PSY 429 Experimental Psychology

Statistical Notation and Statistical Formula

X	a particular score
S X group 1	the sum of scores in a specific group (e.g., group 1)
SS X	the sum of scores for all groups in the experiment
(S X group 1) 2	the sum of a specific group's scores, squared (e.g. group 1)
S X 2 group 1	the sum of the square of each score, in a specific group, (e.g. group 1)
SS X 2	the sum of the square of each score in the experiment
n group 1	the number of scores in a specific group (e.g. group 1)
N	the total number of scores in the experiment
m	mean of the population
M	mean of the sample
s	standard deviation of the population
s	standard deviation of the sample
s2	variance of the population
s2	variance of the sample

Statistical Formula that is used in PSY 429

Mean, Variance and Standard Deviation

M = S X group 1 /n	Mean of a specific group (e.g. group 1)
s2 = S (X - M)2 /n -1	Variance of a sample [Definitional formula]
s2 = [S X 2group 1 - (S X)2 /n ] /n -1	Variance of a sample [Computational formula]
s = square root of s2	Standard deviation of sample

Pearson r Correlation Coefficient Between variable X and variable Y

r = N S XY - (S X)(S Y) /the square root of [N S X 2 - (S X)2 ][N S Y 2 - (S Y)2 ] variable X and variable Y = two variables, X and Y, which reflect the two variables in a correlation; in the correlation equation, X reflects a specific score in variable X and Y reflects a specific score in variable Y.   Testing the significance of r (if you don't have Table 1 handy listing the critical values of r): Two different procedures are used to test the hypothesis that r = 0 (null hypothesis). If N (number of paired scores) is 30 or larger, a critical-ratio z test can easily be done: z = r / square root of N -1 If z is greater than d 1.96, then r is significant at the .05 level of significance using a two-tailed test. If N is smaller than 30, compute t: t = r(the square root of [(N - 2) / 1 - r2 ] ) Look up the critical value of t in Table 3. An observed t value GREATER than the t value in the table means that the correlation coefficient is significant.

Testing Hypotheses for a Single Sample

s M = s / square root of n	Standard error of the mean when population standard deviation is known
z = M - m /s M	z score when population mean and standard deviation are known
s M = s / square root of n	Estimated standard error of the mean when population standard deviation is NOT known
t = M - m /s M	t statistic: to use when testing hypotheses when population standard deviation (s) is NOT known

Testing Hypotheses for Two Independent Samples When Parametric Assumptions are Met

t =

Testing Hypotheses for Two Related Samples When Parametric Assumptions are Met

t =

Calculation of d'

d' = z noise - z signal + noise

In psychophysics, d' is a measure of the ability to detect the signal.

Calculation of F max

F max = maximum s2 /minimum s2

F max is the statistic to use to determine whether the variances of the groups are equal or homogeneous. If the observed (or calculated) F max is larger than that found in the F max table (Table 4), then you must conclude that the variances of your groups are NOT homogeneous. You must either transform your scores so that the differences among the variances are as small as possible (i.e., homogeneous), or use a non-parametric test.

 Calculation of Tukey's HSD Test

HSD = q(square root of [MSerror /n])

Tukey's HSD test is a test to determine whether significant differences occur between any two groups in a multi-group analysis of variance. If the observed HSD is GREATER than the difference between the means of any two groups, you conclude that a significant difference between these two groups has been found. To find q, see Table 6.

 Calculation of eta-squared

h2 = SStreatment /SStreatment + SSerror

Eta-squared is calculated to determine the proportion of variance accounted for by the independent variable. It is used to determine the degree or amount of POWER, or EFFECT Size, of your experiment. See Table 7a & 7b to determine the power.

One-Way Independent-Measures (Between-Subject) Analysis of Variance

(used when your data meet the parametric assumptions of the test, you have only one independent variable, and each participant has only one score)

SStotal = SS X 2 - (SS X)2 /N

SStreatment = (S Xgroup 1 )2 /ngroup 1 + (S Xgroup 2 )2 /ngroup 2 + (S Xgroup 3 )2 /ngroup 3+ ......... to k groups - (SS X)2 /N

SSerror = SStotal - SStreatment

MStreatment = SStreatment/dftreatment

MSerror = SSerror/dferror

dftotal = N - 1

dftreatment = k- 1

dferror = dftotal - dftreatment

F = MStreatment /MSerror

The F ratio tells you whether there are significant differences among your groups. If your observed F ratio is larger than the one in the F table, at a particular level of significance, you reject the null hypothesis (or fail to reject the experimental hypothesis) and conclude sigificant differences were found.

Two-Way Independent-Measures (Between-Subject) Analysis of Variance

(used when your data meet the parametric assumptions of the test, you have two independent variables, and each participant has only one score)

SStotal = SS X 2 - (SSX)2 /N

SSfactor A = (S Xgroup A1 )2 /ngroup A1 + (S Xgroup A2 )2 /ngroup A2+ ......... to k groups for Factor A - (SS X)2 /N

SSfactor B = (S Xgroup B1 )2 /ngroup B1 + (S Xgroup B2 )2 /ngroup B2 + ......... to k groups for Factor B - (SS X)2 /N

SSAxB = (S Xgroup A1 B1 )2 /ngroup A1B1 + (S Xgroup A1B2 )2 /ngroup A1B2 +(S Xgroup A2B1 )2 /ngroup A2B1 + (S Xgroup A2B2 )2 /ngroup A2B2 +......... (to k groups) - (SS X)2 /N

SSerror = SStotal - SSFactor A - SSFactor B - SSAxB

dftotal = N - 1

dfFactor A = kA - 1

dfFactor B = kB - 1

dfAxB =(dfFactor A)(dfFactor B )

dferror = dftotal - dfFactor A - dfFactor B - dfAxB

MSFactor A= SSFactor A/dfFactor A

MSFactor B= SSFactor B/dfFactor B

MSAxB= SSAxB/dfAxB

MSerror = SSerror/dferror

F Factor A = MSFactor A /MSerror	The F ratio for Factor A reflects the main effect for Factor A (IV1). If the observed F ratio is GREATER than the F ratio in the F table, you conclude that IV1 had a significant effect on the dependent variable (i.e., there were significant differences among the groups of Factor A).
F Factor B = MSFactor B /MSerror	The F ratio for Factor B reflects the main effect for Factor B (IV1). If the observed F ratio is GREATER than the F ratio in the F table, you conclude that IV1 had a significant effect on the dependent variable (i.e., there were significant differences among the groups of Factor B).
F AxB = MSAxB /MSerror	The F ratio for AxB reflects the interaction for the two independent variables. If the observed F ratio is GREATER than the F ratio in the F table, you conclude that the effect of IV1 on the dependent variable depended on the scores on IV2 or vice versa.