Measurement Uncertainty
Objective
In this lab we’ll look at the sources and results of measurement error. From this lab, you should
learn to calculate and interpret mean, variance, and standard deviation, and identify component
characteristics including tolerance, price, form factor, and de-rating.
Concepts
An error is defined as the difference between the measured value (xm) and the true value (xtrue).
Error = xm - xtrue
(1)
This relationship begs several questions. What is the “true” value? If I measure very carefully, will
my measurements still contain errors? Where do these errors come from and how can I minimize
their impact on my design?
System accuracy is the magnitude of the maximum expected error. This measure of system
performance is usually specified for a piece of test equipment as a percentage of full-scale value.
System precision is an estimate of repeatability. A precise system, making repeated measurements
of the same thing, will consistently produce the same result. The more precise an instrument, the less
random error affects the result.
System resolution is the smallest possible discernible increment. The higher the resolution, the
smaller the smallest increment is.
--------------------------------------------------Types of Error---------------------------------------------Systematic errors occur in a repeatable way every time a measurement occurs under similar
conditions. They are also known as ‘bias’ errors.
If a measurement is consistently in error by a constant number, added or subtracted to the true value,
it’s called an offset error.
Offset Error: Xm = Xtrue ± Constant
(2)
When the true value can be determined by a ratio of the measured value and a constant (which can be
greater or less than 1), it’s called a scale error.
Scale Error: Xm = Xtrue x Constant
(3)
Nonlinear errors can result from poor design or from inappropriate system use. Many components
and systems of components are designed to produce linear output from linear input. However, if
those components are improperly used, either by the designer or the end user, nonlinear ( y= x 2, y =
cos xt, or y = log x are examples of nonlinear functions) can result.
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Over time, measurement conditions such as ambient temperature and humidity can change.
Electronic components age, and their characteristic values can also change over time. This
phenomenon is called drift.
Random errors are different for every measurement event. The defining characteristic for random
errors is that they have an average value of zero. As a result, it’s often possible to reduce random
error in measurement by repeating the measurement and then averaging the outcomes.
The National Institute of Standards and Technology, www.nist.gov, maintains many national and
international standards of metrology, the science of measurement. Most sophisticated test equipment
needs periodic calibration. For many design, maintenance, and test facilities, a current certificate of
calibration that is “traceable” to NIST is mandatory. That means that the equipment has been
calibrated by a facility that can credibly claim that the work was performed by someone who has
been trained to calibrate that piece of test equipment and who used metrology standards that have
been endorsed by NIST.
____________________Median, Mean, Variance, and Standard Deviation__________________
Suppose for a set of measurements on a small sample of 15 kΩ resistors we have the following data.
15.15 kΩ
14.81 kΩ
15.07 kΩ
14.98 kΩ
15.02 kΩ
We can organize this data in several ways to help us understand the characteristics of the larger
population of 15 kΩ resistors.
First we can determine the sample median value of the sample. The sample median is just the
middle value when the measurements are arranged from smallest to largest. If we know the sample
median value, we can infer that there are as many measurements above the median as below. For our
sample of 14.81, 14.98, 15.02, 15.07, and 15.15, the middle value is15.02.
Another measure of the ‘center’ of a sample of data is the sample mean, also known as an ‘average’
value. The sample mean of a set of measurements is the sum of the measurements divided by the
number of samples:
Sample Mean: X 

n
 xi
i 1
n

(4)
where Xi is the individual measurement and n is the total number of measurements. The sample
mean for this set of data is 15.006K.
How do you chose between the median and mean to determine the center of a sample? The median
isn’t perturbed by a few very small or very large measurements. On the other hand, even one or two
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extreme measurements will dramatically affect the sample mean. If we add one measurement of 10
kΩ and another of 12 kΩ, the median changes only slightly to 14.98 kΩ. The sample mean on the
other hand changes from 15.006 kΩ to 13.86 kΩ. From a practical point of view, this is why it’s so
hard to recover from even a single missed assignment or low test score when a sample mean is used.
The variation on either side of the center or deviation is another important characteristic of our
measurements. If we take the sample mean X as the center of our data, then the sample deviation is
the difference between the measured value and the sample mean:
Deviation = X - X
(5)
Because some deviations are positive and some are negative the deviation is not a good measure of
the “spread” of the data. In fact it can be shown that the sum of the deviations for any data set is
zero. So we calculate the sample variance by summing the square of the deviations and dividing by
the number of measurements minus one.
 X
2
n
Sample Variance: S2 =
i 1
i
X
(6)
n 1
In order to get an estimate of the variability of the sample in the same units as the data, take the
positive square root of the sample variance. This quantity is known as the standard deviation and is
the basic measure of variability.
 X
2
n
Sample Standard Deviation: s 
i 1
i
X
n 1
(7)
For “bell shaped” distributions you should expect approximately 68.3% of the measurements will fall
between x  s , 95.4% of all measurements will fall between x  2s , and 99.7% of all measurements
will fall between x  3s . This bounded range is sometimes called the confidence interval.
Consequences of Error
Every system you design will use components that vary from their nominal value. Your job will be
to create a system that is independent of these variations. This is called robust design. If you have to
pick through the parts bin to find just the right inductor, your design is probably not very robust. A
sensitivity analysis of your design will reveal which components contribute to the overall sensitivity
of the design and which do not. Components that are manufactured to a very close tolerance, i.e. a
very small standard deviation, cost more than components that are manufactured to a looser standard.
