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The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.
The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples.
Statistical inference, however, moves in the opposite direction— inductively inferring from samples to the parameters of a larger or total population.
Experimental and observational studies[ edit ] A common goal for a statistical research project is to investigate causality , and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables.
There are two major types of causal statistical studies: experimental studies and observational studies.
In both types of studies, the effect of differences of an independent variable or variables on the behavior of the dependent variable are observed.
The difference between the two types lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between predictors and response are investigated.
While the tools of data analysis work best on data from randomized studies , they are also applied to other kinds of data—like natural experiments and observational studies  —for which a statistician would use a modified, more structured estimation method e. Experiments[ edit ] The basic steps of a statistical experiment are: Planning the research, including finding the number of replicates of the study, using the following information: preliminary estimates regarding the size of treatment effects , alternative hypotheses , and the estimated experimental variability.
Consideration of the selection of experimental subjects and the ethics of research is necessary. Statisticians recommend that experiments compare at least one new treatment with a standard treatment or control, to allow an unbiased estimate of the difference in treatment effects.
Design of experiments , using blocking to reduce the influence of confounding variables , and randomized assignment of treatments to subjects to allow unbiased estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the experimental protocol that will guide the performance of the experiment and which specifies the primary analysis of the experimental data. Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol.
Further examining the data set in secondary analyses, to suggest new hypotheses for future study. Documenting and presenting the results of the study. Experiments on human behavior have special concerns. The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers.
The researchers first measured the productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved under the experimental conditions.
However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. Nominal-level and ordinal-level variables are also referred to as categorical variables, because each category in the variable can be completely separated from the others. The categories for an interval variable can be placed in a meaningful order, with the interval between consecutive categories also having meaning.
Age, weight, and blood glucose can be considered as interval variables, but also as ratio variables, because the ratio between values has meaning e. Interval-level and ratio-level variables are also referred to as continuous variables because of the underlying continuity among categories.
As we progress through the levels of measurement from nominal to ratio variables, we gather more information about the study participant. The amount of information that a variable provides will become important in the analysis stage, because we lose information when variables are reduced or aggregated—a common practice that is not recommended.
As the terms imply, the value of a dependent variable depends on the value of other variables, whereas the value of an independent variable does not rely on other variables.
In addition, an investigator can influence the value of an independent variable, such as treatment-group assignment. Independent variables are also referred to as predictors because we can use information from these variables to predict the value of a dependent variable.
Building on the group of variables listed in the first paragraph of this section, blood glucose could be considered a dependent variable, because its value may depend on values of the independent variables age, sex, ethnicity, exercise frequency, weight, and treatment group.
Statistics are mathematical formulae that are used to organize and interpret the information that is collected through variables. There are 2 general categories of statistics, descriptive and inferential. Descriptive statistics are used to describe the collected information, such as the range of values, their average, and the most common category.
Knowledge gained from descriptive statistics helps investigators learn more about the study sample. Inferential statistics are used to make comparisons and draw conclusions from the study data.
Knowledge gained from inferential statistics allows investigators to make inferences and generalize beyond their study sample to other groups. Before we move on to specific descriptive and inferential statistics, there are 2 more definitions to review.
Parametric statistics are generally used when values in an interval-level or ratio-level variable are normally distributed i. These statistics are used because we can define parameters of the data, such as the centre and width of the normally distributed curve.
In contrast, interval-level and ratio-level variables with values that are not normally distributed, as well as nominal-level and ordinal-level variables, are generally analyzed using nonparametric statistics. This can be done using figures to give a visual presentation of the data and statistics to generate numeric descriptions of the data.
Selection of an appropriate figure to represent a particular set of data depends on the measurement level of the variable. Data for nominal-level and ordinal-level variables may be interpreted using a pie graph or bar graph. Both options allow us to examine the relative number of participants within each category by reporting the percentages within each category , whereas a bar graph can also be used to examine absolute numbers. For example, we could create a pie graph to illustrate the proportions of men and women in a study sample and a bar graph to illustrate the number of people who report exercising at each level of frequency never, sometimes, often, or always.
Interval-level and ratio-level variables may also be interpreted using a pie graph or bar graph; however, these types of variables often have too many categories for such graphs to provide meaningful information. Instead, these variables may be better interpreted using a histogram. Unlike a bar graph, which displays the frequency for each distinct category, a histogram displays the frequency within a range of continuous categories.
Information from this type of figure allows us to determine whether the data are normally distributed. In addition to pie graphs, bar graphs, and histograms, many other types of figures are available for the visual representation of data.
Figures are also useful for visualizing comparisons between variables or between subgroups within a variable for example, the distribution of blood glucose according to sex. Box plots are useful for summarizing information for a variable that does not follow a normal distribution. The lower and upper limits of the box identify the interquartile range or 25th and 75th percentiles , while the midline indicates the median value or 50th percentile.
Scatter plots provide information on how the categories for one continuous variable relate to categories in a second variable; they are often helpful in the analysis of correlations.