When a list of measurements of the same quantity are made the best estimate of that measurement is the average or mean of the list of measurements. i
N å
=
1 x i x best
= x avg
= x mean
= x
=
N
The standard deviation gives the average uncertainty of a single measurement within the set used to calculate the average/mean. This is also the uncertainty of a single measurement of the same quantity after the average/mean has been calculated. s =
1
N
-
1 i
N
-
1
( x i
x avg
)
2
The standard deviation of the mean gives the error on the best estimate or average/mean for a set of measurements of the same quantity. s mean
= s
N
A student measures the acceleration due to gravity 10 times and has the following data.
10.5, 11.2, 10.8, 11.8, 11.1, 10.9, 10.8, 11.0, 10.4, 10.6 (all in m/sec 2 )
Determine the average, standard deviation, and standard deviation of the mean.
Average: g avg
=
N å i
=
1 g i
N
=
10.5
+
11.2
+
10.8
+
11.8
+
11.1
+
10.9
+
10.8
+
11.0
+
10.4
+
10.6
10
=
10.9
m sec
2

Standard Deviation: s =
1
N
-
1 i
N å
=
1
( g i
g avg
)
2 =
1
9
[(10.5
-
10.91)
2
+
(10.9
-
10.91)
2
+
(11.2
-
10.91)
+
(10.8
-
10.91)
2
2 +
(10.8
-
10.91)
+
(11.0
-
10.91)
2
2 +
(11.8
-
10.91)
+
(10.4
-
10.91)
2
2 +
(11.1
-
10.91)
+
(10.6
-
10.91)
2
]
2 s =
0.4
m sec
2
Standard Deviation of the Mean: s mean
= s
N
=
0.4
10
=
0.1
m sec
2
Final answer in Standard Form: g = 10.9 + 0.1 m/sec 2
Determine the Discrepancy from the known value for the acceleration due to gravity (9.8 m/sec 2 ).
Discrepancy
= g measured
g known
=
1.1
m sec
2
Is there a possibility for Systematic Errors to play a role in the above measurement?
Since the discrepancy is much larger than the uncertainty (Standard Deviation of the Mean) in our average measurement of g, and the fact that all of our measurements are higher than the known value of g, there is evidence to suggest that some type of Systematic Error has occurred.
Almost always, when making many measurements of the same quantity using the same method for each measurement, and the error in each measurement is random/uncorrelated, the resulting set of measurements will occur according to what we call the Normal or Gaussian distribution. This is certainly the case for the experiments in the undergraduate laboratory.
For example, if we made 500 measurements of the acceleration due to gravity and create a histogram of the data we might get the following:
100
80
60
40
20
7 8 9 10 11 12

The data is centered on the accepted value of gravity (9.8 m/sec 2 ) but many measurements have been made and some fall above and some fall below the accepted value. We can do the statistics on this data set to find the average value, the standard deviation, and the standard deviation of the mean: g avg
= 9.76 m/sec 2  = 1.02 m/sec 2  mean
= 0.05 m/sec 2
We see that the discrepancy between our measured average value and the accepted value of g (|g avg
– g known
| =
0.04 m/sec 2 ) is within the error on the mean (Standard Deviation of the Mean). As we have stated before, this tells us that we can be reasonably assured that the measured average is reliable.
Notice the general structure of the histogram above. If we were to take many more (infinitely more) measurements we would obtain a smooth curve histogram that looks something like the one below.
0.20
0.15
0.10
0.05
10 15
This is known as the Normal Distribution and represents the distribution of an infinite number of measurements of a quantity where the peak of the distribution is centered on the actual value of that quantity. In this case
9.8m/sec 2 .
The shaded region in the graph above actually represents the probability of a measurement being made between any two points within the distribution. This distribution is characterized by the following function called the
Gaussian or Normal Function:
G
X , s
( x )
=
1 s
2 p e
-
( x
-
X )
2
/ 2 s 2
Where X is the true value of our measured quantity, and  is the standard deviation.
We don’t actually know the “true value” of any measured quantity so X is some abstract “true value”. The best estimate of the “true value” is the average/mean of the measurements we have made of a specific quantity, thus we replace the X with x avg
and the equation becomes:
G x , s
( x )
=
1 s
2 p e
-
( x
x )/ 2 s 2
To find the probability of making a measurement between two values in the distribution is found by taking the integral of the Gaussian/Normal Function and evaluate between any two values of the distribution. A natural value to look at would be the standard deviation from the mean ( x
± s
) . This integral is as follows:

