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1 Measurements and their precision

Measurements are never exact, and there are several types of errors that are associated with the measurement process. This experiment will examine only one error related limitation on the accuracy of a result computed from experimentally obtained data. These are random errors.
Caution; the next ﬁve or six pages contain quite a bit of applied but basic calculus. I will make no apologies for this. The proper analysis of laboratory data requires an understanding of certain concepts from statistics and probability theory, and this in turn requires the use of calculus. I report all of the details here without pulling any punches with the philosophy that it is best to learn how to do things the right way from the very beginning.
If you wan to skip through this section, there is a summary of the ﬁnal results near the end of the discussion.

1.1 Data and random numbers

Scientiﬁc measurements are assumed to result in a random number distributed around the correct or true value of the quantity being measured, in the absence of systematic errors. The precision of a set of measurements is obtainable by applying statistical methods to the set. The notion that the results of measurements are random numbers distributed about the true value is based on the supposition that each time a measurement is made, the conditions are unique and irreproducible. For example small changes in the temperature of a metal caliper due to handling will cause the metal to expand and contract. This results in a slightly different value for the length of an object measured with the caliper, each time a measurement is made. An example of a systematic error is the use of a gauge whose zero point is off by one unit.
Since the conditions of the experiment are determined by a huge number of factors beyond the control or knowledge of the experimenter, the rules of statistics allow such conditional variations to be very accurately modeled by assuming that the measurement results are distributed random numbers.

1.2 Random number distributions (PDFs)

In most experiments, it turns out that if one were to take thousands of measurements, and from all of these compute the probability $\frac{d P_{x} (ξ)}{d ξ}$ (called the PDF or probability distribution function of $x$ ) that any one measurement results in a value for $x$ between $ξ$ and $ξ + d ξ$ , one would get a probability distribution like the following;

\frac{d P_{x} (ξ)}{d ξ} = \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(ξ - \bar{x})}^{2}}{2 σ^{2}}}

This is called a Normally distributed random variable, and the most probable outcome of a measurement is called $〈 x 〉$ or $\bar{x}$ . This is in fact the average measurement

\bar{x} = \int ξ \frac{d ℘_{x} (ξ)}{d ξ} d ξ

Since we have no idea how the actual physical data is distributed (we can’t know $\frac{d ℘_{x} (ξ)}{d ξ}$ ), we estimate the mean as

\bar{x} = 〈 x 〉 = \frac{x_{1} + x_{2} + . . . + x_{N}}{N} = \frac{\sum_{i = 1}^{N} x_{i}}{N}

Example Suppose that $100, 000$ measurements of quantity $x$ indicate that the measured value of $x$ is Normally random with $\bar{x} = 2.0 c m$ and $σ = 1.0 c m$ . If you were to make a measurement of $x$ , what is the probability that your resulting value would be between $1.9$ and $1.91$ ?
It would be

d P \approx \frac{1}{\sqrt{2 π} \cdot 1.0} e^{- \frac{{(2.0 - 1.9)}^{2}}{2 {(1.0)}^{2}}} \cdot (0.01) = 0.039

The actual distribution is the Bell-curve which looks like the following;

This type of random number is important for a second reason that is very deeply connected with lab measurements; averages of collections of random numbers are always themselves Normally distributed. This is called the Central Limit Theorem of statistics. It means that we do not need to know the probabilities of actually getting a particular measurement result, averages of many measurements will always be Normal random numbers, which is why any proper experiment will always involve many measurements from which averages are computed. Another corollary of the theorem is that averages come closer to the “true” value than single measurements in the absence of systematic errors. If quantity $x$ is randomly distributed with standard deviation (precision) $σ_{x}$ , then averages of sets of $N$ measurements of $x$ are normally distributed with standard deviation

σ_{\bar{x}} = \frac{σ_{x}}{\sqrt{N - 1}} < < σ_{x}

The mean is more “tightly distributed”, and this translates into averages being a more precise measure of a quantity’s actual value.

