Statistics Explained, 3rd Edition

between subjects - Also known as independent measures. In this design, the samples we select for each condition of the independent variable are independent, in that the samples come from different subjects.

bootstrapping - A sample is used to estimate a population. New bootstrap samples are randomly selected from the original sample with replacement (so an item can be selected more than once). The bootstrap samples, often 1000 or more, are then used to estimate the population sampling distribution.

causal relationship - A relationship where variation in one variable causes variation in another. Statistical tests can show a relationship between variables but not that it is causal. Other factors might be involved in the relationship. We might find that it snows more when the leaves have fallen from the trees, but we cannot claim the fallen leaves cause the snow. Factors such as the season and temperature are involved.

component - The term used in the principal components method of factor analysis for a potential underlying factor.

condition - A researcher chooses levels or categories of the independent variable to observe its effect on the dependent variable. These are referred to as conditions, levels, treatments or groups. For example, ‘morning’ and ‘afternoon’ might be chosen as the conditions for the independent variable of time of day.

confidence interval - In statistics we use samples to estimate population values, such as the mean or the difference in means. The confidence interval provides a range of values within which we predict lies the population value (to a certain level of confidence). The 95 per cent confidence interval of the mean worked out from a sample indicates that the estimated population mean would fall between the upper and lower limits for 95 per cent of the samples chosen.

confounding factor - An independent variable (in addition to the one under test) that has a systematic influence on the dependent variable.

control group - A group of subjects or participants matched with the experimental group on all relevant factors except the experimental manipulation. For example, a placebo group (who do not take a particular drug) could be used as a control group for a drug group (who do) to examine the effect of the drug on performance.

correlation - The degree to which the scores (from a set of subjects) on two variables co-relate. That is, the extent to which a variation in the scores on one variable results in a corresponding variation in the scores on the second variable. Usually the relationship we are looking for is linear. A multiple correlation examines the relationship between a combination of predictor variables with a dependent variable.

degrees of freedom - When calculating a statistic we use information from the data (such as the mean or total) in the calculation. The degrees of freedom is the number of scores we need to know before we can work out the rest using the information we already have. It is the number of scores that are free to vary in the analysis.

dependent variable - The variable measured by the researcher and predicted to be influenced by (that is, depend on) the independent variable.

descriptive statistics - Usually we wish to describe our data before conducting further analysis or comparisons. Descriptive statistics such as the mean and standard deviation enable us to summarise a set of data.

deviation - The difference of a score from the mean. When we subtract the mean value from a score the result is the deviation.

discriminant function - A discriminant function is one derived from a set of independent (or predictor) variables that can be used to discriminate between the conditions of a dependent variable.

distribution - The range of possible scores on a variable and their frequency of occurrence. In statistical terms we refer to a distribution as a ‘probability density function’. We use the mathematical formulae for known distributions to work out the probability of finding a score as high as or as low as a particular score.

effect s - ize - The size of the difference between the means of two populations, usually expressed in standard deviation units.

eigenvalue - In a factor analysis - an eigenvalue provides a measure of the amount of variance that can be explained by a proposed factor. If a factor has an eigenvalue of 1 then it can explain as much variance as one of the original independent variables.

equality of variance - see homogeneity of variance.

factor - Another name for ‘variable’, used commonly in the analysis of variance to refer to an independent variable. In factor analysis we analyse the variation in the data to see if it can be explained by fewer factors (i.e. ‘new’ variables) than the original number of independent variables.

frequency - The number of times a score, a range of scores, or a category is obtained in a set of data is referred to as its frequency.

frequency data - The data collected is simply the number of scores that fall into each of certain specified categories. See also ‘nominal data’.

general linear model - The underlying mathematical model employed in parametric statistics. When there are only two variables, X and Y, the relationship between them is linear when they satisfy the formula Y = a+ bX (where a and b are constants). The general linear model is a general form of this equation allowing as many X and Y variables as we wish in our analysis.

histogram - A plot of data on a graph, where vertical bars are used to represent the frequency of the scores, range of scores or categories under study.

homogeneity of variance - Underlying parametric tests is the assumption that the populations from which the samples are drawn have the same variance. We can examine the variances of the samples in our data to see whether this assumption is appropriate with our data or not.

homoscedasticity - The scores in a scatterplot are evenly distributed along and about a regression line. This is the assumption made in linear correlation and regression. (This is the correlation and regression equivalent of the homogeneity of variance assumption.)

hypothesis - A predicted relationship between variables. For example: ‘As sleep loss increases so the number of errors on a specific monitoring task will increase.’

independent measures - A term used to indicate that there are different subjects in each condition of an independent variable.

independent variable - A variable chosen by the researcher for testing, predicted to influence the dependent variable.

inferential statistics - Statistics that allow us to make inferences about the data – for example whether samples are drawn from different populations or whether two variables correlate.

interaction - When there are two or more factors in an analysis of variance then we can examine the interactions between the factors. An interaction indicates that the effect of one-factor is not the same at each condition of another factor. For example, if we find that more cold drinks are sold in summer and more hot drinks sold in winter then we have an interaction of ‘drink temperature’ and ‘time of year’.

