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1

Precision: The Population

Standard Deviation

N

xN

i

i

1

2

Precision: The Sample

Standard Deviation

1

1

2

N

xx

s

N

i

i

Side by side

N

xN

i

i

1

2

1

1

2

N

xx

s

N

i

i

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N -1 = Degrees of Freedom

By calculating the mean, we use up a

degree of freedom.

Thus, only N -1 remain.

As N , N - 1 N

The z - statistic

or iix x x

sz z

The deviation from the mean given

as multiples of the standard deviation.

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3

Watch those calculators!

xn-1 sample

xn population

Watch Excel!

=STDEV(cells) (sample)

=STDEVP(cells) (population)Let’s play some more.

Example 6-1, page 126

Do it with Excel.

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4

The Standard Error, or The

Standard Deviation of the Mean

m NFind the mean many, many times using

N data each time, and then find the

standard deviation of the mean. This

equation gives the same value if is

the standard deviation of any subset.

Pooling Data

If we have a number of different people or

a number of different labs perform an

analysis, we can “pool” the data to get a

better estimate of the standard deviation.

Each set of data has its own sample mean

and its own sample standard deviation, so

we lose some degrees of freedom in the

process.

Calculating spooled

s

x x x x x x

N N N n

i jj

N

m nm

N

i

N

n

n

pooled

FHG

IKJ

FHG

IKJ

FH

IK1

2

2

2

1

2

11

1 2

21L

L

Where

n = the number of data sets

Nn = the number of data in each data set

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5

Hg in Chesapeake Bay Fish

The mercury in sample of seven fish taken

from the Chesapeake Bay was determined

by a method based on the absorption of

light by mercury atoms. We wish to

estimate the standard deviation of the

method by pooling the data from the seven

fish. Why? Let’s look at the data using

Excel.

The Standard Deviation s and

the Variance s 2

s

x x

N

s

x x

N

ii

N

ii

N

FHG

IKJ

FHG

IKJ

2

1

2

2

1

1

1variance

The Relative Standard

Deviation

RSD

RSD,%

RSD,ppt ppt

sx

sxsx

100

1000

%

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6

One of the Data Sets

2.06

1.93

2.12

2.16

1.89

1.95

Mean = 2.018333

= 0.110529

RSD

RSD 0.1105292.018333

RSD,%

RSD,ppt

0.054763

=0.055

5.476253%

=5.5%

t 54.762526 ppt

=55 ppt

0.1105292.018333

0.1105292.018333

pp =

100

1000

%

Standard Deviation of

Computed Results: Addition

and Subtraction

+0.50 ( 0.02)

+4.10 ( 0.03)

-1.97 ( 0.05)

2.63 ( ?????)

What is the uncertainty in the answer?

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Standard Deviation of

Computed Results: Addition

and SubtractionVariances are additive.

y a s b s c s

s s s s

s s s s

a b c

y a b c

y a b c

( ) ( ) ( )

2 2 2 2

2 2 2

Standard Deviation of

Computed Results: Addition

and Subtraction

06.0

0616441.0

0038.0

0025.00009.00004.0

05.003.002.0

222

222

ys

Standard Deviation of

Computed Results: Addition

and Subtraction

+0.50 ( 0.02)

+4.10 ( 0.03)

-1.97 ( 0.05)

2.63 ( 0.06)

What is the uncertainty in the answer?

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8

Standard Deviation of Computed

Results: Multiplication and Division

What is the uncertainty in the answer?

671 003 00071 00004543 006

00087737. ( . ) . ( . ). ( . )

. ( ??)

Standard Deviation of Computed

Results: Multiplication and Division

222

2222

c

s

b

s

a

s

y

s

c

s

b

s

a

s

y

s

c

bay

cbay

cbay

Standard Deviation of Computed

Results: Multiplication and Division

05758523.0

0031606.0

00012210.000317397.000001999.0

011050.0056338.0004471.0

43.5

06.0

0071.0

0004.0

71.6

03.0

0087737.043.5

0071.071.6

222

2222

y

s

y

s

y

y

y

9/23/2009

9

Standard Deviation of Computed

Results: Multiplication and Division

sy

y

s

y

y y

y

0.05758523

0.05758523

0.05758523

0.000505

00087737

00087737

00088 00005

.

.

. .