1.3 Experimental error

The standard deviation $σ$ is a measure of the precision (not accuracy) of the measurement set; the smaller $σ$ , the “tighter” the distribution of outcomes. This means that your chances of making a measurement close to the true value (assumed to be the mean in the absence of systematic errors) is higher as $σ$ gets smaller. The calculation of $σ$ is simple, as it measures the average deviation of each measurement from the mean value; if we actually knew the PDF of $x$ we would use

σ^{2} = \int {(ξ - \bar{x})}^{2} \frac{d ℘_{x} (ξ)}{d ξ} d ξ

but we in general do not, so we estimate the standard deviation as

σ^{2} = \frac{{(x_{1} - \bar{x})}^{2} + {(x_{2} - \bar{x})}^{2} + . . . + {(x_{N} - \bar{x})}^{2}}{N - 1} = \frac{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2}}{N - 1}

This is the quantity that we will associate with the experimental error. We will accept $σ$ as being a measure of experimental precision, which is what we actually mean when we refer to experimental error. If a given set of data has a standard deviation of sigma, then any randomly chosen result has a better than $89 %$ chance of being between $\bar{x} - σ$ and $\bar{x} + σ$ .

1.4 Error propagation

If you measure a quantity $x$ in the lab, and get $N$ values ${x_{1}, x_{2}, . . ., x_{N}}$ with mean $\bar{x} = \int ξ \frac{d ℘_{x} (ξ)}{d ξ} d ξ$ and standard deviation (squared) $σ_{x}^{2} = \int {(ξ - \bar{x})}^{2} \frac{d ℘_{x} (ξ)}{d ξ} d ξ$ , and then use this data to compute some function of $x$ , say $g (x)$ , how do you report your results?
This is the topic of error propagation, what are the means and standard deviations of functions of random numbers?
The answer comes from a major theorem in the mathematics of random numbers (statistics); the Law of the Unconscious Statistician;

\bar{g (x)} = g (\bar{x}) = \int g (ξ) \frac{d ℘_{x} (ξ)}{d ξ} d ξ

which is not difficult to prove.
Using this we can easily compute the standard deviation in $g$ since it is just a mean (of ${(g (ξ) - ḡ)}^{2}$ );

σ_{g}^{2} = \int {(g (ξ) - ḡ)}^{2} \frac{d ℘_{x} (ξ)}{d ξ} d ξ

We use the Taylor theorem to expand $g (ξ)$ about the mean value

g (ξ) = g (\bar{x}) + (\frac{d g}{d ξ})_{\bar{x}} (ξ - \bar{x}) + . . .

and so we get

σ_{g}^{2} = \int (\frac{d g}{d ξ})_{\bar{x}}^{2} {(ξ - \bar{x})}^{2} \frac{d ℘_{x} (ξ)}{d ξ} d ξ = (\frac{d g}{d x})_{\bar{x}}^{2} σ_{x}^{2}

and by taking roots

σ_{g} = | \frac{d g}{d x} |_{\bar{x}} σ_{x}

A detailed analysis using basic statistics indicates that the standard deviation of a quantity that is a function of a set of random variables is of a form that suggests the interpretation that errors combine like perpendicular vectors. In other words errors in independent quantities cannot cancel, but simply combine in such a way as to result in a maximal error.
If

y = f (x_{1}, x_{2}, . . .)

then

σ_{y}^{2} = (\frac{\partial f}{\partial x_{1}})^{2} σ_{x_{1}}^{2} + (\frac{\partial f}{\partial x_{2}})^{2} σ_{x_{2}}^{2} + . . . + (\frac{\partial f}{\partial x_{N}})^{2} σ_{x_{N}}^{2}

in which each function $\frac{\partial f}{\partial x_{i}}$ is computed using the averages ${\bar{x}}_{i}$ .

