Descriptive Statistics: Summary of Formulas 1.6 2008/05/13 11:28:44 GMT-5 2008/07/16 11:51:59.685 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org elementary statistics 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}} ={}} {}∑xn size 12{ { { Sum {x} } over {n} } } {} or x¯= size 12{ {overline {x}} ={}} {}∑f· xn size 12{ { { Sum {f times x} } over {n} } } {} μ= size 12{μ={}} {}∑xN size 12{ { { Sum {x} } over {N} } } {} or μ size 12{μ} {}= ∑f·xN 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)