Pr ob ( x
± s
)
=
x
+ s x
s
1 s
2 p e
-
( x
x )/ 2 s 2 dx
This integral can is usually done on a computer or calculator but we have a nice way of finding the probability of measuring a value between and factor of the standard deviation (t  ) in Appendix A in your textbook on page 287.
When evaluating this integral we get a probability of 68%. This means that there is a 68% chance of measuring a value between ( x
s
) and ( x
+ s
) .
The area under the curve that is shaded in yellow is the probability of a measurement being made between and ( x
+ s
)
( x
s
)
. This means there is a 68% chance an additional measurement will be made in this region.
Suppose you make 100 measurements of the acceleration due to gravity on the moon where the accepted value is
1.624 m/sec 2 and find an average of 1.54 m/sec 2 with a standard deviation of 1.0 m/sec 2 . What is the Standard
Deviation of the Mean? Is the average value found within the discrepancy of the accepted value?
Standard Deviation of the Mean: s mean
= s
N
=
1.0
100
=
0.10
m / sec 2
Discrepancy:
Discrepancy
= x avg
x accepted
=
0.08
If we compare the discrepancy with the average value along with the standard deviation of the mean we see that the discrepancy is less than the average error on the mean and our average is consistent with the accepted value.
Now lets look at some probabilities…
1. If we were to make another measurement what is the probability that it would be within 0.50
 (the green shaded region below) of the average value above?
The diagram to the left shows the area under the curve
(probability) for the regions between ½ and 1 standard
 1/2  is shaded in green, while the 1  is the green and yellow regions together.

Using Appendix A from your textbook, we find the first decimal place of the multiplier (t) in the left hand column and move to the right to the next decimal place to the 0.00. Here we have found the multiplier t = 0.50 and the number at that intersection is the probability of finding a measurement within that fraction of 
We find that the probability of making a measurement within half of the standard deviation is 38.29%. Compare this with the probability of making a measurement within 1 standard deviation (68.27%).
Using Appendix A in our textbook allows us to find the probability of making a measurement within any multiple of the standard deviation up to 5.0
 without actually doing the integral for the probability. With some manipulation we can find any probability we could want to find.
2. What is the probability that we would make a measurement outside of 0.50
 ?
We know that the probability of making a measurement within 0.50
 so the probability must be 100% - 38.29% =
61.79%. There is a 100% chance we will make some measurement, and if we subtract the probability that the measurement we make will be within 0.50
 the result is the probability of making a measurement outside of
0.50

3. What is the probability that a measurement above 0.50
 will be made?
From the previous question we know that there is a 61.79% chance that a measurement will be outside of 0.50

The symmetry of the Gaussian Distribution dictates that half of the measurements outside of 0.50
 will be above 0.50
 and the other half will be below. So we divide the probability by 2 and we get: 30.90% probability that the measurement we make will be above 0.50
 Conversely, there is a 30.90% probability the measurement we make will be below 0.50

4. What is the probability that a measurement will be made: ( x
+
0.50
s
)
£ x measured
£
( x
+ s
) such that it falls above 0.50
 and below  as seen in the diagram below. The region shaded in red represents the probability we are concerned with.

We know that the probability of making a measurement within 0.50
 is 38.29% and the probability of making a measurement within  is 68.27%. By subtracting:
68.27% - 38.29% = 29.98% we obtain the probability of the yellow and red regions in the figure above. To simply eliminate the yellow region and find the probability of the red region only, we can divide by 2:
29.98%
=
14.99%
.
2
So the probability of making a measurement: ( x
+
0.50
s
)
£ x measured
£
( x
+ s
) is almost 15%. Again due to the symmetry of the Gaussian Distribution, we simultaneously know the probability of the yellow region above.