Example of Standard deviation calculation
Question. Suppose that you measure the length, l, of a cylinder. From 5
trials, you have the following data:
Table 1
Trial Measured
#
length,
l (cm)
1
1.52
2
1.56
3
1.48
4
1.50
5
1.54
1. Calculate the mean (or best) value of l, that is, l . Present your
calculated results,
together with the measured data in a tabular form.
2. Calculate the deviation of each measurement from the mean value.
Present your calculated results, together with the measured data in a
tabular form.
3. What is the mean of the deviations?
4. Calculate the standard deviation of the measurements. Present your
calculated results,
together with the measured data in a tabular form.
5. What is the standard deviation of the mean?
6. What is the absolute uncertainty in these measurements?
7. What is the fractional uncertainty of these measurements?
8. What is the percentile uncertainty of these measurements?
Answer
1. The total number of sampling, N = 5 in this case. The mean is defined
as
∑ l
N
l =
i =1 i
N
= ( l1 +
l
2
+
l
3
+
l
4
+
l
5
)/5
= (1.52 + 1.56 + 1.48 + 1.50 + 1.54) /5 = 1.52 cm.
2. The deviation, d i , of the ith measurement is defined as:
d i = li − l .
Using this definition, we can include the d i values in table 1 as follows.
Trial Measured
#
length,
1
2
3
4
5
l (cm)
1.52
1.56
1.48
1.50
1.54
d
i
(cm)
0.00
0.04
-0.04
-0.02
0.02
3. The mean of deviation is defined as:
∑ d
N
i =1
N
i
= (0 .00 + 0.04 –0.04 –0.02 +0.02) = 0 !
Clearly, the calculation of this number doesn’t make any sense. Note,
2
however, the term d i
is non-zero, and it makes sense to calculate this term. This term is used in
the definition
of standard deviation (see below).
4. The standard deviation in the measurement of l is defined as:
N
σ
l
=
∑d
i =1
2
i
N −1
.
Note that in the definition of σ l , we use the number of intervals (N-1)
between
Measurements, instead of the number of measurements (N). We will
2
calculate d i for each i,
and then, we will calculate σ l . These values are tabulated below.
First step: Include
di
2
in the table.
Trial Measured
#
length,
l (cm)
1.52
1.56
1.48
1.50
1.54
1
2
3
4
5
d
= li − l
(cm)
i
0.00
0.04
-0.04
-0.02
0.02
2
di
cm
2
0
1.6
1.6
0.4
0.4
× 10 − 3
× 10 − 3
× 10 − 3
× 10 − 3
Next step: calculate standard deviation:
2
N
1/2
∑
d
σ l = [ i=1 i ]
N −1
= [(0+ 1.6 ×10− 3 + 1.6 ×10− 3 + 0.4 ×10− 3 + 0.4 ×10− 3 )/(5-1)]1/2 cm
= 0.03 cm.
5. Standard deviation of mean (SDOM) is defined as:
SDOM = σ l =
σ
l
N
=
0.03
cm = 0.01 cm.
5
6. Absolute uncertainty = ∆l = SDOM = 0.01 cm.
7. Fractional uncertainty
=
∆l
= (0.01/1.52) = 0.007 ≈ 0.01.
l
8. Percentile uncertainty = fractional uncertainty × 100
= 0.01 × 100 = 1%