STATS 252

Population

the entire collection of all individuals or items under consideration in a statistical study.

• The target ____ should be clearly defined since there are ____ within _____.

Population size

total number of individuals or items in the population under study.

Census

collecting data from the entire population.

• Often too expensive or even impossible to undertake; therefore, a sample is taken.

Sample

a subset of individuals from the population. Data are only recorded on these individuals.

• a relatively small number of observations from the population being investigated

• Can be used to mean one observation or a collection of measurements from a population.

Sample size

number of individuals or observations in a single sample

Inferential statistics

uses information from a sample to make decisions, conclusions, and predictions about the entire population

Parameter

descriptive measure of a population (symbolized by Greek letters),

• population mean (u)

• population standard deviation (omega)

• slope of the populationregression line 1 (beta).

Statistic

a descriptive measure of a sample used to estimate a parameter,

• sample mean y (with line on top)

• sample standard deviation S

• and slope of the sample regression line (beta1 with ^)

Components of research design

• Study units (within the context of the target population and study area/sites)

• Variables

• Spatial Aspects of Design

• Temporal Aspect of Design

• Techniques and Methods of Data Collection

• Sampling strategy & Randomness: More about Where and When

• Overall Type of Research Design: Observational versus Experimental

Study units

individuals, or subjects (people, animals, objects, or things) about which information is required and on which measurements are recorded.

• AKA units of analysis or cases.

• In an experimental study = experimental units

• observation study = units of observation.

• In agricultural research, these are the pre-determined plots where different treatments are applied.

• Social sciences = “who”

Variable

characteristic that varies from one study unit (individual, subject, person, or thing) to another.

• social sciences, these may be opinions, behavior, attitudes, perception, etc.

• In physics – weight, force, energy, light, etc.

• In biology – growth rate, chlorophyll content, height, density, color, etc.

Distribution of a variable

all the values that a variable takes on.

Data

the values of a variable, i.e., actual measurements/observations recorded for each variable.

Datum

individual piece of data (an observation) or a single measurement.

Categorical/Qualitative variable

• categorical variable is a nonnumerically valued variable and does not follow an ordered sequence.

• On nominal scale

• Values of the variable are classified by some quality or attribute, i.e., the values are put into categories.

• cannot be measured, but rather, the frequencies of individuals in the categories are counted to obtain numbers to analyze the data.

• Examples: color, regions of a country (north, south, east, west); marital status; like or dislike a certain hobby or activity; yes, no, or indifferent to something; types of animals; types of items to purchase; languages.

Ordinal scale

• Data or observations which can be put in order from lowest to highest, but which do not have a constant interval between successive units, i.e., the data can be ranked.

• Relative magnitudes are known, so many types of statistical analysis can be applied.

• scale of 1 – 5 can be used, where 1 = very poor, 2 = poor, 3 = moderate, 4 = good, 5 =very good.

Qualitative scale

• numerically valued variable.

• Constant interval size between successive units

• Discreate/continuous quantitative variables

Discrete

quantitative variable that can only take on specific values, usually whole numbers.

• a countable variable.

Continuous quantitative variable

quantitative variable that can have an infinite number of values between any observed range.

• a measurable variable.

Indicator variable

• dummy variable

• Categorical variables that are coded in order to obtain quantitative variables that can be analyzed using hypotheses tests like ANOVA and regression.

• Ex. Responses to a question, Yes or No, can be coded as 1 = Yes, 0 = No.

Derived variable and indices

This is when a combination of categorical, ordinal, and/or quantitative variables are recorded and the values are multiplied, divided, added, or otherwise combined to obtain one derived quantitative variable with units (e.g., basal area of mangroves calculated as cm2/25-m2 plot).

• Or they can be combined to obtain an index (without units).

Explanatory/predictor/independent variable

variables of interest that are hypothesized to explain or affect other variables in the study, but which are not likely to be affected by those other variables

• application of variables must either precede or occur during the same time period as the expected reaction of the response variable.

Response/dependent variable

variable that is hypothesized to be affected by the explanatory or independent variables.

