where
J is the mean of your measurements and each
represents an individual measurement.
Don’t worry about the details of this formula, you can quickly use the Excel function =STDEV()
to calculate
for you. But it is worth pausing now to take a look at what Excel is actually
calculating when you use =STDEV(). The term
G
J
K
is the difference between an individual
measurement and the mean. We then square that term so positive and negative deviations
from the mean are treated the same. We add up the
G
J
K
deviations for each individual
measurement, and then divide by the total number of measurements to get an “average”
deviation. (Note that we divide by
instead of
for complicated reasons that are best
discussed in a statistics class.) Finally, we take the square root to undo the fact that we squared
the
G
J
K
term. Thus, the standard deviation is a measurement of the average amount
that the data deviates from the mean value.
To recap: If your data fluctuates between different trials, take the mean of the data and
calculate the error on the mean. For data that fluctuates, the error on the mean gives you a
better estimate of your precision than the instrumental errors on individual data points. For
fluctuating data, the error on the mean is almost always larger than the instrumental error, and
is thus a more conservative way to measure your experimental uncertainty.
Note: If you take significantly fewer than 10 trials, the error on the mean becomes less
meaningful. Ideally, you should always perform at least 10 trials in any experiment to quantify
your random error. If for some reason you only took a very small data sample (fewer than 10
trials), then a “quick and dirty” way to estimate your error is to use the half-range formula:
G
K
The numerator represents the full range of your data. When you divide by 2, you get “half the
range”. This will give you an idea of the uncertainty in your measurements; however, running
10 or more trials and calculating the error on the mean is much more rigorous.
4.3 Errors in Parameters of Fitted Functions
If you have created a scatter plot with your data and are fitting a function to it, the parameters
of the fitted function will all have errors. In some cases, LoggerPro is able to calculate the
errors in the fit parameters. Please consult your instructor during lab to verify whether or not
LoggerPro is properly estimating fit parameter errors.
In the cases where LoggerPro does not accurately estimate fit parameter errors, you can use
Excel to calculate the errors for you. Excel can only do this for a linear function (i.e., a straight-
line function). The process of using Excel to calculate errors in a linear fit is described below.
The Excel function LINEST (“line statistics”) is able to calculate the errors in the slope and y-
intercept of a linear function of the form
. To do so, follow the directions below: