Heiman G. Basic Statistics for the Behavioral Sciences

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Chapter 13

1. (a) Analysis of variance.

(b) A study that contains one independent variable.

(d) A condition of the independent variable.

(e) Another name for a level.

(f) All samples are independent.

(g) All samples are related, either through a repeated-

measures or matched-samples design.

3. Because the hypotheses require more than two levels, or

because it’s easy to obtain additional information.

5. (a) It is the probability of making a Type I error

after comparing all possible pairs of means in an

experiment.

(b) Multiple t-tests result in an experiment-wise error

rate larger than alpha, but performing ANOVA and

then post hoc tests keeps the experiment-wise error

rate equal to alpha.

7. (a) When is significant and k is greater than 2. The

indicates only that two or more sample means

differ significantly; post hoc tests determine which

levels differ significantly.

(b) When is not significant or when .

9. The effect size, as the proportion of variance in depend-

ent scores accounted for by changing the levels of the

independent variable.

11. The researcher measured participants on more than one

dependent variable in each condition of the independent

variable.

13. (a)

(b) : not all s are equal.

that one is.

15. (a) The is less than the ; either term is a

poor estimate of and is assumed to be true.

(b) He made a computational error— cannot be a

negative number.

17. (a) This is a pretest–posttest design, comparing the

weights of one group before and after they dieted.

(b) A within-subjects design.

19. (a)

Sum of Mean

Source Squares df Square F

Between 134.800 3 44.933 17.117

Within 42.000 16 2.625

Total 176.800 19

(b) With and 16, , so is signifi-

cant,

:, ,

, . Only ages 4 and 10, and

ages 6 and 8 do not differ significantly.

5 5.8X

5 9.40

5 10.8X

5 4.414.052122.625>525 2.93

HSD 5q

5 4.05df

5 16k 5 4

p 6 .05.

obt

crit

5 3.24df 5 3

obt

crror

␮H

: ␮

5 ␮

k 5 2F

obt

448 APPENDIX D / Answers to Odd-Numbered Questions

(d) Because , this relationship

accounts for 76% of the variance, so it’s a very

important relationship.

(e) Label the axis as the factor of age and the axis

as the mean creativity score. Plot the mean score for

each condition, and connect adjacent data points

with straight lines.

21. (a) A Type I error: rejecting a true , thus claiming the

factor influences behavior when really it does not.

(b) It produces scientific “facts” that are wrong.

23. (a) By testing the same participants under all condi-

tions or by testing different participants who are

matched on a variable.

(b) The related-samples t-test and the within-subjects

ANOVA.

for each condition.

(d) The independent-samples t-test and the between-

subjects ANOVA.

25. (a) Both measure effect size, indicating the proportion

of variance accounted for.

(b) The improvement in predicting scores when using

our level means.

27. (a) It is manipulated by the researcher to create the

conditions; it is presumably the cause of a behavior.

(b) They are the different conditions created by the

researcher under which participants are tested.

“caused” by the factor.

29. (a) Levels: statistics grades; dependent variable (DV):

math phobia scores.

(b) Levels: recall vs. recognition. DV: memory score.

(d) Predictor: First exam grade. Criterion: final exam

grade.

(e) Levels: male vs. female. DV: number who voted.

(f) Levels: pilot vs. navigator. DV: reaction time.

Chapter 14

1. To examine the interaction between two independent

variables, or for the efficiency of simultaneously study-

ing multiple factors.

3. A main effect mean is based on scores in a level of one

factor while collapsing across the other factor. A cell

mean is the mean of scores from a particular combina-

tion of a level of factor A with a level of factor B.

5. (a) A confounded comparison involves two cells that

differ along more than one factor. It occurs

with cells that are diagonally positioned in a study’s

diagram.

(b) An unconfounded comparison involves two cells

that differ along only one factor. It occurs with

means within the same column or within the same

row of a diagram.

␩

5 134.8>176.8 5 .76

duced the difference.

7. (a) is that the s represented by the level means

from factor A are all equal; is that not all s are

equal.

(b) is that the s represented by the level means

from factor B are all equal; is that they are not

all equal.

not form an interaction; is that they do form an

interaction.

9. (a) 4 ⫻ 3; (b) 30; (c) 40; (d) 10

11. Study 1: For A, means are 7 and 9; for B, means are

3 and 13. Apparently there are effects for A and B but

not for .