intercept - A linear regression finds the best fit linear relationship between two variables. This is a straight line based on the formula Y = a+ bX, where b is the slope of the line and a is the intercept, or point where the line crosses the Y-axis.

interval data - Data produced by the use of an interval scale. Parametric tests require interval data.

interval scale - A scale of measurement where the interval between consecutive numbers is always the same. Most measuring devices, such as timers, thermometers, tape measures, employ interval scales.

item - When we employ a test with a number of variables (such as questions in a questionnaire) we refer to these variables as items, particularly in reliability analysis where we are interested in the correlation between items in the test.

kurtosis -       The degree to which a distribution differs from the bell-shaped normal distribution in terms of its peakness. A sharper peak with narrow ‘shoulders’ is called leptokurtic and a flatter peak with wider ‘shoulders’ is called platykurtic.

linear correlation - The extent to which two variables correlate in a linear manner. That is, how close their scatterplot is to a straight line.

main effect - The effect of a factor on the dependent variable in an analysis of variance measured separately from other factors in the analysis.

MANOVA - A M - ultivariate AN - alysis O - f VA - riance. An analysis of variance technique where there can be more than one dependent variable in the analysis.

matching subjects - Subjects are matched on relevant criteria across the conditions of the independent variable to control for possible confounding variables. For example, participants may be matched on intelligence or experience to control for these factors.

mean - A measure of the ‘average’ score in a set of data. The mean is found by adding up all the scores and dividing by the number of scores.

mean square - A term used in the analysis of variance to refer to the variance in the data due to a particular source of variation.

median - If we order a set of data from lowest to highest the median is the point that divides the scores into two, with half the scores below and half above the median.

mixed design - A mixed design is one that includes both independent measures factors and repeated measures factors. For example, a group of men and a group of women are tested in the morning and the afternoon. In this test ‘gender’ is an independent measures variable (also known as ‘between subjects’) and time of day is a repeated measures factor (also known as ‘within subjects’), so we have a mixed design.

mode - The score which has occurred the highest number of times in a set of data.

multiple comparisons - The results of a statistical test with more than two conditions will often show a significant result but not where that difference lies. We need to undertake a comparison of conditions to see which ones are causing the effect. If we compare them two at a time this is known as pairwise comparisons. Multiple comparisons are either ‘planned’ and a specific comparison is planned in advance of the main test or ‘unplanned’ where comparisons are undertaken after discovering the significant finding.

multiple correlation - The correlation of one variable with a combination of other variables.

multivariate - Literally this means ‘many variables’ but is most commonly used to refer to a test with more than one dependent variable (as in the MANOVA).

non-parametric test - Statistical tests that do not use, or make assumptions about, the characteristics (parameters) of populations.

normal distribution - A bell-shaped frequency distribution that appears to underlie many human variables. The normal distribution can be worked out mathematically using the population mean and standard deviation.

null hypothesis - A prediction that there is no relationship between the independent and dependent variables.

opportunity sample - An available sample, which is neither randomly chosen nor chosen to be representative of the population.

ordinal data - When we cannot assume that the intervals between consecutive numbers on a scale of measurement are of equal size we have ordinal data and can only use the data to rank order the subjects. Ratings are assumed to be ordinal data. We perform non-parametric tests on ordinal data.

outlier - An extreme value in a dataset – in that it lies outside the main cluster of scores. When calculating a linear correlation or regression an outlier will have a disproportionate influence on the statistical calculations.

parameter - A characteristic of a population, such as the population mean.

parametric tests - Statistical tests that use the characteristics (parameters) of populations or estimates of them (when assumptions are also made about the populations under study).

partial correlation - The correlation of two variables after having removed the effects of a third variable from both. -

participant - A person taking part as a ‘subject’ in a study. The term ‘participant’ is preferred to ‘subject’ as it acknowledges the person’s agency: i.e. that they have consented to take part in the study.

path analysis - - The analysis of the relationships between variables based on a proposed path model. Regression analysis is used to examine the variation in the variables explained by the model.

path model - - A proposed model of the relationship between variables. A one directional path (shown by an arrow) from one variable to a second indicates that variation in the second variable depends on the variation in the first variable.

population - A complete set of objects or events. In statistics this usually refers to the complete set of subjects or scores we are interested in, from which we have drawn a sample.

post hoc tests - When we have more than two conditions of an independent variable a statistical test (such as an ANOVA) may show a significant result but not the source of the effect. We can perform post hoc tests (literally post hoc means ‘after this’) to see which conditions are showing significant differences. Post hoc tests should correct for the additional risk of Type I errors when performing multiple tests on the same data.

power of - a - test - The probability that, when there is a genuine effect to be found, the test will find it (that is, correctly reject a false null hypothesis). As an illustration, one test might be like a stopwatch that gives the same time for two runners in a race but a more powerful test is like a sensitive electronic timer that more accurately shows the times to differ by a fiftieth of a second.

probability - The chance of a specific event occurring from a set of possible events, expressed as a proportion. For example, if there were 4 women and 6 men in a room the probability of meeting a woman first on entering the room is 4/10 or 0.4 as there are 4 women out of 10 people in the room. A probability of 0 indicates an event will never occur and a probability of 1 that it will always occur. In a room of only 10 men there is a probability of 0 (0/10) of meeting a woman first and a probability of 1 (10/10) of meeting a man.

