   < Back Glossary, page 1 of 1 Next >      ## Glossary

Introduction to Statistics contains a large, scrollable and searchable glossary of all the technical terms that occur in it. Whenever one of these terms occurs in the program, it is highlighted and clicking on it brings up its definition. All these glossary terms and explanations are shown below the picture. You can jump to any letter of the alphabet. ## A

Absolute quantitative term: an expression such as 'big' or 'fast' which describes the size or strength of something without comparing it to anything else.

Absolute scale: a ratio scale in which the unit of measurement is fixed. Numbers on an absolute scale are almost always obtained by counting something.

Aggregate conclusion: an answer which is true on average but is not necessarily true of every individual. See also Existential conclusion and General conclusion.

Alternative Hypothesis: the logical opposite of the Null Hypothesis, asserting that those things the Null Hypothesis says are false are true, and that those things the Null Hypothesis says are true are false.

Anecdotal evidence: evidence obtained informally from isolated incidents rather than as part of a systematic investigation.

Assume: take to be true without evidence, or with very inadequate evidence.

Average: a general term for a score considered to be typical of a group of scores. The Mode, Median or Mean might all be described as 'Average' scores, but unless otherwise indicated, the term 'Average' is usually assumed to refer to the Mean.

## B

Bar chart: a graph resembling a histogram but in which the horizontal axis shows the values of a qualitative variable.

Bias: any tendency for results to differ from the true value in some consistent way. Bias should be distinguished from random error, which does not occur in any particular direction.

Biased sample: a sample selected in a way that makes some individuals in the population more likely than others to be included in the sample.

Blind study: one in which the participants do not know the details of the materials they are responding to or the nature of the difference between one experimental condition and another. See also, 'double-blind study',

Box-and-whisker plot: a graph which represents the first and third quartiles of a distribution by the ends of a box, the second quartile (the median) by a line across it and its highest and lowest scores by lines extending beyond the box. Also called a boxplot.

## C

Case study: an investigation based on the detailed study of a single individual.

Chi-squared: The expected distribution of the sum of squares for scores distributed according to the normal curve. The symbol for a statistic calculated in such a way that it is expected to follow the Chi-squared distribution is the squared Greek letter Chi ( χ ). The statistic is used in various tests based on the frequencies with which events occur.

Clinical case study: a case study conducted in connection with therapy.

Cohort study: an investigation based on detailed study, usually over an extended period of time, of a group (or cohort) of individuals.

Confounding: a defect in the design of a study whereby two or more variables change together, with the result that we cannot tell which of them is responsible for any effects that are observed.

Control group: a group of participants in an experiment who are treated in the same way as the experimental group except in one particular respect which is the topic of the experiment. To interpret the outcome of the experiment, we compare the results of the experimental group with those of the control group.

Controlled variable: something that might have varied in a study but has been prevented from varying in order to eliminate any effect it might have had on the results.

Correlation coefficient: a number between -1 and +1 indicating the strength and direction of any tendency for the size of a score on one variable to be related to the size of the corresponding score on another.

Counterbalance: When a nuisance variable (such as participants' improvement with practice) cannot be eliminated from a study it is sometimes possible to counterbalance it. That is, we make its effects operate equally on all the other conditions, for example by administering the various tasks in different orders.

## D

Data set: A collection of results that form a group, for example because they were obtained at the same time or from a particular group of people. In statistical notation, it is quite common to use a single symbol, such as X, to refer to all the scores in a data set.

Data: Results (often, but not necessarily, numerical scores) that can be interpreted to give information about the outcome of an investigation. See also Raw score.

Deduce: arrive at a conclusion by calculation or the use of logic.

Degrees of freedom (df): the number of different ways in which scores are free to vary around some value (such as their mean).

Dependent variable: a variable in a study whose value is influenced by the state of some other variable, called the 'independent variable'. Dependent variables are those that are measured to provide the results in an experiment.

Developmental case study: a case study recording the increasing capabilities of a child or young animal as it matures.

Deviation: the difference between an individual score and the mean of the group of scores it belongs to.

Diary study: a technique of investigation in which participants record their own activities for subsequent analysis and interpretation.

