Statistics

1. Why include statistics as part of Psychology? • Doing psychology research • Reading psychology research articles • Analytical reasoning, critical thinking • Statistics • Fundamental tool for all scientific inquiry • Way of making sense out of data Statistics

2. Population • the group of individuals (or things) of interest in a particular study • For example, a researcher may be interested in the relation between class size (variable 1) and academic performance (variable 2) for the population of third-grade children. • Sample • Usually populations are so large that a researcher cannot examine the entire group • a sample is selected to represent the population in a research study • Sample size depends on the type of research • The goal is to use the results obtained from the sample to help answer questions about the population Populations and Samples

3. Sampling from a Population

4. Variables And Data A variable is a characteristic or condition that can change or take on different values Most research begins with a general question about the relationship between two variables for a specific group of individuals.(similar to forming an hypothesis) Data are measurements or observations The measurements obtained in a research study are called the data or data set Each measurement is a datum (singular) or score The goal of statistics is to help researchers organize and interpret the data.

5. Carrying out psychological research using an empirical approach means the collection of data. Statistics are a way of making use of this data • Descriptive Statistics: used to describe characteristics of our sample • Inferential Statistics: used to generalise from our sample to our population • Any samples used should therefore be representative of the target population Using Statistics in Psychology

6. Make an Inference representative of the population Figure 1-1 The relationship between a population and a sample.

7. Descriptive Statistics Descriptive statistics are methods for organizing and summarizing data. For example, tables or graphs are used to organize data, and descriptive values such as the average score are used to summarize data. A descriptive value for a population is called a parameter and a descriptive value for a sample is called a statistic.

8. Inferential Statistics Inferential statistics methods for using sample data to make general conclusions (inferences) about populations a sample is only a part of the population sample data provide only limited information about the population sample statistics are imperfect representatives of the population parameters because of sampling error

9. Sampling Error The discrepancy between a sample statistic and its population parameter is called sampling error. Defining and measuring sampling error is a large part of inferential statistics.

10. Sampling from a Population

11. Figure 1-2 A demonstration of sampling error. Two samples are selected from the same population. Notice that the sample statistics are different from one sample to another, and all of the sample statistics are different from the corresponding population parameters. The natural differences that exist, by chance, between a sample statistic and a population parameter are called sampling error.

12. Is an example of sampling error • Terminology used in polling data such as political polls • Amount of error between a sample statistic and a population parameter • There will always be sampling error in: • survey research • experiments Margin of Error Box 1.1

13. The role of statistics in experimental research.

14. Data structure is used to classify statistical methods • Correlational Method • Measuring two variables for each individual • Height and Weight • SAT and GPA • Wake-up Time and Academic Performance (Figure 1.4) • Determine the relationship between the variables • Limitations of the correlational method • Can not demonstrate cause-and-effect relationships Relationship Between Variables

15. Figure 1.4 One of two data structures for studies evaluating the relationship between variables. Note that there are two separate measurements for each individual (wake-up time and academic performance). The same scores are shown in a table (a) and in a graph (b).

16. Data structure is used to classify statistical methods • Two or more groups of scores • Experimental see fig 1.5 • One variable defines the groups (violent vs nonviolent video game) • Manipulate the Independent variable • Another variable “aggressive behavior” is the measurement • Dependent variable • Control • Participant variables: age, gender ………. • Environmental variables • Several different statistical tests are used such as t-test or ANOVA based on number of groups Data Structures and Statistical Method

17. Data structure is used to classify statistical methods • Comparing two groups of scores • NonExperimental “quasi-experimental” • Natural or pre-existing groups such as gender which are selected not manipulated • Two common versions see fig 1.6 • 1. Quasi-independent variable: Women compared to Men • 2. Before and after measurements for example before and after therapy • One variable is time point (before vs after) • The other variable is the measurement Relationship Between Variables

18. Two examples of nonexperimental studies that involve comparing two groups of scores. In (a) the study uses two preexisting groups (boys/girls) and measures a dependent variable (verbal scores) in each group. In (b), time is the variable used to define the two groups, and the dependent variable (depression) is measured at each of the two times.