1.4.1 A summary for the impatient

If in the lab you measure several quantities $a$ , $b$ , and $c$ each $N$ times, obtaining the data sets

{a_{1}, a_{2}, . . ., a_{N}}, {b_{1}, b_{2}, . . ., b_{N}}, {c_{1}, c_{2}, . . ., c_{N}}

and must compute some function $f (a, b, c)$ as an experimental result, What must you do?

Step 1. Estimate the means and standard deviations (estimate since you do not know the true PDFs for $a, b$ and $c$ )

ā = \frac{1}{N} \sum_{i = 1}^{N} a_{i}, \bar{b} = \frac{1}{N} \sum_{i = 1}^{N} b_{i}, \bar{c} = \frac{1}{N} \sum_{i = 1}^{N} c_{i}

and

σ_{a} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} {(a_{i} - ā)}^{2}}, σ_{b} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} {(b_{i} - \bar{b})}^{2}}, σ_{c} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} {(c_{i} - \bar{c})}^{2}}

Step 2. Compute the derivatives

f_{a} (a, b, c) = \frac{\partial f (a, b, c)}{\partial a}, f_{b} (a, b, c) = \frac{\partial f (a, b, c)}{\partial b}, f_{c} (a, b, c) = \frac{\partial f (a, b, c)}{\partial c}

Step 3. Compute the mean $f$ ;

\bar{f} = f (ā, \bar{b}, \bar{c})

and its standard deviation (squared)

σ_{f}^{2} = (f_{a} (ā, \bar{b}, \bar{c}) σ_{a})^{2} + (f_{b} (ā, \bar{b}, \bar{c}) σ_{b})^{2} + (f_{c} (ā, \bar{b}, \bar{c}) σ_{c})^{2}

and report as the results of your experiment

\bar{f} \pm σ_{f}

1.4.2 Examples

Suppose that we measure the critical angle for a static equilibrium on an inclined plane, and discover that motion begins at angle $θ = 0.120, 0.123, 0.119, 0.121, 0.120, 0.121$ radians for six trials. The mean of these is estimated to be

\bar{θ} = \frac{0.120 + 0.123 + 0.119 + 0.121 + 0.120 + 0.121}{6} = 0.120667

and the standard deviation is estimated to be

σ_{θ}^{2} = \frac{1}{5} ({(0.120 - 0.120667)}^{2} + {(0.123 - 0.120667)}^{2} + {(0.119 - 0.120667)}^{2}

+ {(0.121 - 0.120667)}^{2} + {(0.120 - 0.120667)}^{2} + {(0.121 - 0.120667)}^{2}), σ_{θ} = 0.00137

The coefficient of friction computed from this is then

{\bar{μ}}_{s} = tan \bar{θ} = tan 0.120667 = 0.12126

The error in its value is then, with

\frac{\partial tan θ}{\partial θ} = \frac{1}{{cos}^{2} θ}

equal to

σ_{μ_{s}} = | \frac{σ_{θ}}{{cos}^{2} \bar{θ}} | = \frac{0.00137}{0.9840} = 0.0014

In our lab report we would write

μ_{s} \pm Δ μ_{s} = 0.12126 \pm 0.0014

Example The error in the area of a circle whose radius measurements have average $\bar{R}$ and standard deviation $σ_{R}$ ; use

A = π R^{2}, \frac{\partial A}{\partial R} = 2 π R

σ_{a} = ∣ 2 π \bar{R} ∣ σ_{R}

Example An experiment measures three quantities $a$ , $b$ , and $c$ repeatedly, getting averages $ā$ , $\bar{b}$ , and $\bar{c}$ and standard deviations $σ_{a}$ , $σ_{b}$ , and $σ_{c}$ for them. These data are used to obtain an experimental value for a quantity $d$ supposedly given by the formula

d = a^{x} b^{y} c^{z}

in which $x$ , $y$ , and $z$ are constants. What is the experimental value of $d$ and its standard deviation?
Constants are assumed to be known to at least one full digit or signiﬁcant ﬁgure than data. We will need only the derivatives

