Introduction To Statistics

Statistics, Science, ad Observations What are statistics? What are statistics? The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information.The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information. Statistics does the job of assigning a numeric value to events that have already taken place. This numeric value can then be interpreted to mean something (description or inference).Statistics does the job of assigning a numeric value to events that have already taken place. This numeric value can then be interpreted to mean something (description or inference). Why do we use statistics? Why do we use statistics? To summarize large amounts of dataTo summarize large amounts of data To support hypothesesTo support hypotheses To understand relationshipsTo understand relationships To answer research questionsTo answer research questions

What do we need in order to do statistics?  What is the elemental content of statistics? DATA!!!! DATA!!!! Data are measurements of observations. A data set is a collection of measurement or observations. Data are measurements of observations. A data set is a collection of measurement or observations. A datum is a single measurement or observation and is commonly called a score or a raw score. A datum is a single measurement or observation and is commonly called a score or a raw score.

Populations and Samples  What do we want to do with data? Summarize it! Summarize it! For this we use descriptive statisticsFor this we use descriptive statistics Statistical procedures used to summarize, organize and simplify data. Statistical procedures used to summarize, organize and simplify data. Use it for predictions, or test hypotheses. Use it for predictions, or test hypotheses. For this we use inferential statisticsFor this we use inferential statistics Techniques that allow us to study samples and then make generalizations about the populations from which they were selected. Techniques that allow us to study samples and then make generalizations about the populations from which they were selected.

Where do we get data?  In general (especially in psychology) data is collected from people, or groups of people. Population Population The set of all the individuals of interest in a particular study.The set of all the individuals of interest in a particular study. Is it feasible to collect data on everyone in a particular population? Is it feasible to collect data on everyone in a particular population? SampleSample A set of individuals selected from a population, usually intended to represent the population in a research study. A set of individuals selected from a population, usually intended to represent the population in a research study.

If we are collecting data from samples of people within a population, can we ever know the exact population values?  We cant usually get data from everyone in a population, so we estimate population values with sample values. Parameter Parameter A value, usually numerical, that describes a population. Represented with Greek letters.A value, usually numerical, that describes a population. Represented with Greek letters. Statistic Statistic A value, usually numerical, that describes a sample. Represented with Roman letters.A value, usually numerical, that describes a sample. Represented with Roman letters.

How do we take into account error?  Sampling error The discrepancy, or amount of error, that exists between a sample statistic and the corresponding population parameter. The discrepancy, or amount of error, that exists between a sample statistic and the corresponding population parameter.

Data Structures, Research Methods, and Statistics  What kinds of research questions (hypotheses) might we have that we can collect data with and run statistics on?  What are the components of these hypotheses? Construct Construct An abstraction that cannot be observed but is useful in explaining behaviorAn abstraction that cannot be observed but is useful in explaining behavior Variable Variable A characteristic or condition that changes or has different values for different individualsA characteristic or condition that changes or has different values for different individuals Methodology Methodology The method by which the experiment will answer the question. Experimental, quasi-experimental, correlationalThe method by which the experiment will answer the question. Experimental, quasi-experimental, correlational

Constructs  Can we name a few constructs? Intelligence Intelligence Friendliness Friendliness Cheerfulness Cheerfulness

What do we do with constructs?  Operationalization An operationalization (or operational definition) defines a construct in terms of behaviors that can be measured and observed. An operationalization (or operational definition) defines a construct in terms of behaviors that can be measured and observed.  How can we operationalize the constructs I listed before? Intelligence Intelligence IQIQ Friendliness Friendliness Number of people you smile at in a dayNumber of people you smile at in a day Cheerfulness Cheerfulness How often you laugh during a dayHow often you laugh during a day

More components of the hypotheses  Variable What are the two types of variables? What are the two types of variables? IndependentIndependent The variable used to measure the CAUSAL construct. The independent variable is the variable that is manipulated by the researcher. The variable used to measure the CAUSAL construct. The independent variable is the variable that is manipulated by the researcher. DependentDependent The variable used to assess the affected construct. The dependent variable is the one that is observed in order to assess the effect of the treatment. The variable used to assess the affected construct. The dependent variable is the one that is observed in order to assess the effect of the treatment.

