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Summary of Formulas

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

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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