19. To form a hypothesis from a research question the researcher needs to define the variables • What are the effects of drug PQX1450 on Anxiety? • Independent variable is drug or no drug • Dependent variable is “anxiety” which is a construct • so we need to define the construct of anxiety • Need an Operational Definition for anxiety Constructs and Operational Definitions

20. Discrete Variable • Discrete categories such as students, cars, houses • Usually a count of the number of individuals or things • number of students in class • number of cars in the parking lot • number of houses along the street • Also called “Categorical variables” • Continuous Variable • Variable can be divided into an infinite number of values • height, weight, time Variables And Measurement

21. Use of Real Limits with Continuous Variables When working with continuous variable Because a variable such as weight is infinitely divisible the researcher needs to set boundaries or limits use real limits which are boundaries located exactly half-way between adjacent categories. Researcher decides as a practical matter to record weight to the nearest pound Someone with a recorded weight of 150 could be 149.6 Each value “150” is an interval with upper and lower limits See fig 1.7 Values that fall on the boundary “150.5” can be rounded up or down just be consistent with the rounding rule

22. Figure 1.7 When measuring weight to the nearest whole pound, 149.6 and 150.3 are assigned the value of 150 (top). Any value in the interval between 149.5 and 150.5 is given the value of 150.

23. Nominal (by name / category) • Ordinal (by order / rank) • Interval (equal interval scaling) • Ratio (interval with a “real”zero point) Four Types of Measurement Scales :

24. Nominal Scale • “Names” • Classifying subjects into categories • No category is “more” or “less,” just different • Categories can be labeled by • words (e.g., Male, Female) or • numbers (e.g., 0, 1) which can be confusing • Nominal scale always yields discrete variable Scales of Measurement

25. Ordinal Scale • “ordered” • Categories are in ordered sequence, ranked • Examples: • Gold, silver, bronze medals • Don’t know how far gold was from silver, or silver from bronze • Class standing (33rd out of 108) • Ordinal scale technically yields discrete variables (can not be ranked 33rd and a half) • Different statistical procedures are required. Scales of Measurement

26. Interval Scales • Distance between two values is the same at any point on the scale • The difference between scores of 6 and 10 is 4 units • The difference between scores of 26 and 30 is 4 units • Interval scale does not have absolute zero • Attitudinal scales, on a scale of from 1(not a all) to 10 (a great deal) how much do you like anchovy pizza? • Example 1.2 page 23: convert ratio scale to interval scale • Height measurements (ratio scale) can be converted to difference scores i.e. difference from the average score • Average height of 50 inches so a height of 52 becomes a difference score of +2 which is an interval scale measurement Scales of Measurement

27. Ratio Scales • In addition to having even intervals we can calculate ratios so a • Ratio scale has meaningful, absolute zero • Distance: zero distance • Weight: zero weight • Temperature: absolute zero but not zero degrees Fahrenheit • Time: zero time ?? • But what about • IQ score? Interval • Score on test of neuroticism? Interval Scales of Measurement

28. Statistical Notation X is a discrete variable, where 0=men and 1=women. Y is a continuous variable, representing years of age. • N refers to number of subjects; N=6. • Xirefers to the ith person’s score on variable X. • For this data set, X4 = 1, Y 2 = 10.

29. Statistical Notation • S Greek letter Sigma symbolizes summation • Gender (x) • Girl • Girl • Boy • Girl • Boy • Boy • Y = ? • X = ? 53 Can not sum categories

30. Statistical Notation • Examples 1.3, 1.4 & 1.5 p 26-27 • ΣX = 3+1+7+4 = 15 • ΣX2 = 9+1+49+16 = 75 • Σ(X-1) = 2+0+6+3 = 11 • Σ(X-1)2 = 4+0+36+9 = 49 • However • ΣX-1 = 15 – 1 = 14 • Because • The order of operations is: • 1. parentheses • 2. exponents • 3. multiply / divide • 4. summation ( Σ) • 5. addition / subtraction

31. Statistical Notation • Example 1.6 p 28 • ΣX = 3+1+7+4 = 15 • ΣY = 5+3+4+2 = 14 • ΣXY = 15+3+28+8 = 54 • Σ(X + Y) = ΣX + ΣY • Σ(X + Y) = 8+4+11+6 = 29 • ΣX + ΣY = 15+14 = 29