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Terms in this set (190)
data
All empirical research is based on observation and measurement, resulting in numbers, or scores, which are our data.
data point
a dot plotted on a graph to represent a pair of X and Y scores.
percent
A proportion multiplied by 100.
proportion
A decimal between 0 and 1 that indicates a fraction of the total. To transform a score to a proportion, divid the score by the total.
statistical notation
The standardized code for symbolizing the mathematical operations performed in the formulas and the answers obtained.
transformation
a procedure for systematically converting one set of scores into a different set of scores. Transformations make scores easier to work with and make different kinds of scores comparable.
Why do researchers need to learn statistics?
To conduct research and to understand the research of others.
If given no other information, what is the order in which to perform mathematical operations?
PERMDAS!
A researcher measures the IQ scores of a group of college students. What four things will the researcher use statistics for?
Organize, Summarize, Communicate, and Conclude
population
the group of all individuals to which research applies
sample
the subset of the population that is actually measured
variable
anything that, when measured, can produce two or more different scores
quantitative
measuring a quantity or amount
qualitative
measuring a quality or category
relationship
as the scores on one variable change, the scores on the other variable tend to change in a consistent fashion
strength
the extent to which one or close to one value of Y tends to be associated with only one value of X
perfectly strong relationship
one value of Y is associated with on X
weaker relationship
one batch of Y scores is associated with one X, and a different batch of Y scores is associated with a different X.
"given"/independent variable
the X variable, dictates the outcome of the dependent variable
descriptive stats
use to organize, summarize, and describe sample data, and to predict and individual's Y score using the relationship with X
inferential stats
for deciding whether the sample data actually represent the relationship that occurs in the population
statistic
a number that describes a characteristic of a sample of scores, symbolized using a letter from the English alphabet.
paramater
a number that describes a characteristic of a population of scores, symbolized using a letter from the Greek alphabet
design
the way in which the study is laid out
experiment
we manipulate the independent variable and then measure participants' scores on the dependent variable.
condition, treatment, level
a specific amount or category of the independent variable
correlational study
neither variable is actively manipulated. SCores on both variables are simply measured and then the relationship is examined
nominal
numbers name or identify a quality or characteristic; bar graph
ordinal
numbers indicate rank order; bar graph
interval
numbers measure a specific amount, but with no true zero; histograph/polygon
ratio scale
numbers measure a specific amount and 0 indicates truly zero amount; histograph/polygon
continuous variable
measure in fractional amounts
discrete variable
measured only in whole numbers
dichotomous variable
discrete variable that has only two amounts or categories
How are samples used to make conclusions about the population? What are researchers really referring to when they talk about the population?
We assume that the relationship found in a sample reflects the relationship found in the population.All relevant individuals in the world, in nature.
What are the two aspects of a study to consider when deciding on the particular descriptive or inferential statistics that you should employ?
The design of the study and the scale of measurement used.
What is the general purpose of all research, whether experiments or correlational studies?
To discover relationships between variables which may reflect how nature operates.
N
the number of scores in the data
f
Simple frequency distribution: shows the frequency (on Y) of each score (on X)
normal distribution forming a normal curve
extreme high and low scores are relatively infrequent, scores closer to the middle score are more frequent, and the middle score occurs most frequently. The low-frequency, extreme low and extreme high scores are in the tails of a normal distribution.
negatively skewed
low-frequency, extreme low scores, but not low-frequency, extreme high scores.
positively skewed
low-frequency, extreme high scores, but not low-frequency, extreme low scores.
bimodal distribution
two areas showing relatively high-frequency scores.
rectangular distribution
all scores have the same frequency
rel. f
relative frequency is the proportion of time that the score occurred
relative frequency distribution
same as simple frequency distribution except that the Y axis is labeled in increments between 0 and 1.
proportion of total area under the normal curve
occupied by particular scores equals the combined relative frequency of those scores
cf
cumulative frequency of a score: the frequency of all scores at or below the score
percentile
percent of all scores at or below a given score, on the normal curve the percentile of s core is the percent of the area under the curve to the left of the score.
ungrouped distribution
the f, rel. f, cf, or percentile of each individual score is reported
grouped distribution
different scores are grouped together, and the f, rel. f, cf, or percentile for each group is reported.
What is the difference between graphing a relationship as we did in Chapter 2 and graphing a frequency distribution?9
The graph showed the relationship where, as scores on the X variable change, scores on the Y variable change. A frequency distribution shows the relationship where, as X scores change, their frequency (shown on Y) changes.
