Common Statistical Formulas

Statistical formulas are used to calculate values related to statistical concepts or analyses.  Here we will discuss common formulas and what they stand for.

Population Mean

The term population mean, which is the average score of the population on a given variable, is represented by:

μ = ( Σ Xi ) / N

The symbol ‘μ’ represents the population mean.  The symbol ‘Σ Xi’ represents the sum of all scores present in the population (say, in this case) X1 X2 X3 and so on.  The symbol ‘N’ represents the total number of individuals or cases in the population.

Population Standard Deviation

The population standard deviation is a measure of the spread (variability) of the scores on a given variable and is represented by:

σ = sqrt[ Σ ( Xi – μ )2 / N ]

The symbol ‘σ’ represents the population standard deviation.  The term ‘sqrt’ used in this statistical formula denotes square root.  The term ‘Σ ( Xi – μ )2’ used in the statistical formula represents the sum of the squared deviations of the scores from their population mean.

Population Variance

The population variance is the square of the population standard deviation and is represented by:

σ2 = Σ ( Xi – μ )2 / N

The symbol ‘σ2’ represents the population variance.

Sample Mean

The sample mean is the average score of a sample on a given variable and is represented by:

x_bar = ( Σ xi ) / n

The term “x_bar” represents the sample mean.  The symbol ‘Σ xi’ used in this formula represents the represents the sum of all scores present in the sample (say, in this case) x1 x2 x3 and so on.  The symbol ‘n,’ represents the total number of individuals or observations in the sample.

Sample Standard Deviation

The statistic called sample standard deviation, is a measure of the spread (variability) of the scores in the sample on a given variable and is represented by:

s = sqrt [ Σ ( xi – x_bar )2 / ( n – 1 ) ]

The term ‘Σ ( xi – x_bar )2’ represents the sum of the squared deviations of the scores from the sample mean.

Sample Variance

The sample variance is the square of the sample standard deviation and is represented by:

s2 = Σ ( xi – x_bar )2 / ( n – 1 )

The symbol ‘s2’ represents the sample variance.

Pooled Sample Standard Deviation

The pooled sample standard deviation is a weighted estimate of spread (variability) across multiple samples.  It is represented by:

sp = sqrt [ (n1 – 1) * s12 + (n2 – 1) * s22 ] / (n1 + n2 – 2) ]

The term ‘sp’ represents the pooled sample standard deviation.  The term ‘n1’ represents the size of the first sample, and the term ‘n2’ represents the size of the second sample that is being pooled with the first sample.  The term ‘s12’ represents the variance of the first sample, and ‘s22’ represents the variance of the second sample.

Tools to Calculate Sample Size and Power Analysis

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The Sample Size/Power Analysis Calculator with Write-up is a tool for anyone struggling with power analysis.  Simply identify the test to be conducted and the degrees of freedom where applicable (explained in the document), and the sample size/power analysis calculator will calculate your sample size for a power of .80 of an alpha of .05 for small, medium and large effect sizes.  The calculator then presents the write-up with references which can easily be integrated in your dissertation document.  Click here for a sample.

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