quartile - If we order a set of scores from the lowest to the highest the quartiles are the points that divide the scores into four equal groups, with a quarter of the scores in each group. The second quartile is the median.

random error - There will always be random factors influencing subjects’ scores in an experiment. Random error is the influence of these random factors on the data. Statistical tests take account of random factors.

random sample - A sample of a population where each member of the population has an equal chance of being chosen for the sample.

range - The difference between the highest and lowest scores in a set of data.

rank - When a set of data is ordered from lowest to highest the rank of a score is its position in this order.

rank order - A method of ordering scores, listing them from lowest to highest.

ratio data - Data measured on a ratio scale.

ratio scale - An interval scale with an absolute zero. A stopwatch has an absolute zero as 0 indicates ‘no time’ and so we can make ratio statements: 20 seconds is twice as long as 10 seconds. The Celsius and Fahrenheit scales of temperature are interval but not ratio scales and indeed have 0 at different temperatures.

regression - The prediction of subjects’ scores on one variable by their scores on a second variable. This prediction is usually based on the relationship between the variables being linear and hence the prediction can be made using the formula Y = a+ bX. The larger the correlation between the variables the more accurate the prediction. A multiple regression predicts the variation in a variable by a number of predictor variables.

reliability - A reliable test is one that will produce the same result when repeated (in the same circumstances). We can investigate the reliability of the items in a test (such as the questions in a questionnaire) by examining the relationship between each item and the overall score on the test.

repeated measures - A term used to indicate that the same subjects are providing data for all the conditions of an independent variable.

representative sample - A subset of a population that shares the same key characteristics of the population. For example, the sample has the same ratio of men to women as the population.

residual - A linear regression provides a prediction of the subjects’ scores on one variable by their scores on a second. The residual is the difference between a subject’s actual score and their predicted score on the first variable. (A linear regression predicts that the data follow a linear model. The residuals indicate the extent to which the data do not fit the model, so are often referred to as ‘errors’.)

scatterplot - A graph of subjects’ scores on one variable plotted against their scores on a second variable. The graph shows how the scores are ‘scattered’.

significance level - The risk (probability) of erroneously claiming a relationship between an independent and a dependent variable when there is not one. Statistical tests are undertaken so that this probability is chosen to be small, usually set at 0.05 indicating that this will occur no more than 5 times in 100. This sets the probability of making a Type I error.

simple main effects - A significant interaction in a two-factor analysis of variance indicates that the effect of one variable is different at the various conditions of the other variable. Calculating simple main effects tells us what these different effects are. A simple main effect is the effect of one variable at a single condition of a second variable.

skew - The degree of symmetry of a distribution. A symmetrical distribution, like the normal distribution, has a skew of zero. The skew is negative if the scores ‘pile’ to the right of the mean and positive if they pile to the left.

standard deviation - A measure of the standard (‘average’) difference (deviation) of a score from the mean in a set of scores. It is the square root of the variance. (There is a different calculation for standard deviation when the set of scores are a population as opposed to a sample.)

standard error of the estimate - A measure of the ‘average’ distance (standard error) of a score from the regression line.

standard error of the mean - The standard deviation of the distribution of sample means. It is a measure of the standard (‘average’) difference of a sample mean from the mean of all sample means of samples of the same size from the same population.

standard normal distribution - A normal distribution with a mean of 0 and a standard deviation of 1.

standard score - The position of a score within a distribution of scores. It provides a measure of how many standard deviation units a specific score falls above or below the mean. It is also referred to as a z score.

statistic - Specifically, a characteristic of a sample, such as the sample mean. More generally, statistic and statistics are used to describe techniques for summarising and analysing numerical data.

structural equation modelling - A proposed model for the relationship between variables. The model can include latent variables (hypothesized variables) as well as manifest variables (measured variables). The model is defined by the multiple regression equations based on their proposed relationships. Using a combination of multiple regression and factor analysis the adequacy of the model to explain the variation in the data is tested.

subject - The term used for the source of data in a sample. If people are the subjects of the study it is viewed as more respectful to refer to them as participants, which acknowledges their role as helpful contributors to the investigation.

sums of squares - The sum of the squared deviations of scores from their mean value.

systematic error - Data that has been systematically influenced by another variable in addition to the independent variable under test is said to contain systematic error. The additional variable is said to confound the experiment.

two-tailed test - A prediction that two samples come from different populations, but not stating which population has the higher mean value.

Type - I error - The error of rejecting the null hypothesis when it is true. The risk of this occurring is set by the significance level.

Type II error - The error of failing to reject the null hypothesis when it is false.

univariate - A term used to refer to a statistical test where there is only one dependent variable. ANOVA is a univariate analysis as there can be more than one independent variable but only one dependent variable.

variance - A measure of how much a set of scores vary from their mean value. Variance is the square of the standard deviation.

within subjects - Also known as repeated measures. We select the same subjects for each condition of an independent variable for a within-subjects design.