Directional test: a statistical test in which any difference – however large – in one direction is attributed to chance and the Null Hypothesis is not rejected. If the results show a difference in the opposite direction, the result may be significant. The terms one-tailed and one-sided are equivalent. The alternative possibility is a non-directional (or two-tailed or two-sided test where a difference in either direction can constitute evidence against the Null Hypothesis.

Dispersion: in statistics, refers to the amount of difference from one score to another in a group of scores. The terms spread and scatter are equivalent.

Distribution: the relative frequencies with which scores of different sizes occur.

Distribution-dependent test: see distribution-free test.

Distribution-free test: a statistical test that makes no assumption about the shape of the distribution of random errors. As a result, such a test gives valid results when we are uncertain about the nature of the distribution or where the data indicate that the distribution is not the one assumed by a distribution-dependent test such as the t-test. Compared with a distribution-dependent test, a distribution-free test such as the Mann-Whitney usually has less statistical power but can be used with confidence in more cases. The terms non-parametric and parametric are often used as synonyms for 'distribution-free' and 'distribution- dependent'.

Double-blind study: one in which neither the participants nor the person who administers the study knows the details of the materials or the conditions being presented in each particular case. See also, 'blind study'.

## E

Existential conclusion: an answer which shows that something exists or can occur, even if it is observed only rarely. See also Aggregate conclusion and General conclusion.

Expected frequencies: in a chi-squared test, the mean values of the frequencies that would be observed if the Null Hypothesis is true and data-collection is repeated many times.

Experiment: a method of investigation in which one or several 'independent variables' are manipulated in order to discover their effect on one or more 'dependent variables'.

Experimental design: the activity of planning data collection (usually, but not always, an experiment) in such a way as to make it possible for the results to be interpreted as clearly and efficiently as possible.

Experimental hypothesis: see Alternative Hypothesis.

## F

Field experiment: an experiment conducted in the normal environment of the participants.

Field study: an investigation, usually involving observation rather than experiment, conducted in the normal environment of the participants.

Formula: in statistics, a mathematical expression describing how a calculation should be carried out.

Frequency polygon: a line graph representing the relationship between the size of scores on a variable and their frequency of occurrence.

Frequency: describes the number of times that something occurs, for example the number of scores of a particular size.

Frequency: describes the number of times that something occurs, for example the number of measurements of a particular size.

## G

General conclusion: an answer which is true of every individual. See also Aggregate conclusion and Existential conclusion.

Graph: a diagram used to represent some aspect of data. It is useful to distinguish between graphs of distributions and graphs of relationships.

## H

Histogram: a graph in which the sizes of scores on a quantitative variable are plotted on the horizontal axis and their frequency is plotted as the height of a bar.

Hypothesis: a prediction or an explanation that is considered to be possible but is not known to be correct. See also Alternative Hypothesis and Null Hypothesis.

## I

Independent variable: a variable which influences one or more 'dependent variables'. Independent variables are those that are manipulated by someone conducting an experiment.

Inference:a conclusion reached by a process of calculation or logical deduction.

Interquartile Range: a measure of spread representing the difference between the third and the first quartile in a group of scores.

Interval scale: a scale of measurement in which we know that equal differences between numbers indicate equal differences in the attribute that is measured, but we do not know what numerical value corresponds to complete absence of the attribute.

Interview: a method of collecting data by face-to-face or telephone questioning of one person by another.

Introspection: a method of collecting data where we record and comment on our own feelings and internal experiences.

Investigation: a broad term referring to any method of collecting data, such as experiments, surveys, interviews and so on (equivalent to 'study').

## L

Legend: a paragraph following the title of a Table or Graph and containing additional explanations.

Level of Measurement: equivalent to Measurement Scale. We usually distinguish between Nominal, Ordinal, Interval and Ratio measurement scales.

Line graph: a graph which shows the relationship between two variables by plotting corresponding values of each variable, one horizontally and the other vertically, and joining up these points by a line.

Longitudinal case study: a case study recording the changes that occur in the characteristics of someone with the passage of time.

## M

Mann-Whitney test: a distribution-free test for two groups of unmatched measurements.