Study 2: For A, means are 7.5 and 7.5; for B, means

are 7.5 and 7.5. There is no effect for A or B but there is

an effect for .

Study 3: For A, means are 8 and 8; for B, means are

11 and 5. There is no effect for A, but there are effects

for B and .

13. Perform Tukey’s post hoc comparisons on each main

effect and the interaction, graph each main effect and

interaction and compute its ; where appropriate, com-

pute confidence intervals for the represented by a cell

or level mean.

15. Only the main effect for difficulty level is significant.

17. (a) For low reward ; for medium ; and for

high . It appears that as reward increases,

performance increases.

(b) For low practice ; for medium ; for

high . It appears that increasing practice

increases performance.

depends on the level of practice, and vice versa.

(d) By comparing the three means within each column

and the three means within each row.

19. (a)

Sum of Mean

Source Squares df Square F

Factor A 7.20 1 7.20 1.19

Factor B 115.20 1 115.20 19.04

Interaction 105.80 1 105.80 17.49

Within groups 96.80 16 6.05

Total 325.00

For each factor, and 16, so : fac-

tor B and the interaction are significant,

(b) For factor A, , ; for factor B,

, ; for the interaction,

contains only two levels, such tests are unnecessary.

5 5.4.X

5 14.8,X

5 8.8, 9.0

5 7.1X

5 11.9

5 10.1X

5 8.9

p 6 .05.

crit

5 4.49df 5 1

5 15

5 8X 5 7

5 12

5 10X 5 8

␮

␩

A 3 B

␮H

For A ⫻ B, adjusted , so , HSD ⫽

; the only significant dif-

ferences are between males and females tested by a

male, and between females tested by a male and

females tested by a female.

(d) Conclude that a relationship exists between gender

and test scores when testing is done by a male, and

that male versus female experimenters produce a

relationship when testing females, .

(e) For B, ; for ,

21.

Sum of Mean

Source Squares df Square F

A: Fantasy 42.467 2 21.233 13.134

B: Meditation .833 1 .833 .515

A ⫻ B: Interaction 141.267 2 70.633 43.691

Within 38.800 24 1.617

Total 223.367 29

The main effect of fantasy and the interaction are signifi-

cant. For the main effect of fantasy, ,

; ; only

low versus high do not differ significantly

. For the interaction, with

meditation the cell means for the low-, medium-, and

high-fantasy levels are 4, 6.4, and 9.4; with no medita-

tion the means are 9.8, 3.2, and 5.8;

; all unconfounded compar-

isons are significant .)

Conclude that: with meditation each increase in fantasy

increases relaxation; without meditation, low fantasy

produces the highest relaxation, which drops drastically

with medium fantasy and then increases slightly with

high fantasy.

23. That the relationship in the sample data was not a

chance pattern; that it reflects a relationship in the pop-

ulation (in nature).

25. Are there related samples (because of matching or

repeated measures) or are there independent samples?

27. (a) The researcher selects one group of participants and

measures their scores on and

(b) Are both normally distributed, interval/ratio vari-

ables?

(d) Spearman .

29. (a) The researcher manipulates two variables, for each

creating at least two or more groups that differ on a

variable or are tested in different situations.

(b) Is it a normally distributed, interval or ratio variable

with homogeneous variance.

design ANOVA.

Y.X

1␩

5 14.267>223.367 5 .63

4.17111.617>5

25 2.37

HSD 5

1␩

5 42.467>223.367 5 192

HSD 5 3.53111.617>10

25 1.419X

5 7.6

5 4.8,X

5 6.9

105.8>325 5 .33.

␩

5A 3 B␩

5 115.2>325 5 .35

p 6 .05

13.652116.05>5

5 4.02

5 3.65k 5 3

APPENDIX D / Answers to Odd-Numbered Questions 449

ANSWERS TO ODD-NUMBERED QUESTIONS

Chapter 15

1. Both types of procedures test whether, due to sampling

error, the data poorly represent the absence of the pre-

dicted relationship in the population.

3. (a) Nominal or ordinal scores.

(b) They form nonnormal distributions, or they do not

have homogeneous variance, so they are trans-

formed to ranks.

5. (a) When categorizing participants on one variable.

(b) When categorizing participants on two variables.

7. (a) That the sample frequencies are unlikely to repre-

sent the frequencies in the population described

by .

