__BACKGROUND__

** The statistical approach
recognizes that it is impossible to precisely state how accurate an
observation might be, or even how accurate an observed average or mean
might be. Aset of observations, which might range from a few
measurements to an extremely large number of measurements, is
represented by a single representative value and an indication of the
reproducibility (uncertainty) of the observations. The value
chosen as the representative value is the average
or arithmetic
mean. The indication of
reproducibility is based on a quantity called the standard deviation (s). This quantity is related to the probability
that a single observation will fall within a specified range about the
mean. The equations for calculating these probabilities are based
on the assumption of an extremely large number of observations, then
they are adapted to smaller numbers of observations(as is the normal
experimental situation). The adaptation to the smaller number of
observations is based on the degreesof
freedom (df). In the case of an arithmetic mean, the degrees
of freedom is simply the number of
observations (N) minus 1 (df= N - 1).**

** The square of the standard deviation (s ^{2}) is defined as the sum of the squares of the
deviations from the mean divided by the degrees of freedom. The
square of the standard deviation of the
mean (s_{mean}^{2}) is defined as the square of the standard deviation
divided by the number of observations ( s_{mean}^{2}= s^{2}/N
). These quantities are more easily
understood by performing calculations on a set of observations:**

** We consider a set of 10
observations: 106,111,108,105,109,115,110,112,114,101 .**

**First we sum the values, then divide by the
number of observations to calculate the mean.**

sum: |
observations106111108105109115110112114101________1091 |
dev.
of obs. frommean106 - 109.1 = - 3.1111 - 109.1 = 1.9108 - 109.1 = - 1.1105 - 109.1 = - 4.1109 - 109.1 = - 0.1115 - 109.1 = 5.9110 - 109.1 = 0.9112 - 109.1 = 2.9114 - 109.1 = 4.9101 - 109.1 = - 8.1________________ 0.0 |
(dev.
of obs. frommean)^{2} 9.61 3.61 1.2116.81 0.0134.81 0.81 8.4124.0165.61______164.90 |

**number of observations: N =
10 degrees of freedom: df = N - 1 = 9;**

**mean = sum/N = 1091/10 = 109.1**

**Then we subtract the mean from each of the
observed values.**

**These values are squared and summed to
calculate the standard deviation and the standard deviation of the mean.**

**s ^{2}
=(sum of squares)/df = 164.9/9 = 18.3**

**s = 4.3**

** s _{mean}^{2}=
s^{2}/N = 18.3/10 = 1.83**

** s _{mean}= 1.4**

**We have carried extra significant figures
through these calculations. In practice, there is little
justification for stating an uncertainty to more than one significant
figure.**

** These statistical
calculations assume that a large number of observations have been used
to calculate the mean and the standard deviation. When smaller
numbers of observations are used, there is less confidence that the
calculated values are really representative of the statistical
probabilities. An additional factor is introduced -
the Student t-factor (W.S. Gosset published his work on this factor under
the pseudonym "Student"). The
standard deviation of the mean is multiplied by the t-factor, which is based on the degrees of freedom and the
desired confidence level to obtain the Confidence
Interval (d) at the
specified percentage (the level
ofconfidence). An abbreviated table
of t-factors is given below.**

** For the set of data above;
Mean= 109.1 , s _{mean}= 1.4 ,
df= 9:**

**These quantities are stated as "the uncertainty is ± 1 at the 50% level of
confidence" or "the mean value is 109 ± 3 at the 95% level of
confidence". This is interpreted to
mean that if you repeated the set of measurements a large number of
times, half of those means would fall in the range 108 - 110, and 95%
of them would fall in the range 106 - 112.**

__COMBINING UNCERTAINTIES IN CALCULATIONS-
PROPAGATION OF UNCERTAINTY__

** Scientific measurements
are often combined to calculate other quantities that may be used
in subsequent calculations. Each of these quantities will have an
uncertainty,and the calculated value must be assigned an
uncertainty. Suppose quantities W,X,and Y are measured with uncertainties d _{W}, d_{X}, and d_{Y}. These will be combined in some equation to
calculate some other quantity Z:**

** There are rigorous rules for
combining the uncertainties to calculate the uncertainty in Z (d _{Z}). Fortunately, most of the situations that you
will encounter may be reduced to two simple relationships.**

**For Multiplication and Division: Z
=WX/Y**

**(d _{Z}/Z)^{2}=(d_{W}/W)^{2 }+(d_{X}/X)^{2}+^{}(d_{Y}/Y)^{2}**

**For Addition and Subtraction: Z = W
-X + Y**

**d _{Z}^{2}=d_{W}^{2}+^{ }d_{X}^{2}+d_{Y}^{2}**

**Those rules also cover the following cases,
but we will include them here for convenience:**

**If there are powers involved in the
equation using multiplication and/or division:**

**If there is multiplication by scalars in
the equation using addition and/or subtraction:**

**In applying these rules, it is important to
have the uncertainties of all of the independent variables (W, X, and
Y) at the same level of confidence. If the uncertainties are at
the 50% level, the uncertainty in Z will be at the 50% level.**

__EXAMPLE__

**Let's assume that you have made several
measurements of the width and height of a rectangle, and have
calculated the mean values, the standard deviations, the standard
deviations of the means, and the uncertainties at the 95% level of
confidence.**

**For the multiplication process: (d _{area}/Area)^{2}=(d_{W}/W)^{2 }+ (d_{H}/H)^{2}**

**Therefore: (d _{area}/11526)^{2}=(3/102)^{2
}+ (4/113)^{2 }= .00087 + .00125 = .00212 ,**

** and: d _{area}= 11526 x (.00212)^{1/2}=11526
x .046 = 530 .**

**The area may be reported as: Area = 11,500 ± 500 at the 95% level of
confidence .**

**The perimeter is calculated: Perimeter = 2W + 2H = 2(102) + 2(113) = 430 .**

**For the addition process (with scalars): d _{perimeter}^{2}=
4 d_{W}^{2} +^{ }4d_{H}^{2}+ = 4(3)^{2}+
4(4)^{2} = 100 ,**

** and: d _{perimeter}= 10.**

**The perimeter may be reported as:Perimeter = 430 ± 10 at the 95% level of
confidence.**