lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
TOPIC 2
INTERPRETATION OF RESULTS
An experiment is a question which science poses to Nature, and a measurement is the recording of
Nature's answer.
Max Planck
I believe in evidence. I believe in observation, measurement, and reasoning, confirmed by
independent observers. I’ll believe anything, no matter how wild and ridiculous, if there is evidence
for it. The wilder and more ridiculous something is, however, the firmer and more solid the evidence
will have to be.
Isaac Asimov
INTRODUCTION
We have now said a lot about precision, the dispersion of measurements about a mean value and
the standard deviation from the mean. The next step is to find a way to estimate the accuracy of a
measurement. In this topic we will show how the best estimate of the precision and the accuracy of
a measurement can be calculated. Thinking back to the soft drink experiment, the following two
questions about errors were asked:
➢ What is the meaning of the errors in the measurements? ➢ How were the errors estimated?
In this topic we answer these questions and show how they apply to the experiments you are going to
conduct.
STUDY UNIT 9
STANDARD ERROR
1
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
Learning Objectives
After having completed this study unit, you should be able to:
1. define
(a) the adjusted standard deviation from the mean s (b) the standard error in the mean 𝜎𝑚 (c)
the adjusted standard error in the mean 𝑠𝑚
2. indicate when and how to apply Bessel's correction 3. calculate the
(a) adjusted standard deviation from the mean s
(b) the standard error in the mean 𝜎𝑚
(c) the adjusted standard error in the mean 𝑠𝑚 of an experimental result
9.1 BEST ESTIMATE OF THE TRUE VALUE: THE MEAN
The limiting frequency curve represents the results of a measurement that was We discussed the
repeated an infinite number of times. When carrying out an experiment we can only limiting
frequency repeat a measurement a finite number of times, so we have to be satisfied with curve in
Study making an estimate of the true value X of any measurement. Suppose some unknown quantity
x is measured N times and this measurement gives the values Unit 6.
𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁.
The value of N is not necessarily large (in the experiments that you are going to conduct, N will
typically lie between 5 and 10). The best estimate of the quantity that we are trying to measure is the
mean value 𝑥 which is given by equation (3.1)
2
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
or (3.5). One always has to keep in mind that this is not a precise determination of BEST
ESTIMATE the true value of the quantity. As consolation, we can expect the mean value 𝑥 to
approach the true value X as N increases. This implies that the best estimate of the
mean of an infinitely repeated measurement is given by the mean of a finite number of
measurements that are made in the actual experiment.
9.2 BEST ESTIMATE OF PRECISION: ADJUSTED STANDARD DEVIATION
In the discussion of the limiting frequency distribution in Study Unit 6, we saw that at a certain point
the frequency distribution takes on a fixed shape that will not change no matter how many more
measurements we make. This means that there is a limit to the precision of a measurement. We have
to decide on the best estimate LIMIT TO of this limiting number that can be calculated from a finite
number of PRECISION measurements.
In Study Unit 7 we evaluated the precision of a measurement that was repeated N times, by first
calculating the mean 𝑥, then the deviation 𝛿𝑖 of each measurement from the mean and finally the
standard deviation 𝜎. The mean approaches the true value X as N increases. There is, however, a flaw
in this argument which, although unimportant for large values of N, becomes significant for small
numbers of measurements.
If we take only one measurement, the value of N = 1, 𝑥 = 𝑥1, 𝛿1 = 0 and 𝜎 = 0. This implies that a
single measurement is the best estimate of the true value. However, as we saw when we initially
compared John’s and Thabo’s first measurements, one measurement does not provide us with
enough information to evaluate its precision. We need at least two independent measurements of
the same variable to enable us to make an estimate of the precision. In general, if N measurements
𝑥1, 𝑥2, … , 𝑥𝑁 are available, there are N ways of choosing 𝑥𝑖 from among them, which leaves (N−1)
measurements for choosing a second value 𝑥𝑗.
To provide for this bias, we define the adjusted standard deviation s by ADJUSTED
STANDARD
2
= 𝑁 𝜎2 DEVIATION
𝑠
3
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
𝑁−1
𝑁 𝛿12 + 𝛿22 + ⋯ + 𝛿𝑁2
from equation (7.4)
𝛿12 + 𝛿22 + ⋯ + 𝛿𝑁2
= .
𝑁−1
Thus
1
=1
𝑠=[ ]
𝑁−1 (9.1)
When comparing equations (7.4) and (9.1), we see that equation (9.1) differs from the mean value is
equation (7.4) in that (N − 1) replaces N. This is known as Bessel’s correction, and
to the true value. If
we make use of it when N is small. It becomes insignificant when N is large. The the frequency
distribution has a
adjusted standard deviation s is almost equal to 𝜎 for large values of N, but gives a
more reasonable interpretation of an experiment involving very few narrow
measurements. We shall therefore adopt s as the best estimate of 𝜎, the standard symmetrical peak,
deviation. This means that s gives the best estimate of the precision of the and therefore a
measurement. small 𝜎, the
measurements will
be close together
9.3 ACCURACY OF THE MEAN VALUE: STANDARD ERROR IN THE and their mean will
MEAN be near the true
value X. If the peak
is broad, the
Let us take a closer look at the N measurements of the unknown quantity measurements will
mentioned above. The best estimate of the true value X of the quantity we are be more widely
trying to measure is the mean value 𝑥 of the N measurements. What we need to do scattered and 𝑥 is
now is to find a way to evaluate how good an estimate 𝑥 is of X, that is how close less likely to be
4
PHY1509 PHYSICS SECTION: STANDARD ERROR
TOPIC 2
INTERPRETATION OF RESULTS
An experiment is a question which science poses to Nature, and a measurement is the recording of
Nature's answer.