What is the advantage of computing relative frequency instead of simple frequency? What is the advantage of computing percentile instead of cumulative frequency? 5
Relative frequency may be easier to interpret than simple frequency. Percentile may be easier to interpret than cumulative frequency.
What is the difference between scores in the left-hand tail and scores in the right-hand tail?11
Scores in the left-hand tail are the lowest scores in the distribution; scores in the right-hand tail are the highest.
measures of central tendency
summarize the location of a distribution on a variable, indicating where the center of the distribution tends to be.
mode
most frequently occurring score or scores in a distribution; used primarily to summarize nominal data
median
symbolized by Mdn, is the score at the 50th percentile. It is used primarily with ordinal data and with skewed interval or ratio data.
mean
the average score located at the mathematical center of a distribution. It is used with interval or ratio data that form a symmetrical distribution, such as the normal distribution. The symbol for a sample mean is X with a line over it, and the symbol for a population mean is mue.
X-X(with a line over it)
the amount a score deviates from the mean
sum of the deviations around the mean
equals zero; makes the mean the best score to use when predicting any individual score, because the total error across all such predictions will equal zero.
What two pieces of information about the location of a raw score does a deviation score convey?9
Deviations convey whether a score is above or below the mean and how far teh score is from the mean.
measures of variability
indicate how much the scores differ from each other and how much the distribution is spread out
range
the difference between the highest and the lowest score
variance
used when the mean to describe a normal distribution of interval or ratio scores. It is the average of the squared deviations of scores around the mean. Variance equals the squared standard deviation.
standard deviation
used with the mean to describe a normal distribution of interval/ratio scores. It can be thought of as somewhat like the "average" amount that scores deviate from the mean. For any roughly normal distribution, the standard deviation will equal about one-sixth of the range. It equals the square root of the variance
normal distribution
34% of the scores are between the mean and the score that is a distance of one standard deviation from the mean. Therefore, approximately 68% of the distribution lies between the two scores that are plus and minus one standard deviation from the mean.
standard deviations
when they're relatively small, the scores in the conditions are similar, and so a more consistent relationship is present (stronger)
Why is describing the variability important?1
It is needed for a complete description of the data, indicating how spread out scores are and how accurately the mean summarizes them.
When is the range used as the sole measure of variability?3
With nominal or ordinal scores or with interval/ratio scores that cannot be accurately described by other measures.
relative standing
of a score reflects a systematic evaluation of teh score relative to a sample or population
z-score
indicates a score's relative standing by indicating the distance the score is from the mean when measured in standard deviations
positive z-score
indicates that the raw score is above the mean
negative z-score
indicates that the raw score is below the mean
the larger the absolute value of z
the farther the raw score is from the mean, so the less frequently the z-score and raw score occur
z-distribution
produced by transforming all raw scores in a distribution into z-scores
standard normal curve
a perfect normal z-distribution that is our model of the z-distribution that results from data that are approximately normally distributed, interval or ratio scores.
sampling distribution of means
the frequency distribution of all possible sample means that occur when an infinite number of samples of teh same size N are randomly selected from one raw score population.
central limit theorem
shows that in a sampling distribution of means the distribution will be approx. normal, the mean of the sampling distribution will equal the mean of the underlying raw score population used to create the sampling distribution, and the variability of the sample means is related to the variability of the underlying raw score population.
standard error of the mean
the standard deviation of the sampling distribution of means.
scatterplot
a graph that shows the location of each pair of X-Y scores in the data.
outlier
a data point that lies outside of the general pattern in the scatterplot
regression line
summarizes a relationship by passing through the center of the scatterplot
linear relationship
as the X scores increase, the Y scores tend to change in only one direction
circular/elliptical scatterplots
produce horizontal regression lines indicate no relationship
correlation coefficient
describes the type of relationship (the direction Y scores change) and the strength of the relationship.
Pearson correlation coefficient
used to describe the type and the strength of the linear relationship between two interval and/or ratio variables.
