Statistics Formulas

Variance for a Population
The mean of the squared deviation scores

Where:
σ² = variance for a population
μ = mean for a population
X = each individual score
N = total scores for the population

This is known as the deviation score formula.

Standard Deviation for Populations
(deviation scores formula)

Where:
σ = standard deviation for a population
μ = mean for a population
X = each individual score
N = total scores for the population

Standard Deviation for Samples
(raw scores formula)

Where:
s = standard deviation for a sample
X = is the raw scores
N = is the population

To change this to the version for populations, change N-1 to N. The s would also become the greek letter σ.

z Score
(for samples)

Where:
X = is the raw score
Xbar = is the mean
s = is the standard deviation

Correlation

r =

Where:
n = is the number of people in the study
X = is the 1st set of scores
Y = is the 2nd set of scores

t-test for Independent Samples

Where:
Xbar₁ is the mean for Group 1
Xbar₂ is the mean for Group 2
n₁ is the number of participants in Group 1
n₂ is the number of participants in Group 2
s²₁ is the variance for Group 1
s²₂ is the variance for Group 2

t-test for Dependent Samples

Where:
∑D is the sum of all the differences between groups
∑D² is the sum of the differences squared between groups
n is the number of pairs of observations

Effect Size

Where:
ES is effect size
Xbar₁ is the mean of Group 1
Xbar₂ is the mean of Group 2
σ²₁ is the variance for Group 1
σ²₂ is the variance for Group 2

Slope

Intercept

Y Prime

Y^´ = a + bX

Standard Error

Where:
Se = is the standard error of estimate
Sy = is the standard deviation of the y scores
r² = is the correlation of the x and y scores

Note: 1 - r² is the formula for coefficient of alienation.

Alternate version:
S_xy = √ ( ( ∑(Y - Y´)² ) / ( n - 2 ) )
but since you need Y´ (y-prime), you might as well just use the correlation value (r) since you already have it.

Chi Square
A comparison between what is observed and what is expected by chance.

Where:
x² = chi square
o = observed frequency
e = expected frequency

Expected frequency for each cell is--

row_total * column_total / grand_total

df = (columns - 1) * (rows - 1)
(if there's only one rwo leave out the columns side of the equation.)

Anova (F Ratio)

The point of the ANOVA is to determine if scores in three (3) or more groups differ significantly. After arriving at the F value (named after R. A. Fisher), it is compared to a two dimentional table consisting of number of groups and number of participants. If the F value is greater than the minimum significant value, the numbers are in fact, different significantly.

Here's a real world example that Dr. Basham did at a Washington medical center that shows how an ANOVA might be used in a management report-- example

Calculate the Sum of Squares Total (SS_T) using all the scores from all samples as if they were from one group.
SS_T = ∑X_T² - ((∑X_T)²) / N)
Calculate the Sum of Squares for each sample.
SS₁ = ∑X₁² - (∑X₁)² / N₁
SS₂ = ∑X₂² - (∑X₂)² / N₂
...
Add up the samples Sum of Squares to get the Sum of Squares Within.
SS_W = SS₁ + SS₂ + SS₃ ...
Subtract the Sum of Squares Within from the Sum of Squares Total to get the Sum of Squares Between.
SS_B = SS_T - SS_W
Divide the sum of Squares Between by it degress of freedom (K-1) to get Mean Squares Between.
MS_B = SS_B / K - 1
Divide the Sum of Squares Within by its degress of fredom (N-K) to get Mean Squares Within.
MS_W = SS_W / N - K
Divide Means Squares Between by Mean Squares Within to get F ratio.
F = MS_B / MS_W
df(K-1, N-K)
Look up the critical value of the F ratio using the appropriate degress of freedom (K-1, N-K). If your F ratio exceeds the C.V., reject the null hypothesis. If it does not; retain the null hypothesis.