(b) That category membership on one variable depends

on or is correlated with category membership on

the other variable.

9. (a) It is the correlation coefficient between the two

variables in a significant chi square design.

(b) indicates the improvement in predicting partici-

pants’ category membership on one variable by

knowing their category membership on the other

variable.

variables in a significant two-way chi square that is

not a design.

(d) indicates the improvement in predicting partici-

pants’ category membership on one variable by

knowing their category membership on the other

variable.

11. (a) The one-way .

(b) : The elderly population is 30% Republican,

55% Democrat, and 15% other; : Affiliations in

the elderly population are not distributed this way.

crats, ; and for others,

(d)

(e) For , so the results are signifi-

cant: Party membership in the population of senior

citizens is different from party membership in the

general population, and it is distributed as in our

samples, .

13. (a) The frequency with which students dislike each

professor also must be included.

(b) She can perform a separate one-way on the data

for each professor to test for a difference between

the frequency for “like” and “dislike,” or she can

perform a two-way to determine if whether stu-

dents like or dislike one professor is correlated with

whether they like or dislike the other professor.

␹

p 6 .05

df 5 2, ␹

crit

5 5.99

␹

obt

5 4.80 1 1.47 1 .60 5 6.87

1.152110025 15

5 1.552110025 55

5 1.302110025 30

␹

2 3 2

␾

2 3 2

450 APPENDIX D / Answers to Odd-Numbered Questions

15. (a) The two-way .

(b) : Gender and political party affiliation are inde-

pendent in the population; : Gender and political

party affiliation are dependent in the population.

Democrat , and other

. For females, Republi-

can , Democrat

and other

(d)

(e) With , so the results are signifi-

cant: In the population, frequency of political party

affiliation depends on gender, .

(f) , indicating a

somewhat consistent relationship.

17.

19. (a) Yes. Because for the Mann–Whitney test,

and ; therefore, . . The

two groups of ranks differ significantly, as do the

original maturity scores, .

(b) Return to the raw scores: For students who have not

taken statistics, , so you would expect

to be around 41.67. For statistics students,

, so you would expect their to be

around 69.67.

21. (a) She should use the Kruskal–Wallis H test, because

this is a nonparametric one-way, between-subjects

design.

(b) She should assign ranks to the 15 scores, assigning

a 1 to the lowest score in the study, a 2 to the next

lowest, and so on.

each possible pair of conditions, she should rerank

the scores and perform the rank sums test.

(d) She will determine which types of patients have

significantly different improvement ratings.

23. (a) The design involved a within-subjects factor with

two conditions.

(b) The raw scores were ordinal scores.

higher or lower than those in the other group.

25. For the independent variable, the number of conditions

we have, the number of independent variables we have,

and whether independent or related samples were

tested. For the dependent variable, whether it requires a

parametric or nonparametric procedure.

␮X

5 69.67

␮X

5 41.67

p 6 .05

crit

5 5U

obt

5 4U

5 4

5 32

␹

12, N 5 4025 13.31, p 6 .05

C 5 114.14>1155 1 14.14 2

5 .29

p 6 .05

df 5 2, ␹

crit

5 5.99

.13 5 14.14

␹

obt

5 3.33 1 3.83 1 .14 1 3.12 1 3.59 1

155 5 16.52

5 18021322>18021662>155 5 34.06

5 18021572>155 5 29.42

5 17521322>155 5 15.48

5 17521662>155 5 31.94

5 17521572>155 5 27.58

␹

Appendix A: Additional Statistical

Formulas

Section A.1: Creating Grouped Frequency Distributions

Score f rel. f cf

53 1 .06 18

52 3 .17 17

51 2 .11 14

50 5 .28 12

49 4 .22 7

48 0 .00 3

47 3 .17 3

Score f rel. f cf

128–131 1 .04 28

124–127 2 .07 27

120–123 6 .21 25

116–119 8 .29 19

112–115 4 .14 11

108–111 3 .11 7

104–107 2 .07 4

100–103 2 .07 2

Section A.2: Performing Linear Interpolation

1. The target z-score is between at .2514 of the

curve and at .2483. With .2500 at .

of the distance between .2514 and .2483, the correspon-

ding z-score is .00452 above .67, at .67452.

