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Level 1 Laboratories
A Rough Summary of Key Error Formulae
for samples of random data.
For details see Physics Lab Handbook
(section 4.5.2 to 4.5.4)
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Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008
Normal, or Gaussian, distribution – a “bell-shaped” curve
Frequency
Mean
x
Standard
Deviation

68.3%
of
area
under
curve
1 n
x   xi
n i 1
( 4  1)
1 n
  lim  xi  x 2
n  n
i 1
(4  5)
2
Quantity x (e.g. rebound height)
2
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008
Normal, or Gaussian, distribution – a “bell-shaped” curve
Frequency
Mean
x
Standard
Deviation

68.3%
of
area
under
curve
Quantity x (e.g. rebound height)
1 n
x   xi
n i 1
( 4  1)
1 n
  lim  xi  x 2
n  n
i 1
(4  5)
1 n
2



x

x
 i
n  1 i 1
(4  6)
2
s
2
n 1
In reality, only a finite amount of measurements can be made.
If we then plotted a histogram,
it would only approximate the true Normal Distribution.
Nevertheless, we can still estimate the mean and standard deviation.
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Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008
Normal, or Gaussian, distribution – a “bell-shaped” curve
Frequency
Mean
x
1 n
x   xi
n i 1
( 4  1)
1 n
  lim  xi  x 2
n  n
i 1
(4  5)
1 n
2



x

x
 i
n  1 i 1
(4  6)
Standard
Deviation

2
68.3%
of
area
under
curve
s
2
n 1
Quantity x (e.g. rebound height)
Note : most simple calculators will provide
x
and
Then one should quote the final result as :
sn 1
(often shown on calc. as “xn-1” )
x  sm
(4  9)
sn 1
(4  8)
Where s m is called the “Standard Error in the Mean”, given by : sm 
n
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Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008
Example calculation of
n
n 1
1 s
1
2


x 8)
smxi  xi(
4
x , sn 1 &
n i n1  1 i 1n
n
1 n
1
x   xi 
4) 6) smn i 1
1
( 4 (1
Suppose we have a set of 10 measurements
of nominally the same thing (e.g. bounce height):
64, 66, 68, 70, 72, 68, 72, 70, 71 and 70 cm
• Mean
s n 1
1 n
1 10
x   xi 
xi  69.1 cm

n i 1
10 i 1
• Variance (or Mean
Squared Deviation)
• Standard Deviation
• Standard Error
(in the Mean)
s
2
n 1
1 n
1 9
2
xi  x    xi  69.12  6.8 cm 2


n  1 i 1
9 i 1
sn1  6.8  2.6 cm
(same as “xn-1” on Casio calculators)
2.6
sn1
sm 

 0.82 cm
n
10
• Quote final result as Mean ± Standard Error : i.e.
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x  sm  69.1  0.8 cm
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2008