Spearman rank-order correlation coefficient
used to describe the type and strength of the linear relationship between two ordinal variables
restriction of range problem
occurs when the range scores from one or both variables is limited. Then the correlation coefficient underestimates the true strength of the relationship that would be found if the range were not restricted.
linear regression
the procedure for predicting unknown Y scores based on correlated X scores
linear regression line
the best-fitting straight line that summarizes a linear relationship
slope
indicates the rate and direction the Y scores change
Y intercept
which is the value of Y when the line crosses the Y axis and the starting value from which the Y scores change
Y1 (Y prime)
can be predicted from each X
predictor variable
X variable
criterion variable
Y variable
standard error of the estimate
interpreted as the "average error" when using Y prime to predict Y scores
homoscedastic
the spread in the Y scores around all Y prime scores are the same and the Y scores at each X are normally distributed around their corresponding value of Y prime.
proportion of variance accounted for
the proportional improvement in accuracy that is achieved by using the relationship to predict Y scores, compared to using Y with a line over it to predict scores.
coefficient of determination
computed by squaring the correlation coefficient
coefficient of alienation
the proportion of variance not accounted for is the proportion of the prediction error that is not eliminated when y prime is the predicted score instead of the Y with a line over it.
multiple correlation and multiple regression
procedures for describing the relationship when multiple predictor variables are simultaneously used to predict scores on one criterion variable.
probability
indicates the likelihood of an event when random chance is operating
random sampling
selecting a sample so that all elements or individuals in the population have an equal chance of being selected.
independent events
the probability of one event is not influenced by the occurrence of the other
dependent events
the probability of one event is influenced by the occurrence of the other.
sampling with replacement
replacing individuals or events back into the population before selecting again
sampling without replacement
not replacing individuals or events back into the population before selecting again.
representative sample
the individuals and scores in the sample accurately reflect the types of individuals and scores found in the population.
sampling error
results when chance produces a sample statistic that is different from the population parameter that it represents.
region of rejection
the extreme tail or tails of a sampling distribution. Sample means here are unlikely to represent the underlying raw score population.
criterion probability
the probability (.05) that defines samples as unlikely to represent the underlying raw score population.
critical value
the minimum z-score needed for a sample mean to lie in the region of rejection.
parametric inferential procedures
require assumptions about the raw score populations being represented. They are performed when we compute the mean.
nonparametric
do not require stringent assumptions about the populations being represented. They are performed when we compute the median or mode.
alternative hypothesis
the statistical hypothesis that describes the population mues being represented if the predicted relationship exists.
null hypothesis
the statistical hypothesis that describes the population mues being represented if the predicted relationship does not exist.
two-tailed test
used when we do not predict the direction in which the dependent scores will change.
one-tailed test
used when the direction of the relationship is predicted
alpha
the symbol for our criterion probability, which defines the size of teh region of rejection and the probability of a Type I erro.
z-test
the parametric procedure used in a one-sample experiment if the population contains normally distributed interval or ratio scores and the standard deviation of the population is known.
z-obt beyond z-crit
then the corresponding sample mean is unlikely to occur when sampling from the population described by the null. Therefore, we reject the null and accept the alternative. This is a significant result and is evidence of the predicted relationship in the population.
z-obt within z-crit
then the corresponding sample mean is likely to occur when sampling the population described by the null. Therefore, we retain the null. This is a nonsignificant result and is not evidence for or against the predicted relationship.
type 1 error
occurs when a true null is rejected. Its theoretical probability equals alpha. The theoretical probability of avoiding a Type 1 error when retaining the null is 1-alpha.
type 2 error
occurs when a false null is retained. Its theoretical probability is beta. The theoretical probability of avoiding a Type 2 error when rejecting the null is 1-beta.
power
the probability of rejecting a false null, and it equals 1-beta.
null is true; we reject the null due to our data
type 1 error
null is true (the relationship does not exist in nature); we retain the null cause of our data
we avoid type 1 error
null is false (the relationship exists in nature); we retain null
type 2 error
null is false (the relationship exists in nature); we reject the null
avoid type 2 error
significant
p<.05
nonsignificant
p>.05
one-sample t-test
for testing a one sample experiment when the standard deviation of the raw score population is not known
t-distribution
a sampling distribution of all possible values of t when a raw score population is infinitely sampled using a particular N. A t-distribution that more or less forms a perfect normal curve will occur depending on the degrees of freedom of the samples used to create it.
point estimation
a mue is assumed to be at a point on the variable equal to average of X. Because the sample probably contains sampling error, a point estimate is likely to be incorrect.
interval estimation
a mue is assumed to lie within a specified interval. Interval estimation is performed by computing a confidence interval.
confidence interval for a single mue
describes a range of mues, one of which the sample mean is likely to represent. The interval contains the highest and lowest values of mue that are not significantly different form the sample mean.
p
Pearson correlation coefficient in the population
ps
Spearman correlation coefficient in the population
sampling distribution of the Pearson (r)
shows all possible values of r that occur when samples are drawn from a population in which p is zero.
