Chapter 1: Introduction to Statistics

What is statistics a set of mathematical procedures to do what? to gather information and to organize and summarize the information and to interpret the information and to answer some questions

Population the set of all the individuals of interest in a research study The entire group of individuals is called the population. 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 data selected from population Usually populations are so large that a researcher cannot examine the entire group. Therefore, a sample is selected to represent the population in a research study. The goal is to use the results obtained from the sample to help answer questions about the population.

Variables A variable is a characteristic or condition that can change or take on different values. e.g. IQ score for the population of third-grade children. Most research begins with a general question about the relationship between two variables for a specific group of individuals.

Data The measurements obtained in a research study are called the data. The goal of statistics is to help researchers organize and interpret the data.

Datum, Data, Score, Raw Score datum: single measurement or observation  commonly called a score or raw score data set: (plural) a collection of measurements or observations

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. e.g. mean, median, mode, variance Every parameter has a corresponding statistic

Inferential Statistics Inferential statistics are methods for using sample data to make general conclusions (inferences) about populations. Because a sample is typically only a part of the whole population, sample data provide only limited information about the population. As a result, sample statistics are generally imperfect representatives of the corresponding population parameters.

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. e.g. political polls was taken from a sample and usually has a margin of error (sampling error).

Research: Descriptive  Inferential Experiment: Compare 2 teaching methods page 10 Population  select 2 samples (1 for Method A, 1 for Method B)  collect test scores from these 2 samples  use table / graph to organize / summarize data (descriptive statistics)  interpret the results (inferential statistics): 1. A ~ B 2. A is better than B What’s your conclusion?

Simply Curious....or I got a score of 81 out of 100 for my entrance exam... Good enough? variable: “test score”  individual variable population: all the contestants in this test sample: survey descriptive feature: average score inferential feature: good enough for me to pass or not?

Two variables... Two variables measured for each individual  correlation relationship between these variables  Correlational Studies e.g. wake-up time vs academic performance survey  sample data  table or graph? How do you interpret the data?

Correlational Studies The goal of a correlational study is to determine whether there is a relationship between two variables and to describe the relationship. A correlational study simply observes the two variables as they exist naturally. But it cannot demonstrate a cause-and-effect relationship

Two or more Groups.... Compare two or more groups  find the relationship between two variables variable 1: to define the groups (X) variable 2: measure the “scores” (Y) for each group e.g. X: control group or treatment group Y: the behavior / choice variable Other things being equal

Experiments The goal of an experiment is to demonstrate a cause-and-effect relationship between two variables; that is, to show that changing the value of one variable (X) causes changes to occur in a second variable (Y).

Experiments (cont'd.) In an experiment, one variable(X) is manipulated to create treatment conditions. A second variable (Y) is observed and measured to obtain scores for a group of individuals in each of the treatment conditions. –The measurements are then compared to see if there are differences between treatment conditions. All other variables are controlled to prevent them from influencing the results. –The manipulated variable (X) is called the independent variable and the observed variable (Y) is the dependent variable.

Other Things being Equal.... These are the “Other factors”(Z): Participant Variables: characteristics of the test subjects, e.g. age, gender, intelligence Environmental Variables: characteristics of the environment, e.g. lighting, time of day, weather

So, Experimental Stduies: To get the cause-and-effect relationship  1. Use random assignment method for X (equal chance of being assign to each group). 2. Use random assignment method to control the “other factors”(Z). e.g. time of day 3. Use “matching” to ensure equivalent groups or environments. e.g. Male40%, Female 60% for every group 4. “Holding them constant”, e.g. use only one type of subjects (age=10, similar socio-economic backgroud)

Independent vs Dependent variables Independent variables (X): can be manipulated and assigned to different groups, not measured.  cause  treatment group vs control group Dependent variables (Y): can be observed and measured  effect Here only 1 variable is measured !! Treatment condition: experimental condition No-treatment condition: control condition (baseline for comparison)

Other Types of Studies Other types of research studies, know as non- experimental or quasi-experimental, are similar to experiments because they also compare groups of scores. These studies do not use a manipulated variable X to differentiate the groups. Instead, the variable that differentiates the groups is usually a pre-existing participant variable (such as male/female) or a time variable (such as before/after).

