ComSt 311 Redmond
Statistics Reference Sheet
Independent Variable: A variable/quality that does not depend on other variables. Sometimes
manipulated, but often simply measured as a state condition (age, sex, education, etc.).
Dependent Variable: A variable (quality) that is affected and changed by independent variables.
In the statement, "women are more empathic than men," sex is the independent variable and empathy is the dependent
variable. Another way of considering the relationship between two variables is to say that by knowing one variable (a
person's sex) we can predict another variable (level of empathy).
Significance p < 0.1
p < .05
p <.01
p < .001
Chance of the statistic happening purely by chance: 0.1 means 10 out of 100 times the result is by
chance, .05 (5 out of 100), .01 (1 out of 100), .001 (1 out of 1,000)
Frequency (number) N = 245
Means (average)
(usually appears to show the number of respondents)
M = 3.45 (on some scale, such as 1-5)
Standard Deviation SD = 2.18 (relative to the mean and scale).
(SD is an indication of how representative the mean is of the sample, the larger the SD relative to
the mean, the less representative it is—also usually, the flatter the bell curve. Just over 68% of
the sample falls within one standard deviation (+/-) of the mean. Thus in the example above, 68%
fall between 1.27 (3.45-2.18) and 5.63 (3.45 + 2.18) which is above the 5.0 scale (thus also
indicating a skewed sample).
Variance: The degree to which a set of numbers are spread out--their distribution. Thus for a sample of
five people responding on a five point scale, if they all answered 3, there is no variance (SD = 0),
if each chose a different number, there is complete variance (SD = 1.6). Notice that for both
groups the means would be 3 and thus for the first group, it would be a very accurate predictor of a
person’s score. For the second group, it would only accurately predict 1 out of the 5 and not be a
very good predictor. Standard deviation reflects this.1
Tests of Differences
T-Test—Tests the difference between means of two groups. Are the two groups under comparison
significantly different on the measured quality?
Sample:
t = (153) = -1.83, p < .05
t (degree of freedom # of participants-1) = value, p < value
This means that in a sample of 154 respondents, a t value of –1.83 was obtained in calculating the
difference in two means, and that that value would occur by chance only 5% of the time.
Analysis of Variance (ANOVA)—Similar to t-test but instead of just two groups, it tests the difference
between three or more groups by examining the variance of each group to a grand mean (between group)
and within each group (within group) on one independent variable (factor)
F (df of categories, df of participants) = value, p < value
(sometimes eta squared 2 is provided to further support the strength—interpreted like r2—as
percent of variation accounted for)
Sample: F (2, 154) = 3.47, p <.05,
2
= .38 (38% of the variance accounted for)
Multiple Factor Analysis of Variance (MANOVA)—the same as ANOVA except that it can handle
more than one independent variable (factor) and determine the interaction effects among those
factors.
Shows as F tests among the various configurations.
1
ComSt 311 Redmond
Interaction Effects
Sex x Age x Education on Affectionate Communication
Sex x Age on Affectionate Communication
Sex x Education on Affectionate Communication
Age x Education Affectionate Communication
F (3, 65) = …
F (2, 65) =
F (2, 65) =
F (2, 65) =
Main Effects (the effect of each independent variable by itself, controlling for the effects of the other
independent variables)
Sex on Affectionate Communication
Age on Affectionate Communication
Education on Affectionate Communication
F (1, 65) =
F (5, 65) =
F (7, 65) =
Chi Square (χ2) Also compares groups when the frequency of responses between nominal/categorical
data (males/females, freshmen/sophomores/juniors/seniors, ComSt majors/nonmajors, etc.). Chi
square simply involves comparing how two groups rated or ranked given options to see if there are
different. The underlying assumption is that the distributions of any responses will be the same.
Chi square can also be used to compare against “expected” distributions of responses such as
equally distributed across the choices.
Example: (reading 3). Number indicting a given quality is part of “What is a date?”
Quality
% of College Students
% of Single Adults (older)
A couple/dyad
72.4%
60.4%
Heterosexual
22%
29.2%
Social
46.8%
29.3%
Chi square is used to answer the question about whether the choices of these two groups are
significantly different. The actual study has more qualities.
