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 represents the average score of the population on a given variable and symbolizes:
μ = ( Σ Xi ) / N
The symbol ‘μ’ stands for the population mean. Meanwhile, ‘Σ Xi’ indicates the sum of all scores in the population (such as 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 equals the square of the population standard deviation and symbolizes:
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 sample standard deviation measures the spread (variability) of the scores in the sample on a given variable and represents:
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 equals the square of the sample standard deviation and represents:
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” stands for the pooled sample standard deviation. “n1” refers to the size of the first sample, while “n2” represents the size of the second sample pooled with the first. “s12” indicates the variance of the first sample, and “s22” denotes the variance of the second sample.