Other Types of Studies (cont'd.) Because these studies do not use the manipulation and control of true experiments, they cannot demonstrate cause and effect relationships. As a result, they are similar to correlational research because they simply demonstrate and describe relationships. Figure 1.7 : 2 types of nonexperimental studies: boys vs girls (nonequivalent groups study) and before vs after (pre-post study)

quasi-independent variable quasi-independent variable: X is used to create different groups X can be: with vs without (eating disorder) single parent vs 2-parent home gloomy vs sunny (env variable) public vs private school before vs after therapy Y is measured  scores

Constructs internal attributes or characteristics e.g. intelligent, anxious, hungry can not be directly observed (intangible) i.e. often called hypothetical constructs useful for describing and explaining behavior

operational definition defines a construct in terms of external behavior behavior that can be observed and measured are representative of the construct e.g. IQ score  intelligence # of hours since last eating  how hungry you are has 2 components: a) it describes a set of operations for measuring construct b) it defines the construct in terms of the resulting measurements

Is This 12-Year-Old Girl Really Smarter Than Einstein? girl-really-achieve-higher-iq-einsteinhttp:// girl-really-achieve-higher-iq-einstein This week saw the news of a 12-year-old girl achieving the highest possible score on a Mensa IQ test, with headlines announcing that she had scored higher than both Albert Einstein and Stephen Hawking. This feat of a perfect score is achieved by only 1% of those who sit the paper, according to Mensa, the society for people with high IQs, and means she joins two other British children who have so far attained the maximum score of 162 this year.the newstwoother

Types of Variables Variables can be classified as discrete or continuous. Discrete variables (such as class size) consist of indivisible categories, and continuous variables (such as time or weight) are infinitely divisible into whatever units a researcher may choose. For example, time can be measured to the nearest minute, second, half-second, etc.

Continuous Variables 1. almost impossible to get equal measurement e.g. your 158 cm height is not exactly equal to my 158 cm height, why? 2. there’s always a middle measurement between two measurements (scores) 3. difficult to get exact measurement  must set up boundaries on the scale of measurement These boundaries are called real limits

Real Limits To define the units for a continuous variable, a researcher must use real limits which are boundaries located exactly half-way between adjacent categories. Precision can be set at 1 cm, or 0.5 cm, or 0.1 cm, or 0.01 cm (to the nearest unit of measurement) Whenever you are free to choose the degree of precision or the number of categories for measuring a variable, the variable must be continuous.

Measuring Variables To establish relationships between variables, researchers must observe the variables and record their observations. This requires that the variables be measured. The process of measuring a variable requires a set of categories called a scale of measurement and a process that classifies each individual into one category.

Four Types of Measurement Scales 1.A nominal scale is an unordered set of categories identified only by name. Nominal measurements only permit you to determine whether two individuals are the same or different. 2.An ordinal scale is an ordered set of categories. Ordinal measurements tell you the direction of difference between two individuals.

Four Types of Measurement Scales (cont’d.) 3.An interval scale is an ordered series of equal- sized categories. Interval measurements identify the direction and magnitude of a difference. The zero point is located arbitrarily on an interval scale. 4.A ratio scale is an interval scale where a value of zero indicates none of the variable. Ratio measurements identify the direction and magnitude of differences and allow ratio comparisons of measurements.

Statistical Notation The individual measurements or scores obtained for a research participant will be identified by the letter X (or X and Y if there are multiple scores for each individual). The number of scores in a data set will be identified by N for a population or n for a sample. Summing a set of values is a common operation in statistics and has its own notation. The Greek letter sigma, Σ, will be used to stand for "the sum of." For example, ΣX identifies the sum of the scores.

Order of Operations 1.All calculations within parentheses are done first. 2.Squaring or raising to other exponents is done second. 3.Multiplying, and dividing are done third, and should be completed in order from left to right. 4.Summation with the Σ notation is done next. 5.Any additional adding and subtracting is done last and should be completed in order from left to right.