Finding: χ2 (25, n= 1,149) = 71.45, p < .001). The 25 is the number of qualities and the 1,149 is
the number of total responses).
Tests of Relationships
Correlation (Pearson’s Product-Moment Correlation)—tests whether two variables vary together
either positively or negatively and is a number between 1.0 and -1.0
Assesses the amount of variation that is common to both variables—in essence, how often does a
response on one quality match up with a quality on another quality.
The square of the correlation coefficient indicates how much variance is accounted for between
the two variables (called the coefficient of determination).
r = .40 produces . r2 = .16, meaning 16% of the variance is accounted for (84% is due to other
factors)
r = .70 produces r2 = .49, meaning 49% of the variance is accounted for (51% by other factors)
r = -.20 becomes r2 = .04, meaning only 4% is accounted for.
A correlation can be statistically significant but account for very little variance between the two
variables. Just as SD tells how well a mean reflects the general scores, a correlation reflects how
well the sloped line reflects the relationship between where two pairs of numbers are plotted in
space. In the example below, the distance of each point away from the line reflects the amount of
variance that is not accounted for. For r = .95, the total distances of all the points away from the
line is small, while for r = .34, a lot more distance (variation) from the line. In essence, the ability
to know what the value of one variable (X) based on knowing the value of another variable (Y) is
not very good in the second correlation. A smaller r2 means that a lot of other factors are affecting
variation in the Y scores. BUT, REMEMBER CORRELATION IS NOT CAUSE AND EFFECT.
ComSt 311 Redmond
5
4.5
4
3.5
r = .95
r = .34
3
2.5
Multiple
Correlation—a correlation calculated between three or more variables; a large R = +1.0 to –1.0
2
70 Correlation—a
90
110
130 between
150 two 170
Partial
correlation
variables (X & Y) in which the effect of one or more
other variables (Y) is removed from between the two variables (X & Y).
Factor Analysis—A method for determining which set of variables are most closely related to one
another, and which ones aren’t. Typically a set of items such as a questionnaire are analyzed to
see which items inter-correlate (multiple correlation) the best. Those items that relate well to one
another constitute a “factor.” Factor lists include the items and their correlation to the factor (the
other items in the factor). Researchers label the factors, and you should carefully examine the
items and the labels to insure you understand what the label represents.
Regression Analysis Regression analysis is a statistical way of demonstrating prediction based only on a
linear model creating a slope much like correlations.
Attempts to answer the question: “What is the best predictor of…?”. Which set of known
independent variables best predicts an dependent variable and with what weighting?
Similar to ANOVA (F-test) in that it attempts to account for the most variance possible by
creating an equation based upon possible combinations and weightings of independent variables.
F-test is used to determine if the regression analysis is statistically significant.
Regression equation format: values of Y = a constant value + some slope + error
Y = a + b*X + e
Y= dependent variable
a = some constant (called an intercept—where the line is intercepted)
b = slope/beta weight the value by which the known value is multiplied (shows relative
contributions when more than one independent variable is included)
X = the value of the independent variable.
e = error
Interpreting Regression Analysis:
Usually, the formula is not reported, but simply the beta weights, R2 change values, F test values.
Beta weights “β” are listed for each variable to indicate its strength of contribution to the overall
prediction of the dependent variable. Each independent variable is evaluated for making a
significant improvement in the prediction (usually an F test), if it fails to reach significance, it is
not included in the final regression equation.
A correlation of determination R2 is calculated comparing the predicted value of the dependent
variable based on the each variable’s contribution is compared with the actual/observed results
reported by participants. This value is between 1.0 and -1.0.
Example: Reading # 8 Coping strategies predict type of commitment in dating relationships.
High moral commitment used reframing, β = .15, R2 change .02, modeling, β = .14, R2
change .01, and punishing, β = -.16, R2 change .02. (F results and significance are reported for
each as well). Notice the Beta and R2 are fairly small making this marginally meaningful.