\frac{\partial d}{\partial a} = x a^{x - 1} b^{y} c^{z} = x \frac{d}{a}, \frac{\partial d}{\partial b} = y a^{x} b^{y - 1} c^{z} = y \frac{d}{b}

and

\frac{\partial d}{\partial c} = z a^{x} b^{y} c^{z - 1} = z \frac{d}{c}

which we compute using the averages

{ā, \bar{b}, \bar{c}}, and \bar{d} = ā^{x} {\bar{b}}^{y} {\bar{c}}^{z}

All of these factors go into the formula

σ_{d}^{2} = (x \frac{\bar{d}}{ā})^{2} σ_{a}^{2} + (y \frac{\bar{d}}{\bar{b}})^{2} σ_{b}^{2} + (z \frac{\bar{d}}{\bar{c}})^{2} σ_{c}^{2}

Remember that you can use the binomial theorem

{(1 + δ)}^{n} \approx 1 + n δ + . . .

to compute the derivatives. In fact the binomial theorem itself can be used to quickly arrive at a formula very close to the correct error expression without the use of calculus in the following way.

Example Obtain an error formula for $A = \frac{a^{2}}{b}$
Partial derivatives of multi-variable functions are the same as ordinary derivatives

\frac{\partial A}{\partial a} = {lim}_{δ a \to 0} \frac{A (a + δ a, b) - A (a, b)}{δ a} = {lim}_{δ a \to 0} \frac{(a^{2} + 2 a δ a + {(δ a)}^{2}) b^{- 1} - a^{2} b^{- 1}}{δ a}

= {lim}_{δ a \to 0} (2 a b^{- 1} + δ a b^{- 1}) = 2 a b^{- 1} = 2 \frac{a}{b}

and

\frac{\partial A}{\partial b} = {lim}_{δ b \to 0} \frac{A (a, b + δ b) - A (a, b)}{δ b} = {lim}_{δ b \to 0} \frac{1}{δ b} (\frac{a^{2}}{b + δ b} - \frac{a^{2}}{b})

Use synthetic division or the binomial theorem

\frac{1}{b + δ b} = \frac{1}{b (1 + \frac{δ b}{b})} \approx \frac{1}{b} - \frac{δ b}{b^{2}} + . . .

to get

\frac{\partial A}{\partial b} = {lim}_{δ b \to 0} \frac{1}{δ b} (- \frac{a^{2} δ b}{b^{2}} + . . .) = - \frac{a^{2}}{b^{2}}

1.5 Signiﬁcant ﬁgures

All of your measurements made with the same instrument should have the same number of signiﬁcant ﬁgures. For a measured number, the ﬁrst nonzero digit that is read directly from the measuring device as well as all the digits that follow it, up to and including the estimated one, are called signiﬁcant digits or signiﬁcant ﬁgures.
The most precise digit (the one furthest to the right) is usually estimated in a measurement. For example a particular vernier caliper is read, resulting in $0.0407 m$ , the seven was probably estimated by eye since this vernier scale is have labeled divisions smaller than $0.001 m$ . The seven is labeled least reliable, and this measurement has three signiﬁcant ﬁgures.
The rules for manipulating signiﬁcant ﬁgures are;

1. A result obtained by addition or subtraction should be rounded to the same decimal place as the least precise number.

2. A result obtained by multiplication or division should be rounded to the same number of signiﬁcant ﬁgures as the number in the product or quotient that has the fewest.

Example

0.04 + 26.704 - 18.7 = 8.044, \frac{(0.030) (152)}{13.20} = 0.3454545 . . .

In the ﬁrst calculation, $0.04$ , has one signiﬁcant digit, but its precision is high (second decimal place). The $26.704$ has the highest precision (three decimal places), and most signiﬁcant digits with ﬁve. The $18.7$ is least precise, only good to one decimal place, so the answer reported must be rounded to one decimal place; $8.0$ .
In the second example, $0.030$ has two signiﬁcant ﬁgures, $152$ has three, and $13.20$ has four. Therefore, the result should be rounded to two signiﬁcant ﬁgures; $0.35$ .
During all of your experimental work, take care to observe these rules.

