Statistical Techniques for Analyzing Quantitative Data

1. Statistical Techniques for Analyzing Quantitative Data Maryam Ramezani Values in Computer Technology CSC 426

2. Outline

3. Role of Statistics in Research With Statistics , we can summarize large bodies of data, make predictions about future trends ,and determine when different experimental treatments have led to significantly different outcomes. Statistics are among the most powerful tools in the research's toolbox.

4. How statistics come to research? In quantitative research we use numbers to represent physical or nonphysical phenomena We use statistics to summarize and interpret numbers

5. Exploring and Organizing a Data Set Look at your data and find the ways of organizing them example: Scores of test for 11 children: What do you see?

6. Exploring and Organizing a Data Set

7. Using Computer Spreadsheets to Organize and Analyze Data Sorting Graphing Formulas What Ifs Save, Store, recall, update information

8. Functions of Statistics Descriptive Statistics: describes what the data look like Inferential Statistics : inference about a large population by collecting small samples.

9. Considering the Nature of the Data Continuous or discrete Nominal, ordinal, interval or ratio scale Normal or non-normal distribution

10. Continuous versus Discrete Variables Continuous Data :takes on any value within a finite or infinite interval. You can count, order and measure continuous data. Example :height, weight, temperature, the amount of sugar in an orange, the time required to run a mile. Discrete Data : values / observations belong are distinct and separate, i.e. they can be counted (1,2,3,....). Example: the number of kittens in a litter; the number of patients in a doctors surgery; the number of flaws in one metre of cloth; gender (male, female); blood group (O, A, B, AB).

11. Nominal Data the numbers are simply labels. You can count but not order or measure nominal data Example: males could be coded as 0, females as 1; marital status of an individual could be coded as Y if married, N if single. classification data, e.g. m/f no ordering, e.g. it makes no sense to state that M > F arbitrary labels, e.g., m/f, 0/1, etc

12. Ordinal Data ordered but differences between values are not important e.g., Like scales, rank on a scale of 1..5 your degree of satisfaction rating of 2 rather than 1 might be much less than the difference in enjoyment expressed by giving a rating of 4 rather than 3. You can count and order, but not measure, ordinal data.

13. Interval Data ordered, constant scale, but no natural zero differences make sense, but ratios do not e.g.: 30°-20°=20°-10°, but 20°/10° is not twice as hot! e.g.: Dates: the time interval between the starts of years 1981 and 1982 is the same as that between 1983 and 1984, namely 365 days. The zero point, year 1 AD, is arbitrary; time did not begin then

14. Ratio Data  Like interval data but has true zero Ordered, Constant scale, natural zero e.g., height, weight, age, length

15. Normal and Non-Normal Distributions

16. Normal Distribution

17. Non-Normal Distributions

18. Leptokurtic and Platykurtic Distributions

19. Descriptive Statistics Descriptive Statistics describes data Points of Central Tendency Amount of Variability Relation of different variables to each other

21. Measure of Central Tendency

22. Measures of Variability

23. Measures of Variability

24. Measure of Relationship: Correlation correlation indicates the strength and direction of a linear relationship between two variables. See page 266 for other examples or correlation statistics

25. Notes about Correlation Substantial correlations between two characteristics needs reasonable Validity and Reliability in measuring Correlation does not indicate causation

26. Examples of using Statistics in Computer Science Conceptual Representation of User Transactions or Sessions

27. Inferential Statistics We use the samples as estimate of population parameter. The quality of all statistical analysis depends on the quality of the sample data

28. Some definitions Parameter: describes a population Statistic: describes a sample

29. Inferential Statistics Estimate a population parameter from a random sample Test statistically hypotheses

30. Inferential Statistics: Estimate a Population Parameter from Sample All sample statistics have some error in estimating population parameters Example: estimate mean height of 10 year old boys in Chicago, Sample:200 boys How close the sample mean is to the population mean? we don’t know but we know: The mean from an infinite number of samples form a normal distribution. The population mean equals the average (mean) of all samples. The Standard deviation of sample distribution ( standard error) is directly related to the std of the characteristic in question for the overall population.

31. Standard Error Standard error tell us how much the particular mean vary from one sample to another when all samples are the same size and drawn randomly from the sample population. Standard Error: n is size of all samples and s is the population std which we don’t have! We use the std of sample:

32. Accuracy of the Estimator

33. Point versus Interval Estimate A point estimate is a single value--a point--taken from a sample and used to estimate the corresponding parameter of a population , s, s2 and r estimate µ, s, s2, ? respectively An interval estimate is a range of values--an interval— within whose limits a population parameter probably lies. we say that we are 95% confident that the unknown population mean lies in the interval

34. Testing Hypothesis Confidence intervals are used when the goal of our analysis is to estimate an unknown parameter in the population. A second goal of a statistical analysis is to verify some claim about the population on the basis of the data. Research Hypothesis =/=Statistical hypothesis A test of significance is a procedure to assess the truth about a hypothesis using the observed data. The results of the test are expressed in terms of a probability that measures how well the data support the hypothesis.

36. Stating an hypotheses

37. General comments on stating hypotheses It is not easy to state the null and the alternative hypothesis! The hypotheses are statements on the population values. The alternative hypothesis Ha is often called “researcher hypothesis”, because it is the hypothesis we are interested about. A significance test is a test against the null hypothesis Often we set Ha first and then Ho is defined as the “opposite” statement!

38. Errors in Hypothesis testing Type I Error : the null hypothesis is rejected when it is in fact true; that is, H0 is wrongly rejected. Type II Error :the null hypothesis H0, is not rejected when it is in fact false

39. Meta- Analysis Meta-analysis refers to the analysis of analyses...the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings. (Glass, 1976, p. 3) Conduct a fairly extensive search for relevant studies Identify appropriate studies to include in meta-analysis Convert each study’s results to a common statistical index

40. Using Statistical Software Packages SPSS SAS Matlab Statistics toolbox SYSTAT, Minitab, Stat View, Statistica

41. Interpreting the Data Relating the findings to the original research problem and to the specific research questions and hypothesis Relating the findings to preexisting literature, concepts, theories and research results. Determining whether the findings have practical significance as well as statistical significance Identifying limitations of the study