Statistical Symbols and Formulas
Taken from: Symbols and Formulas.doc
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Statistical Symbols and Formulas
Statistical Symbols
Mean = M
Beta weight or beta coefficient =
Chi square = 2
Coefficient of variation = V
Degrees of freedom = d
Frequency =
Gamma (also G) =
Kendall’s tau-b = b
Lambda ( also called the Guttman coefficient of predictability) =
Level of significance =
Pearson’s correlation coefficient = r
Population mean score =
Population size = N
Population variance = 2
Rho =
Sample size = n
Sample variance = s2
Standard deviation or standard error of sample = s
Standard Deviation = SD
Standard Error of the mean = SE
The summation sign =
t distribution for rejecting or accepting the null hypothesis = t
Z Score = z
Standardized scores,
and Z distribution = Z
Correlation = r
Point Biserial Correlation = rpb
Statistical significance = p
Statistical significance = Sig.
The 95% Confidence Interval = 95%CI
T-test = t
One Way ANOVA or Comparison Test of Between Group Differences = F
Correlation Coefficient for ANOVA (3+ categories) = eta
ANOVA test of significance of difference among means
= Tukey-b procedure
Multiple correlation coefficient symbol
( 2 or more I.V.). = R
In SPSS Program
Beta = (Standardized regression slope). Slope based on standard scores = average change in D.V. in S.D. units, associated with 1 S.D. increase in I.V. (1 point increase in I.V. = 1 S.D.)
B = (Unstandardized regression slope). Slope based on raw scores = Indicates average change in D.V. for 1 point increase in I.V.
= Beta weight or beta coefficient
R2change = tells whether or not variables entered at that point add anything over and above variables that have been added previously.
Variance =
Covariance (of x and y) =
Statistical Formulas
Measures of Central Tendency
(Simple Distribution)
Mean = =
N
Population Mean =
(Frequency Distribution)
Mean = = f
N
Mdn = simple distribution = center score
odd population
Mdn = simple distribution =
Even population
2 center scores divided by 2
Mdn = frequency distribution =
LRL – (PN-CFL .h)
FI
Mode = Most Frequent Score
Variance Measures in a Population
(Simple Distribution and Frequency Distribution)
Population Standard Deviation
___
= 2 , or
(Simple Distribution)
Population Variance
2 = 1( X2 – ( X )2
N – 1
(Frequency Distribution)
Population Variance
2 = 1 ( fX2 – ( fX )2
N N – 1
Variance =
Sum of squared distributions from the mean for all cases
(number of cases –1)
or,
(X-M)2
N – 1
Note:
col:[XMX-M (X-M)2] (X-M)2
N – 1
or,
Variance Measures in a Sample
(Simple Distribution and Frequency Distribution)
Population Standard Deviation
___
s = s2
(Simple Distribution)
Sample Variance
s2 = 1( x2 – ( x )2
n – 1
(Frequency Distribution)
Sample Variance
s2 = 1 ( fx2 – ( fx )2
n n – 1
Statistical Average Formula:
M + 1 = Mean + 1 Standard Deviation
Raw Score Transformation Formula:
Z Standard Scores
scores – mean = X - M
standard deviation
Probability Density Formula:
M + 1SD = 68%
M + 1.96SD = 95%
Coefficient of Variation Formula:
Coefficient of Variation = SD x 100
Mean
Standard Error of The Mean Formula:
SEM = SD
N – 1
Confidence Interval at 95% Formula:
M + (1.96)(SEM)
Calculating Percent Formula:
3 x X = 3x100 = 300/6 = x = 50
6100
Z Statistic Formula:
Z = M -
m
m = SD
N –1
t-test Statistic Formula:
t = M1 – M2
SE diff
Confidence Interval for t-test Formula:
CI = M1 – M2+ (?) (SE diff)
Point Biserial Correlation Coefficient:
______
rpb = _____t2______
t2 + ( n1 + n2 – 2)
Note: Used for t test for independent groups
EtaCorrelation Coefficient Formula:
(for ANOVA 3+ Categories)
eta2 = SS Between / SS Total, or
between = eta2
total
____
eta = eta2
D Index Calculation Formula:
(Measure of effect size)
Difference between 2 group means
Avg. SD of the 2 groups
Note: Used in Meta Analysis
(or SD of Control group)
Note: less accurate
Formula for Calculating Covariance of two variables (x and y):
Covariance =
Therefore,
The coefficient of correlation of X an Y is then stated as:
Then to get a better measure of correlation calculate:
Formula for Calculating a 2x2 Covariance Matrix:
Formula for Calculating the Eigenvalues of the Covariance Matrix:
Formula for Characterizing a Straight Line:
y = a + Bx
y = Predicted value of Dependent Variable (D.V.)
a = Intercept value of D.V. when the Independent Variable (I.V.) = 0
B = Slope = Average change of D.V. associated with a 1 point increase in the I.V.
x = Value of I. V.
Residual = for a particular person their actual value minus their predicted value.
Formula for Calculating a Residual:
Residual = D.V. – y
Residual = actual value of D.V. – predicted value.
Conversion Formula for Standard Scores: Bivariate
Value of I.V. _ -Mean of I.V. or, z = x - x S.D. of I.V. S.D. x
Note: Does not change shape of distribution.
Formula for a Bivariate Normal Distribution:
Formula for Calculating the Slope of a Regression Line:
Multivariate Interaction EffectsFormula:
Computation formula for data points
Y=a+X1B1 + X2B2 + X12B12
Example:
Yd=a+Xagbag + Xincbinc + X(ag)(inc)b(ag)(inc)
Constant = a
Unit of associated change =1, or 0 =X
Predicted value of 1st slope = bag
Predicted value of 2nd slope = binc
Predicted value of score = Y
Spearman Correlation Formula:
rs = 1- 6D2
n(n2 – 1)
Pearson Correlation Formula:
r = _SP_
SSxSSy
__ __
where SP = (X – X)(Y – Y) =
XY - (X) (Y)
n
Partial Correlation Ratio Formula:
r2 = a
a+d
Part Correlation RatioFormula:
r2 = a
a+b+c+d
Multivariate ANCOVA (Computation formula for adjusted means):
= Y=a+X1(pre)B(pre) + X2(group)B(group)
Intercept Constant = a
Descriptive mean of pretest condition = X1
Unit of associated change =Experimental group =1, or Control group =0 =X2
Predicted value of 1st slope = B(pre)
In Bivariate regression only: ( see SPSS symbol definitions below).
Beta = r B, or
Beta = r
Note: r B
Predicted value of 2nd slope = B (post)
Predicted value of score = Y adjusted experimental mean =
Y adjusted control mean =
Logistic Regression Odds Ratio Formula:
100 (OR – 1) = % Change
(i.e. 100 (.5 – 1) = 50% decrease in odds of participation)
Chi-Square Statistic Formula:
2 = ( o - e)2
e
To be continued …
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