Download Estimating Uncertainties worksheet File

Dealing with Uncertainties:
Means, Standard Deviation and Standard Error
Measured values are always subject to Error, or Uncertainty. To work with errors we can use some
statistical techniques; these will help us understand how confident we can be in our data.
1.
The mean is widely used and easily calculated. It is simply the sum of all the readings
divided by the number of readings. It is sometimes just called the average; but mean is a
better term as there are different types of average. (Note – the word mean has other
meanings in English as well, so watch out for this)
The mean is often given the symbol µ (Greek letter “mu”). This is used in Loughborough
University. 𝑥̅ and <x> may also be seen in books.
Mathematically,
𝑛
1
𝜇 = ∑ 𝑥𝑖
𝑛
𝑖=1
2. The Standard Deviation is also useful; for a large population of readings, 68% of the readings
would be expected to lie within one standard deviation of the mean. This is also called the
RMS or Root Mean Square deviation; it is calculated by
 finding the difference (deviation) from the mean of each reading:
= Deviation
 squaring each deviation
= Square Deviation
 finding the total of these square values
 dividing this by the number of values
= Mean Square Deviation
 taking the root
= Root Mean Square Deviation
This is generally given the symbol σn (“Sigma n”), and is properly called the “population
standard deviation”.
In symbols this is
1 n
xi   2
n 

n i 1
3. Slightly more useful is the Sample Standard Deviation
σn-1 which is almost identical but you
divide by n-1 not n. This is used in Loughborough University, and is the one given on the
formula sheet, as
𝑛
𝜎𝑛−1
1
= √
∑(𝑥𝑖 − 𝜇)2
𝑛−1
𝑖=1
Finally, to get to standard error, which has the symbol s , simply divide the sample standard
deviation by root of n, i.e.
𝑆 =
𝜎𝑛−1
√𝑛
Quote your answer to an appropriate number of decimal places - look at the data you have
to get an idea. Normally only one or two significant figures can be justified.
How do I use these?
It’s easier than it sounds, but it does need practice. You can do them manually or use the built in
statistical functions on your calculator. These are very good but you look up how to do it!
Your calculator will offer you xσn and xσn-1. Choose xσn-1 as it is the sample standard deviation.
Practice:
For these numbers:
11.47, 11.31, 11.12, 11.06 and 11.10.
Calculate the




Mean,
Population Sample Deviation
Sample Standard Deviation and
Standard Error
Try it out manually (draw a table), then try to get the same results on your calculator. You should
get:
Mean = 11.212 (round to 11.21)
Population standard Deviation:
Sample Standard Deviation = 0.173; round to 0.17
Standard Error = 0.07755 – round to 0.08.
So the best answer is 11.21 ± 0.08. It is meaningless to add more decimal places.
deviation
deviation
Number (number minus
squared
mean)
11.47
11.31
11.12
11.06
11.1
total
0.258
0.098
-0.092
-0.152
-0.112
0.066564
0.009604
0.008464
0.023104
0.012544
56.06
There are 5 values so n = 5
mean
11.212
sum of squared deviations
0.12028
sum of squared deviations / (n-1)
0.03007
root
0.173407
this is the Sample Standard Deviation
=
0.17 (two 2 s.f.)
=
0.08 (rounded)
divide this by root n to get standard error: (s)
0.07755