Extraneous variables

possible explanatory variables that are NOT of interest or are NOT related to the purpose of the study, though they could be of interest in a different study.

• May potentially affect the response variable, interfere with the study and lead to “experimental error“

• Lurking, confounding, hidden variables

Factors

explanatory variables, which are categorical, are applied as treatments in an experiment or considered as levels in an observational study

Spatial aspects of design

• The “Where” component of research design.

• Linked to the study units – where are they sampled and measured?

• Involves the way the observations or replicates are arranged in space (distance, area, or volume).

• Study sites and geographical location of target populations.

Temporal aspects of design

• The “When” component of research design – the way observations or replicates are arranged in time.

• Time period (year, month, time of day) and frequency of observations.

• Start, end, frequency of recording the variables

Techniques and methods of data collection

• The “How” component of research design.

• Specific methods and techniques used to take measurements of the variables or to record data.

• The specific techniques to be applied will differ from one field of natural or social science to the other

Random sampling

the selection of individuals or study units from a population without bias, such that:

1. All individuals have an equal chance of selection (or each possible sample of a given size is equally

likely to be the one obtained).

2. The selection of individuals is independent, i.e., the selection of one does not affect the selection of others.

• ensures that the sample is as representative as possible of the entire population.

• simple random sampling (SRS), systematic random sampling,

  stratified random sampling, and cluster random sampling.

Sampling strategies

• Study units or individuals on which data are recorded must be randomly sampling from the target population so that they truly represent the population

• in a given study must specify the way observations are recorded in space and time.

• must eliminate bias as much as possible because bias over-emphasizes or under-emphasizes some characteristics of the population.

Observational studies

• when conducted in order to get opinions from people it is often called a sample survey or social survey

• opinions from people it is often called a sample survey or social survey.

• Aims at estimating population parameters.

• Researcher collects data about a particular phenomenon as it occurs in nature or in society.

• Randomness

• Variables of interest are measured or recorded for the study units.

• No imposing of treatments on the subjects or individuals.

• No manipulation or control of any variables or conditions.

• Extraneous variables cannot be controlled.

Population inferences

• can be made if there is random selection from the target population in observational studies.

• Can be made if there is random selection from the target population in experiments

Causal inferences

• can NOT be made in observational studies, that is, causation or cause-and-effect relationships among variables can NOT be established because there are so many unmeasured factors (or extraneous variables) that may affect the variable being measured.

• can be made if there is random assignment to treatment and control groups In experiments

Experimental studies sampling randomness

• First, study units (experimental units) are randomly selected from the target population.

• Secondly, the experimental units are randomly assigned to treatment and control groups

Treatment groups

exposed to new conditions, that is, one or more levels of the predictor variable or factor being manipulated; the treatments are imposed upon these groups.

Control groups

exposed to the usual level of the manipulated variable or not exposed to it at all

• case of human subjects, the group receives a placebo, so the subjects don’t know whether they are receiving a treatment or not

Replication is required to

• Check or confirm the results,

• Apply statistical analysis – analysis is based on replicates, and

• Increase the power of the test.

No. of replicates = no. of observations in a sample or sample size (n).

Describing quantitative data

1. Shape – symmetric, left skewed, or right skewed.

2. Center – the middle of the distribution.

3. Spread – variation or dispersion of the distribution.

Mean

center of gravity of the distribution (e.g., histogram).

• not a resistant measure of center, because it is seriously influenced by skewness (pulled in the direction of a few extreme observations).

• Best for symmetric distributions

Median

divides the area under the curve into two equal halves.

• resistant measure because it is more robust to extreme values or skewness than the mean and therefore is a better measure of center for a very skewed distribution.

Mode

only measure of center that can also be used for qualitative data. Distributions may be unimodal, bimodal, or multimodal.

Five-number summary

Min, Q1, Q2, Q3, Max

Outliers

observations that lie outside the overall pattern of the data.

• May be due to recording error, may belong to a different population, or may just be unusually extreme observations.

Graphs used for quantitative data

1. Histograms

2. Dotplots

3. Stemplots

4. Boxplots

5. Normal probability plots

6. Scatter diagrams (xy graphs)