3. The of 50 is bracketed by with ,

and with . Because 50 is at .5 of

the distance between 40 and 60, the target is .5 of

the .021 between the brackets, which is 2.0105.

crit

5 2.000df 5 60

crit

5 2.021df 5 40df

0014>.0031z 5 .680

z 5 .670

Section A.3: The One-Way, Within-Subjects Analysis

of Variance

1. (a) It tells you that the researcher tracked participants’

weight gain at different times and, at each, meas-

ured their mood.

(b) On some occasions when participants’ weight

increased, their mood significantly decreased.

3. (a) ; : Not all s are equal.

(b) ; ;

and

Sum of Mean

Source Squares df Square F

Subjects 9.63 7

Factor A 53.08 2 26.54 16.69

A ⫻ Subjects 22.25 14 1.59

Total 84.96 23

The is significant.

(d) The and HSD 1.65. The means for 0,

1, and 2 hours are 2.13, 4.25, and 5.75, respectively.

Significant differences occurred between 0 and 1

hour and between 0 and 2 hours, but not between 1

and 2 hours.

(e) Eta squared .

(f) The variable of amount of practice is important in

determining performance scores, but although 1 or

2 hours of practice significantly improved perform-

ance compared to no practice, 2 hours was not sig-

nificantly better than 1 hour.

1␩

25 53.08>84.96 5 .62

5 3.70

obt

crit

A3subs

5 14df

5 2

subs

5 1205>3 2 392.04

5 445.125 2 392.04SS

tot

5 477 2 392.04

␮H

: ␮

5 ␮

APPENDIX D / Answers to Odd-Numbered Questions 451

ANSWERS TO ODD-NUMBERED QUESTIONS

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453453

GLOSSARY

Alpha The Greek letter , which symbolizes the criterion,

the size of the region of rejection of a sampling distribu-

tion, and the theoretical probability of making a Type I

error

Alternative hypothesis The statistical hypothesis describ-

ing the population parameters that the sample data repre-

sent if the predicted relationship does exist; symbolized

Analysis of variance The parametric procedure for deter-

mining whether significant differences exist in an experi-

ment containing two or more sample means; abbreviated

ANOVA

ANOVA Abbreviation of analysis of variance

As a function of A way to describe a relationship using

the format “changes in Y as a function of changes in X”

Bar graph A graph in which a free-standing vertical bar

is centered over each score on the X axis; used with

nominal or ordinal scores

Beta The Greek letter , which symbolizes the theoretical

probability of making a Type II error

Between-subjects ANOVA The type of ANOVA that

is performed when a study involves between-subjects

factors

Between-subjects factor An independent variable that is

studied using independent samples in all conditions

Biased estimator A formula for a sample’s variability

that involves dividing by N that is biased toward under-

estimating the corresponding population variability

Bimodal distribution A symmetrical frequency polygon

with two distinct humps where there are relatively high-

frequency scores and with center scores that have the

same frequency

Bivariate normal distribution An assumption of the

Pearson correlation coefficient that the Y scores at each

X form a normal distribution and the X scores at each Y

form a normal distribution.

Cell In a two-way ANOVA, the combination of one level

of one factor with one level of the other factor

␤

␣

Central limit theorem A statistical principle that defines

the mean, standard deviation, and shape of a theoretical

sampling distribution

␹

-distribution The sampling distribution of all possible

values of that occur when the samples represent

the distribution of frequencies described by the null

hypothesis

Chi square procedure The nonparametric inferential

procedure for testing whether the frequencies of category

membership in the sample represent the predicted fre-

quencies in the population

Class interval The name for each group of scores in a

grouped frequency distribution

Coefficient of alienation The proportion of variance not

accounted for by a relationship; computed by subtract-

ing the squared correlation coefficient from 1

Coefficient of determination The proportion of variance

accounted for by a relationship; computed by squaring

the correlation coefficient

Cohen’s d A measure of effect size that reflects the

magnitude of the difference between the means of

conditions

Collapsing In a two-way ANOVA, averaging together all

scores from all levels of one factor in order to calculate

the main effect means for the other factor

Complete factorial design A two-way ANOVA design in

which all levels of one factor are combined with all

levels of the other factor

Condition An amount or category of the independent

variable that creates the specific situation under which

subjects’ scores on the dependent variable are measured

Confidence interval for a single A range of values of

, one of which is likely to be represented by the sample

mean

Confidence interval for ␮

A range of values of ␮

, one

of which is likely to be represented by the sample mean

( ) in a related-samples t-testD

␮

␹

454 GLOSSARY

Confidence interval for the difference between two s

A range of differences between two population , one

of which is likely to be represented by the difference

between two sample means

Confounded comparison In a two-way ANOVA, a com-

parison of two cells that differ along more than one factor

Contingency coefficient The statistic that describes the

strength of the relationship in a two-way chi square

when there are more than two categories for either vari-

able; symbolized by C

Continuous scale A measurement scale that allows for

fractional amounts of the variable being measured

Correlation coefficient A number that describes the type

and the strength of the relationship present in a set of data

Correlational study A procedure in which participants’