sampling distribution of the Spearman (rs)
shows all possible values of rs that occur when samples are drawn from a population in which ps is zero.
maximize the power of experiments
-creating large differences in scores between the conditions of the independent variable
-minimizing the variability of the scores within each condition
-increasing the N of small samples
maximize the power of a correlation coefficient
-avoiding a restricted range
-minimizing the variability in Y at each X
-increasing the N of small samples
independent samples
participants are randomly selected for each, without regard to who else has been selected, and each participant is in only one condition
independent-samples t-test
two independent samples, normally distributed interval or ratio scores, and homogeneous variance
homogeneity of variance
that the variances in the populations being represented are equal
confidence interval for the difference between two mues
contains a range of differences between two mues, any one of which is likely to be represented by the difference between our two sample means.
related samples
we match each participant in one condition to a participant in the other condition, or when we use repeated measures of one group of participants tested under both conditions
confidence interval for mueD
contains a range of values of mue D, any one of which is likely to be represented by our sample's D with a line over it.
power of a two-sample t-test
larger differences in scores between the conditions, smaller variability of scores within each condition, and larger ns. the related-samples t-test is more powerful than the independent-samples t-test.
effect size
indicates the amount of influence that changing the conditions of the independent variable had on the dependent scores.
Cohen's d
measures effect size as the magnitude of the difference between the conditions.
proportion of variance accounted for
measures effect size as the consistency of scores produced within each condition. The larger the proportion, the more accurately the mean of a condition predicts individual scores in that condition.
ANOVA factor
independent variable
ANOVA level or treatment
condition
ANOVA sum of squares (SS)
sum of squared deviations
ANOVA mean square
variance
ANOVA treatment effect
effect of independent variable
one-way analysis of variance
tests for significant differences between the means from two or more levels of a factor.
between-subjects factor
the conditions involve independent samples.
within-subjects factor
the conditions involve related samples
experiment-wise error rate
the probability that a Type 1 error will occur in an experiment. ANOVA keeps the experiment-wise error rate equal to alpha.
Anova tests...
null hypothesis that all mues being represented are equal; alternative hypothesis that not all mues are equal
mean square within groups
measures the differences among the scores within teh conditions
mean square between groups
measures the differences among the level means
F-obt
computed using the F-ratio, which equals the mean square between groups divided by the mean square within groups
If F-obt is significant w/ more than two levels
perform post hoc comparisons to determine which means differ significantly.
If all ns are not equal,
perform Fisher's protected t-test on each pair of means
if all ns are equal
perform Tukey's HSD test
eta squared (n^2)
describes the effect size - the proportion of variance in dependent scores accounted for by the levels of the independent variable.
two-way, between-subjects ANOVA
two independent variables, and all conditions of both factors contain independent samples.
two-way, within-subjects ANOVA
performed when both factors involve related samples
two-way, mixed design ANOVA
performed when one factor has independent samples and one factor has related samples
complete factorial design
all levels of one factor are combined with all levels of the other factor
cell
each cell is formed by a particular combination of a level from each factor
two-way ANOVA
we compute an F-obt for the main effect of A, for the main effect of B, and for the interaction of A X B
main effect means
for a factor are obtained by collapsing across (combining the scores from) the levels of the other factor. Collapsing across factor B produces the main effect means for factor A. Collapsing across factor A produces the maine effect means for factor B.
significant main effect
indicates differences between the main effect means, indicating a relationship is produced when we manipulate one independent variable by itself.
significant two-way interaction effect
indicates that the cell means differ such that the relationship between one factor and the dependent scores depends on the level of the other factor that is present. When graphed, an interaction effect produces nonparallel lines.
post hoc comparisons
perform on each significant effect having more than two levels to determine which specific means differ significantly.
unconfounded
the means from two cells are unconfounded if the cells differ along only one factor.
confounded
the cells differ along more than one factor.
chi square
used with one or more nominal variables, and the data are the frequencies with which participants fall into each category
one-way chi square
compares teh frequency of category membership along one variable
two-way chi square tests
whetehr category membership for one variable is independent of category membership for the other variable.
phi coefficient
2 X 2 design with a significant two-way chi square
contingency coefficient (C)
if the design is not a 2 X 2.
Mann-Whitney U test
nonparametric, independent-samples t-test
Wilcoxon T test
nonparametric, related-samples t-test for ranks
Kruskal-Wallis H test
nonparametric, one-way, between-subjects ANOVA for ranks
Friedman X^2 test
nonparametric, one-way, within-subjects ANOVA for ranks
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