2 Plotting scientiﬁc data

You will use two tools in the lab section of Physics 201 to analyze experimental data; statistics and curve-ﬁtting. In this section we will address these two topics, ﬁrst curve ﬁtting.

2.1 Linear regression

Several of the experiments test a hypothesis that some quantity $y$ is related to another quantity $x$ according to a linear relation $y = a x + b$ . The experiment is usually about testing the hypothesis by taking a set of data ${(x_{i}, y_{i}) ∣ i = 1, 2, . . ., N}$ and using the data to compute $a$ the slope and $b$ the intercept of the relation and comparing the results to the theoretical values. We perform such an analysis by computing $a$ and $b$ for the line that most closely conforms to the set of data taken in the lab, through the method of linear regression.

Consider the two collections of points ${(x_{i}, y_{i}) ∣ i = 1, 2, . . ., N}$ , which is your lab data, and ${(x_{i}, a x_{i} + b) ∣ i = 1, 2, . . ., N}$ which is the “true” set of points gotten by using the actual theoretical relation between $x$ and $y$ , which is assumed to be the straight line $y = a x + b$ . Consider then the sum of squares of the distances between corresponding points in the two sets;

σ^{2} = \frac{1}{N - 1} \sum_{i = 1}^{N} {(y_{i} - (a x_{i} + b))}^{2}

The line that best conforms to the set of data is the line for which this quantity is smallest, in other words the line from which the points deviate the least from vertically.

To ﬁnd the slope $a$ and $b$ of the best ﬁt line we simply minimize $σ^{2}$ with respect to $a$ and $b$ ; set the derivatives to zero

\frac{\partial σ^{2}}{\partial a} = 0 = - 2 \sum_{i = 1}^{N} x_{i} (y_{i} - (a x_{i} + b))

\frac{\partial σ^{2}}{\partial b} = 0 = - 2 \sum_{i = 1}^{N} (y_{i} - (a x_{i} + b))

which results in two equations in $a$ and $b$

\sum_{i} (x_{i} y_{i}) - a \sum_{i} x_{i}^{2} - b \sum_{i} x_{i} = 0, \sum_{i} y_{i} - a \sum_{i} x_{i} - N b = 0

which we solve for the best ﬁtting line parameters

a = \frac{N \sum_{i} (x_{i} y_{i}) - \sum_{i} x_{i} \sum_{i} y_{i}}{N \sum_{i} x_{i}^{2} - {(\sum_{i} x_{i})}^{2}}, b = \frac{\sum_{i} x_{i}^{2} \sum_{i} y_{i} - \sum_{i} x_{i} \sum_{i} (x_{i} y_{i})}{N \sum_{i} x_{i}^{2} - {(\sum_{i} x_{i})}^{2}}

and obtain standard deviations as well, by using the shorthand

S_{x} = \sum_{i} x_{i} = x \cdot u, S_{x^{2}} = \sum_{i} x_{i}^{2} = x \cdot x, N = u \cdot u, x = (x_{1}, x_{2}, . . ., x_{N})

and $u = (1, 1, . . ., 1)$ . The formulas for $a$ and $b$ deﬁne them to be linear functions of the random deviates $y = (y_{1}, y_{2}, . . ., y_{N})$

a = \sum_{i = 1}^{N} (\frac{N x_{i} - S_{x} u_{i}}{D}) y_{i}, b = - \sum_{i = 1}^{N} (\frac{S_{x} x_{i} - S_{x^{2}} u_{i}}{D}) y_{i}, D = N S_{x^{2}} - {(S_{x})}^{2}

To get the standard deviations we regard $a$ and $b$ as random deviates, and expand each around their means

\bar{a} = \sum_{i = 1}^{N} (\frac{N x_{i} - S_{x} u_{i}}{D}) \bar{y_{i}}, \bar{b} = - \sum_{i = 1}^{N} (\frac{S_{x} x_{i} - S_{x^{2}} u_{i}}{D}) \bar{y_{i}}