scores on two variables are measured, without manipula-

tion of either variable, to determine whether they form a

relationship

Criterion probability The probability that defines

whether a sample is too unlikely to have occurred

by chance and thus is unrepresentative of a particular

population

Criterion variable The variable in a relationship whose

unknown scores are predicted through use of the known

scores on the predictor variable

Critical value The value of the sample statistic that

marks the edge of the region of rejection in a sampling

distribution; values that fall beyond it lie in the region

of rejection

Cumulative frequency The frequency of the scores at or

below a particular score; symbolized by cf

Curvilinear relationship See Nonlinear relationship

Data point A dot plotted on a graph to represent a pair of

X and Y scores

Degree of association See Strength of a relationship

Degrees of freedom The number of scores in a sample

that are free to vary, and thus the number that is used

to calculate an estimate of the population variability;

symbolized by df

Dependent events Events for which the probability of

one is influenced by the occurrence of the other

Dependent variable In an experiment, the variable that

is measured under each condition of the independent

variable

Descriptive statistics Procedures for organizing and sum-

marizing data so that the important characteristics can be

described and communicated

Design The way in which a study is laid out so as to

demonstrate a relationship

Deviation The distance that separates a score from the

mean and thus indicates how much the score differs from

the mean

Dichotomous variable A discrete variable that has only

two possible amounts or categories

␮s

␮

Discrete scale A measurement scale that allows for meas-

urement only in whole amounts

Distribution An organized set of data

Effect size A measure of the amount of influence that

changing the conditions of the independent variable had

on dependent scores; may be computed as the proportion

of variance accounted for

Error variance The inherent variability within a popula-

tion, estimated in ANOVA by the mean square within

groups

Estimated population standard deviation The unbiased

estimate of the population standard deviation, calculated

from sample data using degrees of freedom ( );

symbolized by

Estimated population variance The unbiased estimate of

the population variance, calculated from sample data

using degrees of freedom ( ); symbolized by

Estimated standard error of the mean An estimate of

the standard deviation of the sampling distribution of

means, used in calculating the one-sample t-test; sym-

bolized by

Eta squared The proportion of variance in the dependent

variable that is accounted for by changing the levels of a

factor, and thus a measurement of effect size; symbol-

ized by

Expected frequency In chi square, the frequency expected

in a category if the sample data perfectly represent the

distribution of frequencies in the population described by

the null hypothesis; symbolized by

Experiment A research procedure in which one variable

is actively changed or manipulated, the scores on

another variable are measured, and all other variables

are kept constant, to determine whether a relationship

exists

Experimental hypotheses

Two statements made before a

study is be

gun, describing the predicted relationship that

may or may not be demonstrated by the study

Experiment-wise error rate The probability of making a

Type I error when comparing all means in an e

xperiment

Factor In ANOVA, an independent variable

F-distribution The sampling distribution of all possible

values of F that occur when the null hypothesis is true

and all conditions represent one population

Fisher’s protected t-test The post hoc procedure per-

formed with ANOVA to compare means from a factor in

which all levels do not have equal ns

F-ratio In ANOVA, the ratio of the mean square between

groups to the mean square within groups

Frequency The number of times each score occurs within

a set of data; also called simple frequency; symbolized

by f

Frequency polygon A graph that shows interval or ratio

scores (X axis) and their frequencies (Y axis), using data

points connected by straight lines

␮

␩

N 2 1

GLOSSARY 455

GLOSSARY

Friedman ␹ test The nonparametric version of the one-

way, repeated-measures ANOVA for ranked scores

Goodness-of-fit test A name for the one-way chi square,

because it tests how “good” the “fit” is between the data

and

Grouped distribution A distribution formed by combin-

ing different scores to make small groups whose total

frequencies, relative frequencies, or cumulative frequen-

cies can then be manageably reported

Heteroscedasticity An unequal spread of Y scores around

the regression line (that is, around the values of )

