distribution. Continuous random distribution: the continuous random variable, uniform probability distribution, normal probability distribution Chapter Three: Testing of Hypothesis Sampling distributions – Testing of hypothesis for mean, variance, proportions and differences using Normal, t, Chi-square and F distributions - Tests for independence of attributes and Goodness of fit. Chapter Four: Simple linear regression Regression model, methods of least squares, the coefficient of correlation, The coeffiecnt of determination Chapter Five Design of Experiments Experimental design terminology, factorial design Completely randomized design – Randomized block design
- 3. Chapter One: introduction Statistics: is concerned with scientific methods for collecting, organizing, summarizing, presenting and analyzing data as well as drawing valid conclusions and making reasonable decisions. The field of statistics teaches us how to make intelligent judgments and informed decisions in the presence of uncertainty and variation.
- 4. Three Reasons to Study Statistics • The First Reason Being Informed:-How do we decide whether claims based on numerical information are reasonable? We are bombarded daily with numerical information in news, in advertisements, an even in conversation. For example, here are a few of the items employing statistical methods that were part of just two weeks’ news. • The Second Reason Making Informed Judgments:-Decide whether existing information is adequate or whether additional information is required. If necessary, collect more information in a reasonable and thoughtful way. Summarize the available data in a useful and informative manner. • The Third Reason Evaluating Decisions That Affect Your Life:- Many companies now require drug screening as a condition of employment, Insurance companies, Medical researchers use statistical methods, University offices survey.
- 5. Types of data: Quantitative data : are those that represent the quantity or amount of something, measured on a numerical scales. For example; the power frequency Qualitative data: it’s the data that can only classified i.e. posses no numerical representation
- 6. Key Definitions Population: refers to all the persons, objects, source or measurements under consideration, or it is a data set that is our target of interest. Sample: refers to any portion of the population. Variables: is a symbol such as X, Y ,H…. which can assume any of the prescribed set of values. It contains qualitative and quantitative variables Continuous variable: can theoretically assume any value between two given values depending on accuracy of measurements Discrete variable: all data can be obtained from counting Parameter: the measures which describe population characteristics.
- 8. Nominal and Ordinal data
- 9. Discrete and Continuous Data
- 11. Descriptive Statistics Descriptive statistics: can be divided into two major categories: Measures of Central Tendency; and Measures of Dispersion or Variability. Both kinds of measures focus on different essential characteristics of distributions. A very complete description of a distribution can be obtained from a relatively small set of central tendency and dispersion measures from the two categories. 1-Measures of Central Tendency • The measures of central tendency describe a distribution in terms of its most “frequent”, “typical” or “average” data value. But there are different ways of representing or expressing the idea of “typicality”. The descriptive statistics most often used for this purpose are the Mean (the average), the Mode (the most frequently occurring score), and the Median (the middle score).
- 12. 1-Measures of Central Tendency • Mean: commonly referred to as the average or arithmetic mean. Most widely used measure of central location. • Median: the value of the middle item in a set of observations which has been arranged in an ascending or descending order of magnitude. • Mode: The median is the middle score in a distribution, determined after all the values in a distribution have been rank-ordered from lowest to highest, or vice- versa. The median can also be defined as that point in a distribution above which and below which lie 50% of all the cases or observations.
- 13. Example
- 14. Median
- 15. Median
- 18. where: L is the true lower limit of the class in which we expect to find the median (in this case, 6.5) N is the total number of observations in the distribution (here 54) cf is the cumulative frequency UP TO but NOT INCLUDING the class in which we expect the median to fall (in this case, 24) f(class) is the number of cases in the class in which we expect the median to fall (in this case, 15), and I is the width or size of the class or interval (in this case equal to 1.0, for instance, from 6.5 to 7.5)
- 19. MODE
- 20. MEAN Advantages of the MEAN: ❖takes into account all observations. ❖can be used for further statistical calculations and mathematical manipulations. Disadvantages of the MEAN: ❖easily affected by extreme values. ❖cannot be computed if there are missing values due to omission or non-response. ❖ in grouped data with open-ended class intervals, themean cannot be computed.
- 21. MEDIAN Advantages of the MEDIAN: ❖not affected by extreme values. ❖can be computed even for grouped data with open-ended class intervals. Disadvantages of the MEDIAN: ❖Observations from different data sets have to be merged to obtain a new median, whether group or ungrouped data are involved.
- 22. MODE Advantage of the MODE: ❖can be easily identified through ocular inspection. Disadvantages of the MODE: ❖does not possess the desired algebraic property of the mean that allows further manipulations. ❖like the median, observations from different data sets have to be merged to obtain a new mode, whether group or ungrouped data are involved.
- 23. 2-Measuring Dispersion • Range • Mean Deviation • Variance • Standard Deviation • The first two measures we will discuss, the range and the mean deviation, may be thought of as building blocks for understanding the variance and standard deviation.
- 24. • To begin our discussion, let us suppose that in a penology class, three teaching assistants—Tom, Dick, and Harriet—had their respective discussion groups role- play court-employed social case workers who read the files of convicted criminals and recommended to the judge the penalty to be imposed for each criminal. The teaching assistants then compared each student’s recommended sentence to the one actually imposed by the real judge. The teaching assistants then rated each student on a 0 to 10 scale, with 10 being a totally accurate reproduction of the sentences that were actually handed down. There were four students in each discussion group.
- 26. THE RANGE • The range is the simplest measure of dispersion. It compares the highest score and the lowest score achieved for a given set of scores. The range can be expressed in two ways: (a) with a statement such as, “The scores ranged from (the lowest score) to (the highest score),” or (b) with a single number representing the difference between the highest and lowest score. ❖Harriet’s Group: Scores ranged from 6 to 10. Range = 10 − 6 = 4 ❖Dick’s Group: Scores ranged from 7 to 9. Range = 9 − 7 = 2 ❖Tom’s Group: Scores ranged from 8 to 8. Range = 8 − 8 = 0
- 27. THE MEAN DEVIATION The mean deviation (M.D.) (also called the average deviation or the mean absolute deviation) is sensitive to every score in the set. It is based on a strategy of first finding out how far each score deviated from the mean of the scores (the distance from each score to the mean), summing these distances to find the total amount of deviation from the mean in the entire set of scores, and dividing by the number of scores in the set. The result is a mean, or “average,” distance that a score deviates from the mean.
- 28. • To get the mean deviation, we first find the distance between each score and the mean by subtracting the mean from each score. Let us use Harriet’s group as an example.
- 30. THE VARIANCE AND STANDARD DEVIATION
- 32. References • Ott, R. Lyman, and Micheal T. Longnecker. An introduction to statistical methods and data analysis. Cengage Learning, 2015. • Peck, R., Olsen, C. and Devore, J.L., 2008. Introduction to statistics and data analysis. Geneva, Switzerland: Thomson Brooks/Cole. • Smith, Michael J. "Statistical analysis handbook: a comprehensive handbook of statistical concepts, techniques and software tools." (2021).