This leads to

\bar{{(a - \bar{a})}^{2}} = \sum_{i, j = 1}^{N} (\frac{N x_{i} - S_{x} u_{i}}{D}) (\frac{N x_{j} - S_{x} u_{j}}{D}) \bar{(y_{i} - \bar{y_{i}}) (y_{j} - \bar{y_{j}})}

We assume that each $y_{i}$ measurement is independent of $y_{j \neq i}$ , so that all non-diagonal correlation coefficients vanish,

\bar{(y_{i} - \bar{y_{i}}) (y_{j} - \bar{y_{j}})} = 0, if i \neq j, \bar{{(y_{i} - \bar{y_{i}})}^{2}} = σ_{i}^{2}

and so

σ_{a}^{2} = \sum_{i}^{N} (\frac{N x_{i} - S_{x} u_{i}}{D})^{2} σ_{i}^{2}

In practice we use the same $σ_{i}^{2}$ for each $y_{i}$ ;

σ_{i}^{2} = σ^{2} = \frac{1}{N - 1} \sum_{i = 1}^{N} {(y_{i} - (a x_{i} + b))}^{2}, \forall i

resulting in

σ_{a}^{2} = σ^{2} (\frac{N^{2} x \cdot x - 2 N S_{x} x \cdot u + S_{x}^{2} u \cdot u}{D^{2}}) = \frac{N}{D} σ^{2}

We can do the same for $σ_{b}$ , starting with

b = - \sum_{i = 1}^{N} (\frac{S_{x} x_{i} - S_{x^{2}} u_{i}}{D}) y_{i}

\bar{{(b - \bar{b})}^{2}} = \sum_{i, j = 1}^{N} (\frac{S_{x} x_{i} - S_{x^{2}} u_{i}}{D}) (\frac{S_{x} x_{j} - S_{x^{2}} u_{j}}{D}) \bar{(y_{i} - \bar{y_{i}}) (y_{j} - \bar{y_{j}})} = \frac{S_{x^{2}}}{D} σ^{2}

The two standard deviations are therefore

σ_{b}^{2} = \frac{σ^{2} \sum_{i} x_{i}^{2}}{N \sum_{i} x_{i}^{2} - {(\sum_{i} x_{i})}^{2}}, σ_{a}^{2} = \frac{σ^{2} N}{N \sum_{i} x_{i}^{2} - {(\sum_{i} x_{i})}^{2}}

In this formula $σ$ is the standard deviation of the $y_{i}$ measurements, all assumed to be the same for each $i$ . This can be estimated, or you could measure $y_{i}$ several times for each $i$ , and compute it more rigorously.

2.1.1 A summary for the impatient

If you have a collection of data points

{(x_{1}, y_{1}), (x_{2}, y_{2}), . . ., (x_{N}, y_{N})}

that presumably fall on a straight line $y = a x + b$ , the slope $a$ and intercept $b$ of the line that best ﬁts this data are gotten as follows;

Step 1. Compute

S_{x} = \sum_{i = 1}^{N} x_{i}, S_{x^{2}} = \sum_{i = 1}^{N} x_{i}^{2}, S_{y} = \sum_{i = 1}^{N} y_{i}, S_{x y} = \sum_{i = 1}^{N} x_{i} y_{i}

Step 2. The best ﬁt slope and intercept are

a = \frac{N S_{x y} - S_{x} S_{y}}{N S_{x^{2}} - {(S_{x})}^{2}}, b = \frac{S_{x^{2}} S_{y} - S_{x} S_{x y}}{N S_{x^{2}} - {(S_{x})}^{2}}

respectively. To get estimates of the error in $a$ and $b$ take another step;

Step 3. Compute

σ^{2} = \frac{1}{N - 1} \sum_{i = 1}^{N} (y_{i} - (a x_{i} + b <