Max Planck
I believe in evidence. I believe in observation, measurement, and reasoning, confirmed by
independent observers. I’ll believe anything, no matter how wild and ridiculous, if there is evidence
for it. The wilder and more ridiculous something is, however, the firmer and more solid the evidence
will have to be.
Isaac Asimov
INTRODUCTION
We have now said a lot about precision, the dispersion of measurements about a mean value and
the standard deviation from the mean. The next step is to find a way to estimate the accuracy of a
measurement. In this topic we will show how the best estimate of the precision and the accuracy of
a measurement can be calculated. Thinking back to the soft drink experiment, the following two
questions about errors were asked:
➢ What is the meaning of the errors in the measurements? ➢ How were the errors estimated?
In this topic we answer these questions and show how they apply to the experiments you are going to
conduct.
STUDY UNIT 9
STANDARD ERROR
1
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
Learning Objectives
After having completed this study unit, you should be able to:
1. define
(a) the adjusted standard deviation from the mean s (b) the standard error in the mean 𝜎𝑚 (c)
the adjusted standard error in the mean 𝑠𝑚
2. indicate when and how to apply Bessel's correction 3. calculate the
(a) adjusted standard deviation from the mean s
(b) the standard error in the mean 𝜎𝑚
(c) the adjusted standard error in the mean 𝑠𝑚 of an experimental result
9.1 BEST ESTIMATE OF THE TRUE VALUE: THE MEAN
The limiting frequency curve represents the results of a measurement that was We discussed the
repeated an infinite number of times. When carrying out an experiment we can only limiting
frequency repeat a measurement a finite number of times, so we have to be satisfied with curve in
Study making an estimate of the true value X of any measurement. Suppose some unknown quantity
x is measured N times and this measurement gives the values Unit 6.
𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑁.
The value of N is not necessarily large (in the experiments that you are going to conduct, N will
typically lie between 5 and 10). The best estimate of the quantity that we are trying to measure is the
mean value 𝑥 which is given by equation (3.1)
2
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
or (3.5). One always has to keep in mind that this is not a precise determination of BEST
ESTIMATE the true value of the quantity. As consolation, we can expect the mean value 𝑥 to
approach the true value X as N increases. This implies that the best estimate of the
mean of an infinitely repeated measurement is given by the mean of a finite number of
measurements that are made in the actual experiment.
9.2 BEST ESTIMATE OF PRECISION: ADJUSTED STANDARD DEVIATION
In the discussion of the limiting frequency distribution in Study Unit 6, we saw that at a certain point
the frequency distribution takes on a fixed shape that will not change no matter how many more
measurements we make. This means that there is a limit to the precision of a measurement. We have
to decide on the best estimate LIMIT TO of this limiting number that can be calculated from a finite
number of PRECISION measurements.
In Study Unit 7 we evaluated the precision of a measurement that was repeated N times, by first
calculating the mean 𝑥, then the deviation 𝛿𝑖 of each measurement from the mean and finally the
standard deviation 𝜎. The mean approaches the true value X as N increases. There is, however, a flaw
in this argument which, although unimportant for large values of N, becomes significant for small
numbers of measurements.
If we take only one measurement, the value of N = 1, 𝑥 = 𝑥1, 𝛿1 = 0 and 𝜎 = 0. This implies that a
single measurement is the best estimate of the true value. However, as we saw when we initially
compared John’s and Thabo’s first measurements, one measurement does not provide us with
enough information to evaluate its precision. We need at least two independent measurements of
the same variable to enable us to make an estimate of the precision. In general, if N measurements
𝑥1, 𝑥2, … , 𝑥𝑁 are available, there are N ways of choosing 𝑥𝑖 from among them, which leaves (N−1)
measurements for choosing a second value 𝑥𝑗.
To provide for this bias, we define the adjusted standard deviation s by ADJUSTED
STANDARD
2
= 𝑁 𝜎2 DEVIATION
𝑠
3
, lOMoAR cPSD| 30970899
PHY1509 PHYSICS SECTION: STANDARD ERROR
𝑁−1
𝑁 𝛿12 + 𝛿22 + ⋯ + 𝛿𝑁2
from equation (7.4)
𝛿12 + 𝛿22 + ⋯ + 𝛿𝑁2
= .
𝑁−1
Thus
1
=1
𝑠=[ ]
𝑁−1 (9.1)
When comparing equations (7.4) and (9.1), we see that equation (9.1) differs from the mean value is
equation (7.4) in that (N − 1) replaces N. This is known as Bessel’s correction, and
to the true value. If
we make use of it when N is small. It becomes insignificant when N is large. The the frequency
distribution has a
adjusted standard deviation s is almost equal to 𝜎 for large values of N, but gives a
more reasonable interpretation of an experiment involving very few narrow
measurements. We shall therefore adopt s as the best estimate of 𝜎, the standard symmetrical peak,
deviation. This means that s gives the best estimate of the precision of the and therefore a
measurement. small 𝜎, the
measurements will
be close together
9.3 ACCURACY OF THE MEAN VALUE: STANDARD ERROR IN THE and their mean will
MEAN be near the true
value X. If the peak
is broad, the
Let us take a closer look at the N measurements of the unknown quantity measurements will
mentioned above. The best estimate of the true value X of the quantity we are be more widely
trying to measure is the mean value 𝑥 of the N measurements. What we need to do scattered and 𝑥 is
now is to find a way to evaluate how good an estimate 𝑥 is of X, that is how close less likely to be
4
