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A summary of useful formulas used in examining descriptive statistics

    Commonly used symbols

  • The symbol Σ means to add or to find the sum.
  • n = the number of data values in a sample
  • N = the number of people, things, etc. in the population
  • x ¯ size 12{ {overline {x}} } {} = the sample mean
  • s = the sample standard deviation
  • μ = the population mean
  • σ = the population standard deviation
  • f = frequency
  • x = numerical value

    Commonly used expressions

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

Mean Formulas:

  • x ¯ = size 12{ {overline {x}} ={}} {} x n size 12{ { { Sum {x} } over {n} } } {} or x ¯ = size 12{ {overline {x}} ={}} {} f · x n size 12{ { { Sum {f times x} } over {n} } } {}
  • μ = size 12{μ={}} {} x N size 12{ { { Sum {x} } over {N} } } {} or μ size 12{μ} {} = f · x N size 12{ { { Sum {f times x} } over {N} } } {}

Standard Deviation Formulas:

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

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Source:  OpenStax, Collaborative statistics homework book: custom version modified by r. bloom. OpenStax CNX. Dec 23, 2009 Download for free at http://legacy.cnx.org/content/col10619/1.2
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