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Edison S Statistics

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  • 1. A review of Statistics Computational Linguistics Edison Reyes Prieto
  • 2. Descriptive statistics
  • 3.
    • Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.
    • Descriptive statistics are typically distinguished from inferential statistics. With descriptive statistics you are simply describing what is or what the data shows. With inferential statistics, you are trying to reach conclusions that extend beyond the immediate data alone. For instance, we use inferential statistics to try to infer from the sample data what the population might think. Or, we use inferential statistics to make judgments of the probability that an observed difference between groups is a dependable one or one that might have happened by chance in this study. Thus, we use inferential statistics to make inferences from our data to more general conditions; we use descriptive statistics simply to describe what's going on in our data.
  • 4.
    • Descriptive Statistics are used to present quantitative descriptions in a manageable form. In a research study we may have lots of measures. Or we may measure a large number of people on any measure. Descriptive statistics help us to simply large amounts of data in a sensible way. Each descriptive statistic reduces lots of data into a simpler summary. For instance, consider a simple number used to summarize how well a batter is performing in baseball, the batting average. This single number is simply the number of hits divided by the number of times at bat (reported to three significant digits). A batter who is hitting .333 is getting a hit one time in every three at bats. One batting .250 is hitting one time in four. The single number describes a large number of discrete events. Or, consider the scourge of many students, the Grade Point Average (GPA). This single number describes the general performance of a student across a potentially wide range of course experiences.
  • 5.
    • Every time you try to describe a large set of observations with a single indicator you run the risk of distorting the original data or losing important detail. The batting average doesn't tell you whether the batter is hitting home runs or singles. It doesn't tell whether she's been in a slump or on a streak. The GPA doesn't tell you whether the student was in difficult courses or easy ones, or whether they were courses in their major field or in other disciplines. Even given these limitations, descriptive statistics provide a powerful summary that may enable comparisons across people or other units.
  • 6. Examples of descriptive statistics
    • Most statistics can be used either as a descriptive statistic, or in an inductive analysis. For example, we can report the average reading test score for the students in each classroom in a school, to give a descriptive sense of the typical scores and their variation. If we perform a formal hypothesis test on the scores, we are doing inductive rather than descriptive analysis.
    • Some statistical summaries are especially common in descriptive analyses. Some examples follow.
    • Measures of central tendency
    • Measures of dispersion
    • Measures of association
    • Cross-tab, contingency table
    • Histogram
    • Quantile, Q-Q plot
    • Scatterplot
    • Box plot
  • 7. Descriptive statistics
    • Continous data
      • Location
        • Mean (arithmetic, geometric, harmonic)
        • Median
        • Mode
      • Dispersion
        • Range
        • Standard deviation
        • Coefficient of variation
        • Percentile
      • Moments
        • Variance
        • Semivariance
        • Skewness
        • Kurtosis
    • Categorical data
      • Frequency
      • Contingency table
  • 8. Inferential statistics
  • 9.
    • With inferential statistics, you are trying to reach conclusions that extend beyond the immediate data alone. For instance, we use inferential statistics to try to infer from the sample data what the population might think. Or, we use inferential statistics to make judgments of the probability that an observed difference between groups is a dependable one or one that might have happened by chance in this study. Thus, we use inferential statistics to make inferences from our data to more general conditions; we use descriptive statistics simply to describe what's going on in our data.
  • 10.
    • Here, I concentrate on inferential statistics that are useful in experimental and quasi-experimental research design or in program outcome evaluation. Perhaps one of the simplest inferential test is used when you want to compare the average performance of two groups on a single measure to see if there is a difference. You might want to know whether eighth-grade boys and girls differ in math test scores or whether a program group differs on the outcome measure from a control group. Whenever you wish to compare the average performance between two groups you should consider the t-test for differences between groups.
  • 11.
    • Most of the major inferential statistics come from a general family of statistical models known as the General Linear Model. This includes the t-test, Analysis of Variance (ANOVA), Analysis of Covariance (ANCOVA), regression analysis, and many of the multivariate methods like factor analysis, multidimensional scaling, cluster analysis, discriminant function analysis, and so on. Given the importance of the General Linear Model, it's a good idea for any serious social researcher to become familiar with its workings. The discussion of the General Linear Model here is very elementary and only considers the simplest straight-line model. However, it will get you familiar with the idea of the linear model and help prepare you for the more complex analyses described below.
  • 12.
    • One of the keys to understanding how groups are compared is embodied in the notion of the "dummy" variable. The name doesn't suggest that we are using variables that aren't very smart or, even worse, that the analyst who uses them is a "dummy"! Perhaps these variables would be better described as "proxy" variables. Essentially a dummy variable is one that uses discrete numbers, usually 0 and 1, to represent different groups in your study. Dummy variables are a simple idea that enable some pretty complicated things to happen. For instance, by including a simple dummy variable in an model, I can model two separate lines (one for each treatment group) with a single equation. To see how this works, check out the discussion on dummy variables.
  • 13.
    • One of the most important analyses in program outcome evaluations involves comparing the program and non-program group on the outcome variable or variables. How we do this depends on the research design we use. research designs are divided into two major types of designs: experimental and quasi-experimental. Because the analyses differ for each, they are presented separately.
    • Experimental Analysis . The simple two-group posttest-only randomized experiment is usually analyzed with the simple t-test or one-way ANOVA. The factorial experimental designs are usually analyzed with the Analysis of Variance (ANOVA) Model. Randomized Block Designs use a special form of ANOVA blocking model that uses dummy-coded variables to represent the blocks. The Analysis of Covariance Experimental Design uses, not surprisingly, the Analysis of Covariance statistical model.
  • 14.
    • Quasi-Experimental Analysis . The quasi-experimental designs differ from the experimental ones in that they don't use random assignment to assign units (e.g., people) to program groups. The lack of random assignment in these designs tends to complicate their analysis considerably. For example, to analyze the Nonequivalent Groups Design (NEGD) we have to adjust the pretest scores for measurement error in what is often called a Reliability-Corrected Analysis of Covariance model. In the Regression-Discontinuity Design, we need to be especially concerned about curvilinearity and model misspecification. Consequently, we tend to use a conservative analysis approach that is based on polynomial regression that starts by overfitting the likely true function and then reducing the model based on the results. The Regression Point Displacement Design has only a single treated unit. Nevertheless, the analysis of the RPD design is based directly on the traditional ANCOVA model.
    • When you've investigated these various analytic models, you'll see that they all come from the same family -- the General Linear Model. An understanding of that model will go a long way to introducing you to the intricacies of data analysis in applied and social research contexts.
  • 15. Regression analysis
  • 16.
    • In statistics, regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps us understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables — that is, the average value of the dependent variable when the independent variables are held fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function . In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.
  • 17.
    • Regression analysis is widely used for prediction (including forecasting of time-series data). Use of regression analysis for prediction has substantial overlap with the field of machine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variable, and to explore the forms of these relationships. In restricted circumstances, regression analysis can be used to infer causal relationships between the independent and dependent variables.
    • A large body of techniques for carrying out regression analysis has been developed. Familiar methods such as linear regression and ordinary least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data. Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.
  • 18.
    • The performance of regression analysis methods in practice depends on the form of the data-generating process, and how it relates to the regression approach being used. Since the true form of the data-generating process is not known, regression analysis depends to some extent on making assumptions about this process. These assumptions are sometimes (but not always) testable if a large amount of data is available. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However when carrying out inference using regression models, especially involving small effects or questions of causality based on observational data, regression methods must be used cautiously as they can easily give misleading results.

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