Mean: the total of all the scores in a group divided by the number of scores in the group.

Measurement model: the relationship that we believe or assume to exist between an attribute that we wish to measure and the numbers describing the attribute.

Measurement: the process of representing attributes or aspects of people, events or things by numbers in a systematic way. See also Scale.

Median test: a nonparametric significance test of the Null Hypothesis that two populations of measurements have the same median.

Median: the middle score when the scores in a group are arranged in order of size.

Mode: the most frequent score in a group of scores.

Model: a representation of an organism, a relationship, an event or a process in a way that is simpler than the original. Models can be concrete (composed of actual substances) such as 'model boats' or abstract (consisting only of theoretical elements, numbers and relationships, for example) such as the 'null model'.

## N

Nominal scale: the use of numbers purely as names for the category that an object or event belongs to. The sizes of the numbers tell us nothing about the properties of the object or event.

Non-directional test: see directional test.

Nonparametric test: a statistical test which does not involve calculating the values of attributes (called parameters) such as the mean and the standard deviation of distributions of scores. Examples of nonparametric tests of significance are the Wilcoxon (T) test and the Mann-Whitney (U) test. The term is often considered equivalent to Distribution-free test.

Normal curve: a bell-shaped mathematical curve which is important in statistics because it describes the frequency distribution to be expected when many independent causes add together to produce each of the outcomes in a data set. Many attributes of humans and animals, such as their heights and weights, have frequency distributions that approximately follow a normal curve. (Also known as the Normal Distribution)

Nuisance variable: a variable we do not want to know about but which is capable of influencing the variables we do want to know about. In designing studies, we either aim to prevent nuisance variables from influencing the results or we design the study in such a way that the effects of nuisance variables can be measured and allowed for.

Null Hypothesis: a statistical model which attributes the observed results to chance alone. The probability of obtaining such results if the Null Hypothesis is correct is called the 'significance' of the results. If the probability is small (by convention, anything less than 0.05) the result is said to be significant and the Null Hypothesis is rejected in favour of some other model which attributes at least part of the effect to something other than chance. See significance test and statistical significance.

Null model: another term for the null hypothesis.

Numerical model: a model using numbers and the relations among them. See also Measurement model which is the most common type of numerical model.

## O

Objective: describes something whose truth or falsity can be demonstrated convincingly to anyone. It is contrasted with 'subjective'.

Observational study: an investigation where data is collected by observing events as they occur and, so far as possible, without influencing them. An observational field study is conducted in the natural environment of the subjects while an observational laboratory study is conducted under controlled conditions.

Observed frequencies: see expected frequencies.

One-sample t-test: a t-test for discovering the significance of the departure of the mean of a single sample of measurements from some specified value. It can also be used to discover if the differences between pairs of measurements from single or matched individuals have a mean that differs significantly from zero. When used in that way it is usually called a related t-test.

One-sided test: see directional test.

One-tailed test: see directional test.

Opportunity sample: a sample consisting of a group that already exists, such as a social club or a school class.

Order effect: an influence on results brought about by the order in which procedures are administered or measurements are taken.

Ordinal scale: a measurement scale in which larger numbers indicate more of the measured attribute, but neither the size of the difference between two numbers nor the ratio of two numbers tells us anything useful about the attributes.

Outlier: a score so different from the pattern of the other scores that we suspect it of arising in a different way from the others.

## P

Panel: a group of people organized to judge something in a situation where individuals might give different answers. Even when the judgement of an individual would be subjective, the average judgment of the panel, and especially any judgments they all agree about, is closer to being objective.

Parameter: an element of a statistical model. The term is often applied to an attribute of a population, such as its mean, whereas the mean of a sample of scores is called a statistic.

Parametric test: a statistical test which involves calculating the values of attributes (called parameters) such as the mean and the standard deviation of distributions of measurements. The commonest parametric test of significance is the t-test. See Nonparametric test.

Participant observation: a method of data collection where the investigator takes part in the activities of a group while recording events that occur.

Participant: a person whose actions or attributes form part of the results. The term 'subject' is more usual in the case of animals.

Pearson correlation: a correlation coefficient calculated from scores. It is also referred to as product-moment correlation.