Specifying a component as a ± 1% instead of a ±10% can significantly affect product profitability
when that small difference in cost is multiplied over a production run of hundreds of thousands. In
his book, “The Chip,” T. R. Reid describes the first engineering job held by Jack Kilby, co-inventor
of the monolithic integrated circuit. “It was an intensely competitive business, where a cost
differential of one dollar per thousand parts—a tenth of a penny per part—could win or lose huge
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contracts. ‘It was sort of a crash course in sensitivity to cost,’ Kilby recalled later.” Your ability to
create a clever design may make the difference in management decisions that affect many other
people.
Equipment and Components
This experiment will require the use of the Fluke Digital Multimeter, several resistors with a nominal
value of 15 k, Microsoft Excel, and Matlab.
Prelab (25 points) – Due at the beginning of lab
1. Find the cost for 10 kΏ ± 1% and ±10% resistors in quantities of 1,000. Provide cost estimates
from two vendors.
2. Write a paragraph identifying and describing three different ways to manufacture resistors. These
variations can be the “form factor” or the technique used to establish the resistance.
3. Provide an example of a resistor derating curve from a manufacturer. Briefly explain the meaning
the chart.
Part 1: Measurement Analysis
First we’ll take some measurements and do some statistical analysis to determine the type(s) of
measurement errors involved. You’ll also be plotting a histogram, which is a chart that has the
measured values on the horizontal axis and the number of times that value occurred on the vertical
axis, as shown in Figure 1.
Figure 1: A quick example histogram
In this example histogram, there are 11 samples (total number of boxes). The number of resistors that
fall between values of 15.2 kΏ and 15.3 kΏ, for example is 3. For the histogram you create the x-axis
values shown here will probably not fit your data, since it is just for a hypothetical data set. When
you make your own, choose x-axis values and increments that best allow you to express your data.
Now on to the procedure proper:
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1. As a class, measure and record the resistance of each of 40 resistors on your data sheet.
2. Using Excel, calculate the sample mean, the deviation for each value, the deviation squared
for each value, as shown in the table on the data sheet. You can just turn in your Excel
spreadsheet rather than writing all the values in. Also calculate the sample variance, and the
sample standard deviation, and include it on both your spreadsheet and your data sheet.
3. Plot the result in a histogram either by hand or using Excel. Label the mean, and draw
vertical lines indicating +/-s, +/-2s, and +/-3s.
4. On your data sheet, provide a brief description of why fitting data to statistics such as
variance and standard deviation is important.
5. In terms of the types of error discussed in this lab, what type(s) of measurement error are
evident in your resistor sample? Answer this on the data sheet.
Part 2: A Simple Derating Simulator
In your prelab assignment, you investigated derating curves. For this part of the lab, you’ll write a
simple Matlab program that shows the output of the series circuit shown in Figure 2 in terms of
resistor derating.
R1
5.1k
V1
24Vdc
R2
13.8k - 15.2k
Heater
Figure 2: Circuit near a heater
The input voltage is 24 V. R1 is 5.1 kΏ. R2, however, is affected by the heater. Its value varies
between 13.8 kΏ and 15.2 kΏ. Write a Matlab program that plots the current through R1 for
each value of R2, and plot the voltage across R1 for values of R2 between 13.8 kΏ and 15.2 kΏ.
First, set the values for V1, R1, and R2. R2 will be an array. Calculate the voltage value across R1
for each value of R2 using a voltage divider (ratio of R1 to total resistance). Also calculate the
current for each resistor value (total series voltage over total series resistance). These calculations are
most easily done in a ‘for’ loop. Once you have the V and I values, plot V vs. R2 and I vs. R2.
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Lab 4: Measurement Uncertainty
Name_____________________________
Section________
Prelab (due at the beginning of lab)
1. Find the cost for 10 kΏ ± 1% and ±10% resistors in quantities of 1,000. Provide cost estimates
from two vendors.
2. Include a paragraph identifying and describing three different ways to manufacture resistors.
These variations can be the “form factor” or the technique used to establish the resistance.
3. Provide an example of a resistor de-rating curve from a manufacturer. Briefly explain the
meaning the chart.
Part 1: Measurement of Time Varying Signals
1. Measured data – calculate values in Excel and print out your spreadsheet to turn in.
Measured
Deviation
Deviation2
Measured
Deviation
Deviation2
Value X
Value X
R1
R21
R2
R22
R3
R23
R4
R24
R5
R25
R6
R26
R7
R27
R8
R28
R9
R29
R10
R30
R11
R31
R12
R32
R13
R33
R14
R34
R15
R35
R16
R36
R17
R37
R18
R38
R19
R39
R20
R40
Total =
Sum =
Sum =
Total =
Sum =
Sum =
2. Sample Mean:________
Sample Variance:_________ Sample Standard Deviation:_________
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3. Provide a brief description of why fitting data to statistics such as variance and standard deviation
is important.
4. In terms of the types of error discussed in this lab, what type(s) of measurement error are evident
in your resistor sample, and why?
Part 2: A Simple Derating Simulator
1. Turn in your Matlab code and plots.
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