Histogram A graph similar to a bar graph but with adja-

cent bars touching, used to plot the frequency distribution

of a small range of interval or ratio scores

Homogeneity of variance A characteristic of data

describing populations represented by samples in a study

that have the same variance

Homoscedasticity An equal spread of Y scores around the

regression line and around the values of

Incomplete factorial design A two-way ANOVA design

in which not all levels of the two factors are combined

Independent events Events for which the probability of

one is not influenced by the occurrence of the other

Independent samples Samples created by selecting each

participant for one sample, without regard to the partici-

pants selected for any other sample

Independent-samples t-test The t-test used with data

from two independent samples

Independent variable In an experiment, a variable that is

changed or manipulated by the experimenter; a variable

hypothesized to cause a change in the dependent variable

Individual differences Variations in individuals’ traits,

backgrounds, genetic makeup, etc., that influence their

behavior in a given situation and thus the strength of a

relationship

Inferential statistics Procedures for determining whether

sample data represent a particular relationship in the

population

Interval estimation A way to estimate a population

parameter by describing an interval within which the

population parameter is expected to fall

Interval scale A measurement scale in which each score

indicates an actual amount and there is an equal unit of

measurement between consecutive scores, but in which

zero is simply another point on the scale (not zero

amount)

Interval size The number of values spanned by each class

interval in a grouped frequency distribution

Kruskal–Wallis H test The nonparametric version of the

one-way, between-subjects ANOVA for ranked scores

Level In ANOVA, each condition of the factor (independ-

ent variable); also called treatment

Linear regression The procedure for describing the

best-fitting straight line that summarizes a linear

relationship

Linear regression equation The equation that defines the

straight line summarizing a linear relationship by

describing the value of at each X

Linear regression line The straight line that summarizes

the scatterplot of a linear relationship by, on average,

passing through the center of all Y scores

Linear relationship A correlation between the X scores

and Y scores in a set of data in which the Y scores tend to

change in only one direction as the X scores increase,

forming a slanted straight regression line on a scatterplot

Line graph A graph of an experiment when the independ-

ent variable is an interval or ratio variable; plotted by

connecting the data points with straight lines

Main effect In a two-way ANOVA, the effect on the

dependent scores of changing the levels of one factor

while ignoring (collapsing over) the other factor

Main effect mean The overall mean of one level of a

factor while ignoring (collapsing over) the influence of

the other factor

Mann–Whitney U test The nonparametric version of the

independent-samples t-test for ranked scores when n is

less than or equal to 20

Margin of error Expressing the expected error when

estimating a population parameter as plus or minus some

amount

Matched-samples design An experiment in which each

participant in one sample is matched on an extraneous

variable with a participant in the other sample

Mean The score located at the mathematical center of a

distribution

Mean square In ANOVA, an estimated population vari-

ance, symbolized by MS

Mean square between groups In ANOVA, the variability

in scores that occurs between the levels in a factor or the

cells in an interaction

Mean square within groups In ANOVA, the variability

in scores that occurs in the conditions, or cells; also

known as the error term

Measure of central tendency A score that summarizes

the location of a distribution on a variable by indicating

where the center of the distribution tends to be located

Measures of variability Measures that summarize the

extent to which scores in a distribution differ from one

another

Median The score located at the 50th percentile; symbol-

ized by Mdn

Mode The most frequently occurring score in a sample

Multiple correlation coefficient The correlation that

describes the relationship between multiple predictor (X)

variables and one criterion (Y) variable

Multiple regression equation The procedure for simulta-

neously using multiple predictor (X) variables to predict

scores on one criterion (Y) variable

Multivariate statistics Procedures applied to a study that

measures two or more dependent variables

Negative linear relationship A linear relationship in

which the Y scores tend to decrease as the X scores

increase

Negatively skewed distribution A frequency polygon

with low-frequency, extreme low scores but without cor-

responding low-frequency, extreme high ones, so that its

only pronounced tail is in the direction of the lower

scores

Nemenyi’s procedure The post hoc procedure performed

with the Friedman test

Nominal scale A measurement scale in which each score

is used simply for identification and does not indicate an

amount

Nonlinear relationship A relationship in which the Y scores

change their direction of change as the X scores change;