Placebo effect: any effect produced by a participant's knowledge or belief about the nature of an investigation. The most typical placebo effect is an improvement produced by giving a dummy therapy or a dummy medication which (unknown to the participants) has no real effect.

Placebo group: a group of participants who are given a placebo. Groups given a therapy or medicine must improve more than the placebo group before the treatment can be considered to have any beneficial effect.

Placebo: a dummy therapy or medicine which, so far as the participants in the study are concerned, is indistinguishable from a real therapy or medicine.

Population: a group to which the results of an investigation are intended to apply. Studies typically investigate smaller groups, called 'samples' drawn from the appropriate population. A population need not include every individual in a country; we can quite well speak of the population of ten-year-old girls, for example.

Power: the ability of a statistical test to reject the Null Hypothesis when it is false. Some tests have more power than others when used with the same data, but all tests increase in power as the amount of data is increased.

Probability: a number between 0 and 1 which expresses how likely it is that some event will occur. If the probability is near 0, the event is unlikely. If the probability is near 1, the event is almost certain to occur.

## Q

Qualitative: having to do with what kind something is. For example, red, blue and green lights are qualitatively different. Compare Quantitative.

Quantitative: having to do with how much there is of something. For example, bright and dim light are quantitatively different (bright light is more of the same thing). Compare Qualitative.

Quartile: a score which is one quarter (for the first quartile, whose symbol is Q1) or three quarters (for the third quartile, whose symbol is Q3) of the way through a group of scores when they have been arranged in order of size. The second quartile is two quarters (that is, half-way) through the group so it is exactly equivalent to the Median.

Questionnaire: a list of questions in written form, usually requiring fairly brief answers.

Quota sample: a type of sample obtained by selecting available individuals until a sufficient number (a quota) of each of a number of specified types has been obtained.

## R

Random sample: a sample in which every member of the relevant population has the same chance of being included.

Randomize: to arrange things, such the order in which tasks are carried out, in an order determined by chance.

Range: a measure of spread representing the difference between the largest and the smallest (or most negative) score in a group of scores.

Ranks: numbers 1, 2, 3, and so on, often used in place of measurements that have been made on an ordinal scale. Ranks are allocated so that the smallest score gets the rank 1, the next is 2, and so on. If two or more scores are equal, all of them are given the average of the ranks they would have had if they had been slightly different from each other.

Ratio scale: a scale of measurement in which we know that equal differences between numbers indicate equal differences in the attribute that is measured, and also know what numerical value corresponds to complete absence of the attribute. It is only if we have measured on a ratio scale that it is meaningful to speak about one score being, say, ten times as great as another.

Raw score: a numerical result in the form in which it was recorded. A raw score is often used as it is, but sometimes it is better to replace all the raw scores in a set by a new value calculated from each of them (for example, by its rank or by its logarithm) before use is made of the scores. See also Data.

Reactivity: a change in behaviour caused by the fact that the behaviour is being observed or by the participant's expectations.

Related t-test: see one-sample t-test.

Relative quantitative term: an expression such as 'bigger' or 'faster' which describes the size or strength of something by comparing it to something else.

Representative sample: a sample intended to reflect the composition of the population.

Respondent: a person who answers questions.

## S

Sample: a smaller group selected for study from a larger group, called a 'population', to which the results are intended to apply.

Scale: a rule governing the relationship between numbers that are assigned and magnitudes of some property that is being measured. See also Nominal scale, Ordinal scale, Interval scale, Ratio scale.

Scatter plot: a graph which shows the relationship between two variables by plotting corresponding values of each variable, one horizontally and the other vertically, with each point shown separately (not joined by lines). Scattergram is an equivalent term.

Scatter: in statistics, refers to the amount of difference from one score to another in a group of scores. The terms dispersion and spread are equivalent.

Score: A numerical result representing the magnitude of something.

Semi-interquartile range: half of the inter-quartile range. It is a measure of spread but is no longer much used.

Sigma: the letter of the Greek alphabet corresponding to S. In statistics, a capital sigma ( Σ ) is an instruction to add up (or 'sum') a group of numbers while the lower-case (small) sigma ( σ ) is the symbol for the Standard Deviation of the scores of a population.