Our hypotheses  Of the hypotheses we have listed on the board, which are experimental and which are correlational Correlational Correlational A method where two variables are observed in order to determine whether there is a relationship between themA method where two variables are observed in order to determine whether there is a relationship between them Experimental Experimental A method where one variable is manupulated while another variable is observed and measuredA method where one variable is manupulated while another variable is observed and measured

What do you need for an experiment?  Causation What are the three conditions for causation What are the three conditions for causation Time priorityTime priority (co) Relation(co) Relation Non-spurious relationship (eliminate alternative hypothesesNon-spurious relationship (eliminate alternative hypotheses

Non-Spurious Relationships  What are the three conditions to eliminate alternative hypotheses? Initial equivalence of groups (random assignment) Initial equivalence of groups (random assignment) A manipulated variable (Control over the IV) A manipulated variable (Control over the IV) What kinds of variables might we not be able to manipulate?What kinds of variables might we not be able to manipulate? When groups differ by a participant variable (e.g., gender) as opposed to an experimentally manipulated variable, the variable that determines the groups is called a quasi-independent variable. This leads to a quasi-experiment. When groups differ by a participant variable (e.g., gender) as opposed to an experimentally manipulated variable, the variable that determines the groups is called a quasi-independent variable. This leads to a quasi-experiment. Control groups Control groups What does a control group mean?What does a control group mean? Individuals in a control condition do not receive the experimental treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental condition Individuals in a control condition do not receive the experimental treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental condition Individuals in the experimental condition do receive the experimental treatment. Individuals in the experimental condition do receive the experimental treatment.

More on variables  What a the scales of measurement for variables? Levels of measurement and scales have to do with the way we measure variables. There are four levels of measurement commonly distinguished: Nominal scales, ordinal scales, interval scales, and ratio scales. Basically, these scales differ on the specificity and precision of measurement. Levels of measurement and scales have to do with the way we measure variables. There are four levels of measurement commonly distinguished: Nominal scales, ordinal scales, interval scales, and ratio scales. Basically, these scales differ on the specificity and precision of measurement.

Scales of Measurement  Nominal scales These consist of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations These consist of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations  Ordinal scales These scales consist of a set of categories that are organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size or magnitude These scales consist of a set of categories that are organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size or magnitude

Scales of Measurement Cont’d  Interval scales These scales consist of ordered categories that are all intervals of exactly the same size. With an interval scale, equal differences between numbers on the scale reflect equal differences in magnitude. However, ratios of magnitudes are not meaningful. These scales consist of ordered categories that are all intervals of exactly the same size. With an interval scale, equal differences between numbers on the scale reflect equal differences in magnitude. However, ratios of magnitudes are not meaningful.  Ratio Scales These scales are interval scales with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude. These scales are interval scales with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.

What are some examples of variables?  Nominal Scales Male versus female Male versus female Registered voter? (Yes/No) Registered voter? (Yes/No)  Ordinal Scales How much autonomy do you have at work? How much autonomy do you have at work? 1 – A little; 2 – Some; 3 – A lot1 – A little; 2 – Some; 3 – A lot  Interval Scales Temperature on Celsius of Fahrenheit Temperature on Celsius of Fahrenheit Income as a measure of social status Income as a measure of social status Is the difference in status the same for 20 – 50k as it is for 1m – 1.03m?Is the difference in status the same for 20 – 50k as it is for 1m – 1.03m?  Ratio Scales Temperature on Kelvin Temperature on Kelvin Measures of income using salary Measures of income using salary Weight, height Weight, height

Scales of Measurement Cont’d  What scale do they sell popcorn at a movie (small, medium, large)? Ordinal Ordinal  The SAT measures aptitude on what scale? Interval Interval  What color are your eyes? Nominal Nominal  How many peanuts have you eaten today? Ratio Ratio

Statistical Notation  If you are worried about your math skills, please review appendix A  Variables are represented as X (IV’s) and Y (DV’s)  N’s will represent the number of scores Upper case N is the number of total scores Upper case N is the number of total scores Lower case n is the number of group scores Lower case n is the number of group scores  Subscripts will not group differences n 1 notes the number of observations in group 1, while n 2 notes the number of observations in group 2. n 1 notes the number of observations in group 1, while n 2 notes the number of observations in group 2.

Summation notation  Σ represents a summation. This means we will take the sum of the mathematical expression that follows.  For example Σ X Σ X Σ (X-1) Σ (X-1) Σ (X2 – 1) Σ (X2 – 1) Σ X2 – 1 Σ X2 – 1 (Σ X)2 (Σ X)2 Σ (XY) Σ (XY)  Play close attention to the order of operations and especially parentheses  Find the value for the above expressions where: X {1,3,4,5} X {1,3,4,5} Y {2,4,3,7} Y {2,4,3,7}