also called a curvilinear relationship

Nonparametric statistics Inferential procedures that do

not require stringent assumptions about the parameters

of the raw score population represented by the sample

data; usually used with scores most appropriately

described by the median or the mode

Nonsignificant Describes results that are considered

likely to result from chance sampling error when the pre-

dicted relationship does not exist; it indicates failure to

reject the null hypothesis

Normal curve The symmetric, bell-shaped curve produced

by graphing a normal distribution

Normal distribution A set of scores in which the middle

score has the highest frequency, and proceeding toward

higher or lower scores the frequencies at first decrease

slightly but then decrease drastically, with the highest

and lowest scores having very low frequency

Null hypothesis The statistical hypothesis describing the

population parameters that the sample data represent

if the predicted relationship does not exist; symbolized

Observed frequency In chi square, the frequency with

which participants fall into a category of a variable;

symbolized by

One-sample t-test The parametric procedure for a one-

sample experiment when the standard deviation of the

raw score population must be estimated

One-tailed test The test used to evaluate a statistical

hypothesis that predicts that scores will only increase or

only decrease

One-way ANOVA The analysis of variance performed

when an experiment has only one independent variable

One-way chi square The chi square procedure for testing

whether the sample frequencies of category membership

on one variable represent the predicted distribution of

frequencies in the population

Ordinal scale A measurement scale in which scores indi-

cate rank order

Outlier A data point that lies outside of the general pat-

tern in a scatterplot; created by an unusual X or Y score

␹

Parameter See Population parameter

Parametric statistics Inferential procedures that require

certain assumptions about the parameters of the raw

score population represented by the sample data; usually

used with scores most appropriately described by the

mean

Participants The individuals who are measured in a

sample; also called subjects

Pearson correlation coefficient The correlation coeffi-

cient that describes the linear relationship between two

interval or ratio variables; symbolized by r

Percent A proportion multiplied times 100

Percentile The percentage of all scores in the sample that

are at or below a particular score

Phi coefficient The statistic that describes the strength

of the relationship in a two-way chi square when there

are only two categories for each variable; symbolized

Point-biserial correlation coefficient The correlation

coefficient that describes the linear relationship between

scores from one continuous interval or ratio variable and

one dichotomous variable; symbolized by

Point estimation A way to estimate a population parame-

ter by describing a point on the variable at which the

population parameter is expected to fall

Pooled variance The weighted average of the sample vari-

ances in a two-sample experiment; symbolized by

Population The infinitely large group of all possible

scores that would be obtained if the behavior of every

individual of interest in a particular situation could be

measured

Population parameter A number that describes a charac-

teristic of a population of scores, symbolized by a letter

from the Greek alphabet; also called a parameter

Population standard deviation The square root of the

population variance, or the square root of the average

squared deviation of scores around the population mean;

symbolized by ␴

Population variance The average squared deviation of

scores around the population mean; symbolized by ␴

Positive linear relationship A linear relationship in

which the Y scores tend to increase as the X scores

increase

Positively skewed distribution A frequency polygon

with low-frequency, extreme high scores but without

corresponding low-frequency, extreme low ones, so that

its only pronounced tail is in the direction of the higher

scores

Post hoc comparisons In ANOVA, statistical procedures

used to compare all possible pairs of sample means in

a significant effect, to determine which means differ

significantly from each other

Power The probability that a statistical test will detect a

true relationship and allow the rejection of a false null

hypothesis

pool

␾

456 GLOSSARY

GLOSSARY 457

GLOSSARY

Predicted Y score In linear regression, the best prediction

of the Y scores at a particular X, based on the linear rela-

tionship summarized by the regression line; symbolized

Predictor variable The variable from which known

scores in a relationship are used to predict unknown

scores on another variable

Probability A mathematical statement indicating the

likelihood that an event will occur when a particular

population is randomly sampled; symbolized by p

Probability distribution The probability of every event

in a population, derived from the relative frequency of

every event in that population

Proportion A decimal number between 0 and 1 that

indicates a fraction of a total

Proportion of the area under the curve The proportion

of the total area beneath the normal curve at certain

scores, which represents the relative frequency of those

scores

Proportion of variance accounted for The proportion of

the error in predicting scores that is eliminated when,

instead of using the mean of Y, we use the relationship

with the X variable to predict Y scores; the proportional

improvement in predicting Y scores thus achieved

Qualitative variable A variable that reflects a quality or