Sign test: a nonparametric test of the Null Hypothesis that differences between two populations of paired scores (or one population of scores and a predicted value) are as likely to be in one direction as the other.

Significance test: calculation of the probability of obtaining results such as those observed if the null hypothesis is true. If the probability is less than 0.05, the result is said to be 'significant' and we reject the null hypothesis as an adequate explanation for the results. When that happens, the alternative hypothesis that there is a real difference between groups is preferred to the null hypothesis that there is no such difference.

Significance test: calculation of the probability of obtaining results as extreme as those observed if the Null Hypothesis is true. If the probability is less than 0.05, the result is said to be significant and we reject the Null Hypothesis as an adequate explanation for the results. When that happens, the Alternative Hypothesis (that there is a real difference between the groups) is preferred to the Null Hypothesis that there is no such difference. Equivalent to Statistical test.

Significant: see statistical significance.

Spearman correlation: a correlation coefficient calculated from the ranks of scores.

Spread: in statistics, refers to the amount of difference from one score to another in a group of scores. The terms dispersion and scatter are equivalent.

Standard Deviation (SD): a measure of spread which is typical of the deviations in a group of scores. It is the square root of the average squared deviation in a group of scores.

Standard error (SE): a measure of our certainty about the value of some parameter such as, the mean of a distribution. It is equivalent to the standard deviation of our estimate of the parameter. If it is large, we are uncertain about the true value of the parameter (the mean, for example).

Standard score: the deviation of a measurement divided by the standard deviation of the group of measurements it belongs to. The usual symbol is z so it is also called a z-score.

Statistic: a numerical value calculated from data to represent some property of the data. For example, a mean, a range and a correlation coefficient are all statistics.

Statistical inference: reaching probabilistic conclusions based on the statistical analysis of data.

Statistical significance: refers to a result showing that results like those observed have a small probability of occurring by chance alone. By convention, results are usually regarded as statistically significant if the probability is less than 0.05 (1 in 20). If a result is statistically significant we feel fairly confident that something other than chance is needed to explain it but we may not know what the 'something' is. Nor does it necessarily follow that the result is of any theoretical or practical importance.

Statistical table: a table of values of a statistic. Often, a statistical table shows the size that a statistic needs to be for the outcome of a statistical test to be significant.

Statistical test: see significance test.

Stem-and-leaf diagram: a method of tabulating data where the first part of each number determines the row to be used and the final digit of the number is written on that row.

Sum of squares (SS): the sum of the squares of the deviations of a set of scores around their mean. When divided by the degrees of freedom ( df ), it gives an estimate of the standard deviation of the population the scores were drawn from.

## T

Test statistic: a statistic whose value determines if the result of a statistical test is significant.

Trend: a systematic relationship between two variables following a straight line or a fairly smooth curve.

t-test: a parametric and distribution-dependent test of the Null Hypothesis that two populations of measurements follow the normal distribution and have the same mean, or that one population has a particular mean. The simplest form of t-test also assumes (if there are two populations) that they have the same standard deviation.

Two-sample t-test: a t-test for comparing two groups of unmatched (independent) measurements.

Two-sided test, or Two-tailed test: see non-directional test.

Type-1 error: concluding that the Null Hypothesis is false when in fact it is true.

Type-2 error: failing to conclude that the Null Hypothesis is false when in fact it is false. (Note that we never conclude that the Null Hypothesis is true.)

## U

U statistic: statistic calculated when performing a Mann-Whitney test.

## V

Variable: in an investigation, it is some feature that is capable of varying, whether it does vary or not.  See also Dependent variable and Independent variable.

## W

Wilcoxon test: a non-parametric and distribution-free significance test based on the ranks of the differences between two sets of related measurements. It is used with measurements that occur in pairs (two scores from each participant, for instance).

## Y

Yates's correction for continuity: a modification to the calculation of the chi-squared statistic calculated from frequencies. It is appropriate only when there is just one degree of freedom. It adjusts for the fact that frequencies must be whole numbers whereas the chi-squared distribution is a continuous mathematical function, but it usually makes too great an adjustment.

## Z

z-score: see standard score.