Statistical Analysis and Expression of Data Reading: Lab Manual Today: Some basics that will help you the entire year The mean or average =  true value,  measurements ≈  true value, finite # of measurements Uncertainty given by standard deviation

σ ≈ S = [  (x i - ) 2 /(n-1)] 1/2 Where is average of x For finite # of measurements Standard deviation: S (Calculators can calculate & σ quite easily!!! Learn how to do this on your calculator.) For small number of measurements σ ≈ S is very poor. Must use Student t value. σ ≈ tS; where t is Student t Usually use 95% Confidence Interval So, 95% confident that if we make a measurement of x it will be in the range ± t 95 s Uncertainty of a SINGLE MEASUREMENT

Usually interested in mean (average) and its uncertainty Standard Deviation of the mean Then average and uncertainty is expressed as ± t 95 S m Often want to know how big uncertainty is compared to the mean: Relative Confidence Interval (C.I.) = (S m / )(t 95 ) Expression of experimental results: 1. Statistical Uncertainties (S, S m, t 95 S, S m (t 95 ) / ) always expressed to 2 significant figures 2. Mean (Average) expressed to most significant digit in S m (the std. dev. of the mean)

Example: Measure 3 masses: , , grams Average = grams; Std. Dev. S = =.13 grams Then average = ?= grams S m = /  3 = =.072 grams 95% C.I. = t 95 S m = * = =.31 grams Relative 95% C.I. = ? = 95 % C.I. / Average = / = =.029 Usually expressed at parts per thousands (ppt) =.029 * 1000 parts per thousand = 29 ppt = Relative 95% C.I. Now work problems. What if measure , , grams? Ave. = ; S m = Ave. = ? Ave. = , not because limited by measurement to.0001 grams place t 95 for 3 measurments