Study Guides (247,941)
United States (123,244)
Chemistry (134)
CAS CH 131 (24)

Gen Chem Appendix.pdf

14 Pages
155 Views

School
Department
Chemistry
Course
CAS CH 131
Professor
Mark Grinstaff
Semester
Fall

Description
Appendix A  Statistical treatment  of data  Including statistical analysis using Microsoft® Excel®  Mean and standard deviation    Suppose that we make N measurements of the same quantity x.  For the measurements to be comparable we usually arrange  for the conditions under which x is measured to be as closely  matched as possible. For example, if you wanted a meaningful  idea of how much you weigh, you wouldn’t jump on the bath‐ room scale right after the big Thanksgiving dinner and then try  to compare that result with how much you weigh after running  a marathon, would you? Of course not – instead, you might try  to weigh yourself first thing every morning for a week just af‐ ter you wake up.    Despite our best efforts, however, N measurements of the  same  quantity  x  can  never  be  made  under  exactly  matched  conditions. There are many reasons for this fact. Some are psy‐ chological and physiological: the more times we repeat an op‐ eration, the better (or worse) we get at it and the manner in  which a scientist executes a measurement (i.e., technique) in‐ fluences the experimental outcome. Some are physical: the sys‐ tem under study changes with the passage of time in ways we  cannot fully control. Given that individual measurements of the  same quantity vary, what is the best way to report the data?    One approach  is  to determine the  mean  of  the measure‐ ments of x and to report the variation in the data as the stan‐ dard deviation. The mean of the N measurements of x is de‐ noted by the symbol    nd is defined by    Appendix A ∙ Statistical treatment of data   A‐1  i=N x x + x + x ++ x ∑ i x = 1 2 3 N = i=1        N N   where  the  x  irepresent  the  individual  measurements  of  the  quantit▯ x and the standard deviation σ is defined by    i=N ∑ (xi− x )2 (x1− x ) +(x −2x ) +(x − 3 ) ++(x − x ) N 2 i=1 σ= =        N N   The mean expresses the central tendency in a set of data. The  ▯ standard deviation expresses the theoretical expectation that  68.27% of the measurements of x will lie within one standard  deviation on either side of the mean when x is measured an in‐ finite number of times.    Example  Table A‐1 presents the rainfall measured at Boston during Sep‐ tember from 1999 to 2003 and the quantities needed to de‐ termine the mean September rainfall and its standard devia‐ tion. The mean is    Table A­1 September rainfall at Boston, 1999–2003  Quantities needed in the calculation of the mean and of the standard deviation  i    x i xi –    (xi –        Year  Rainfall  Deviation from the mean  Deviation squared      [inch]  [inch]  [inch ]            ▯ ▯ 1  1999  9.86  5.65  31.92  2  2000  2.87  –1.34  1.80  3  2001  2.29  –1.92  3.69  4  2002  3.39  –0.82  0.67  5  2003  2.65  –1.56  2.43              Sum  21.06    40.51  Appendix A ∙ Statistical treatment of data   A‐2  9.86+2.87+2.29+3.39+2.65 21.06 x = = = 4.2 (to one decimal)        5 5   The standard deviation is  ▯   2 2 2 2 2 σ= (9.86−4.212) +(2.87−4.212) +(2.29−4.212) +(3.39−4.212) +(2.65−4.212) 5   =2.8 (to one decimal)      Thus, the best way to report the mean September rainfall is  4.2 ± 2.8 inch.  ▯   Recall  that  σ  expresses  the  theoretical  expectation  that  68.27% of an infinite number of measurements will lie within  one standard deviation on either side of the mean, that is, be‐ tween 4.2 – 2.8 = 1.4 inch and 4.2 + 2.8 = 7.0 inch for the rain‐ fall data set we are considering. Because the rainfall was meas‐ ured only five and not an infinite number of times, four (2.87,  2.29, 3.39, 2.65) of the five measurements (80%) fall within  one standard deviation on either side of the mean. Agreement  with  the  theoretical  expectation  improves  as  the  number  of  measurements increases.      The t‐test    Calculating  the  standard  deviation  of  a  mean  is  one  way  of  quantitatively assessing experimental variance. As useful as it  is, the standard deviation suffers from the weakness of its be‐ ing  rigorously  defined  only  when  N  =  ∞  and  we  can  never  measure anything an infinite number of times. Another tech‐ nique, the t‐test, is useful when the mean is calculated from a  small (N
More Less

Related notes for CAS CH 131
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.