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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

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