# OpenStax-CNX

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Inside Collection (Textbook):

Textbook by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

# Summary of Formulas

Summary: A summary of useful formulas used in examining descriptive statistics

## Commonly Used Symbols

• The symbol ΣΣ means to add or to find the sum.
• nn = the number of data values in a sample
• NN = the number of people, things, etc. in the population
• x¯x¯ size 12{ {overline {x}} } {} = the sample mean
• ss = the sample standard deviation
• μμ = the population mean
• σσ = the population standard deviation
• ff = frequency
• xx = numerical value

## Commonly Used Expressions

• x*fx*f = A value multiplied by its respective frequency
• xx size 12{ Sum {x} } {} = The sum of the values
• x*fx*f size 12{ Sum {x} times f} {} = The sum of values multiplied by their respective frequencies
• ( x x ¯ ) (x x ¯ ) or ( x μ ) (xμ) = Deviations from the mean (how far a value is from the mean)
• ( x x ¯ ) 2 ( x x ¯ ) 2 or ( x μ ) 2 ( x μ ) 2 = Deviations squared
• f ( x x ¯ ) 2 f ( x x ¯ ) 2 or f ( x μ ) 2 f ( x μ ) 2 = The deviations squared and multiplied by their frequencies

Mean Formulas:

• x¯=x¯= size 12{ {overline {x}} ={}} {}xnxn size 12{ { { Sum {x} } over {n} } } {} or x¯=x¯= size 12{ {overline {x}} ={}} {}f· xnf· xn size 12{ { { Sum {f times x} } over {n} } } {}
• μ=μ= size 12{μ={}} {}xNxN size 12{ { { Sum {x} } over {N} } } {} or μμ size 12{μ} {}= f·xNf·xN size 12{ { { Sum {f times x} } over {N} } } {}

Standard Deviation Formulas:

• s=s= size 12{s={}} {} Σ ( x x ¯ ) 2 n 1 Σ ( x x ¯ ) 2 n 1 or s=s= size 12{s={}} {} Σ f · ( x x ¯ ) 2 n 1 Σ f · ( x x ¯ ) 2 n 1
• σ=σ= size 12{σ={}} {} Σ ( x μ ¯ ) 2 N Σ ( x μ ¯ ) 2 N or σ=σ= size 12{σ={}} {} Σ f · ( x μ ¯ ) 2 N Σ f · ( x μ ¯ ) 2 N

Formulas Relating a Value, the Mean, and the Standard Deviation:

• value = mean + (#ofSTDEVs)(standard deviation), where #ofSTDEVs = the number of standard deviations
• xx size 12{x} {} = x¯ x + (#ofSTDEVs)(ss)
• xx size 12{x} {} = μμ size 12{μ} {} + (#ofSTDEVs)(σσ)

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