One example of a variable in univariate analysis might be "age". Another might be "height". Univariate analysis would not look at these two variables at the same time, nor would it look at the relationship between them.
Some ways you can describe patterns found in univariate data include looking at mean, mode, median, range, variance, maximum, minimum, quartiles, and standard deviation. Additionally, some ways you may display univariate data include frequency distribution tables, bar charts, histograms, frequency polygons, and pie charts.
Bivariate analysis is used to find out if there is a relationship between two different variables. Something as simple as creating a scatterplot by plotting one variable against another on a Cartesian plane think X and Y axis can sometimes give you a picture of what the data is trying to tell you.
If the data seems to fit a line or curve then there is a relationship or correlation between the two variables. For example, one might choose to plot caloric intake versus weight. Multivariate analysis is the analysis of three or more variables. University Press Scholarship Online. Sign in. Not registered? Sign up.
Publications Pages Publications Pages. Recently viewed 0 Save Search. Users without a subscription are not able to see the full content. Basic Statistics in Multivariate Analysis. Find in Worldcat. Go to page:. Your current browser may not support copying via this button. Search within book. It uses probabilities and models to test predictions about a population from sample data.
The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one. To indirectly reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power. The risk of making a Type I error is the significance level or alpha that you choose.
The significance level is usually set at 0. In statistics, ordinal and nominal variables are both considered categorical variables. Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them. In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative a Type II error.
Your study might not have the ability to answer your research question. While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world. Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes. There are dozens of measures of effect sizes.
Effect size tells you how meaningful the relationship between variables or the difference between groups is. A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications. Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates.
Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie. Standard error and standard deviation are both measures of variability. The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population.
The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
To figure out whether a given number is a parameter or a statistic , ask yourself the following:. If the answer is yes to both questions, the number is likely to be a parameter.
For small populations, data can be collected from the whole population and summarized in parameters. If the answer is no to either of the questions, then the number is more likely to be a statistic.
The arithmetic mean is the most commonly used mean. But there are some other types of means you can calculate depending on your research purposes:. You can find the mean , or average, of a data set in two simple steps:. This method is the same whether you are dealing with sample or population data or positive or negative numbers. Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.
The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset. Descriptive statistics summarize the characteristics of a data set.
Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population. In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data.
The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision. AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting. In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable.
You can test a model using a statistical test. The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters K used to reach that likelihood.
The AIC function is 2K — 2 log-likelihood. Lower AIC values indicate a better-fit model, and a model with a delta-AIC the difference between the two AIC values being compared of more than -2 is considered significantly better than the model it is being compared to. The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables parameters as a way to avoid over-fitting.
AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result. Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares the variance explained by the independent variable to the mean square error the variance left over.
If the F statistic is higher than the critical value the value of F that corresponds with your alpha value, usually 0. If you are only testing for a difference between two groups, use a t-test instead.
The formula for the test statistic depends on the statistical test being used. Generally, the test statistic is calculated as the pattern in your data i. Linear regression most often uses mean-square error MSE to calculate the error of the model.
MSE is calculated by:. Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE. Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative. For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands.
This linear relationship is so certain that we can use mercury thermometers to measure temperature. A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line or a plane in the case of two or more independent variables. A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.
A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.
A one-sample t-test is used to compare a single population to a standard value for example, to determine whether the average lifespan of a specific town is different from the country average. A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time for example, measuring student performance on a test before and after being taught the material.
A t-test measures the difference in group means divided by the pooled standard error of the two group means. In this way, it calculates a number the t-value illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance p-value.
Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means. If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test. If you want to know only whether a difference exists, use a two-tailed test. If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test.
A t-test is a statistical test that compares the means of two samples. It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test. Significance is usually denoted by a p -value , or probability value. Statistical significance is arbitrary — it depends on the threshold, or alpha value, chosen by the researcher. When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups. The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis.
Different test statistics are used in different statistical tests. The measures of central tendency you can use depends on the level of measurement of your data. Ordinal data has two characteristics:. Nominal and ordinal are two of the four levels of measurement.
Nominal level data can only be classified, while ordinal level data can be classified and ordered. If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups. If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data. In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:. The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies. The z -score and t -score aka z -value and t -value show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution. These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is.
If your test produces a z -score of 2. The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.
To calculate the confidence interval , you need to know:. Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate e. The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.
The confidence interval is the actual upper and lower bounds of the estimate you expect to find at a given level of confidence. These are the upper and lower bounds of the confidence interval. Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way. For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.
The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. Statistical tests commonly assume that:. If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences. Measures of central tendency help you find the middle, or the average, of a data set.
Some variables have fixed levels.
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