Chemistry 141

Standard Deviation: Measurement of Volumes Delivered




Because it is an experimental science, chemistry will always involve the taking of measurements.  These measurements may be multiple measurements of the same object, measurements of a single by multiple observers, or one observer’s measurements of many different objects.  What is the interpretation of each set of measurements?  Are the measurements giving the correct answer (ie. are they accurate).  Are they consistent (ie. precise)?  One thing scientists must bear in mind is that no matter how much care is taken when measuring; uncertainty in measurement is always present. 


The rules that apply to significant figures are basically an elementary form of error analysis.  For the most part, following the rules that govern significant figures are sufficient when measurements are of a single, or at the most a duplicate trial.  However, there may be times when we have the opportunity to carry out more exacting experimentation, where uncertainties in measurements can be estimated quite accurately.


Suppose you want to determine the percent chlorine in a sodium chloride sample.  Your balance is good to four significant figures and five observations were taken.  Your results are as follows:


Table 1

Trial Number

Percent Chlorine












Some questions may come to mind.


·      Why didn’t you get the same values each trial?

·      What should you report for the percent chlorine?



Hopefully, the following discussion will help you answer these questions.





Errors in Observational Data


Before we begin to analyze measured results we must first consider how errors occur in the laboratory.  Here we must acknowledge the different types of error that exist. 


Random error

The reality is, no matter how carefully we take a series of measurements, there is always a certain amount of “scatter” in the data.  This scatter is due to the inability of an instrument or an observer of that instrument to discriminate between readings differing by less than some small amount.  For example in filling a graduated cylinder to the 10.0 mL mark, sometimes you will slightly overfill and other times you will slightly under fill the cylinder or when performing a titration you will titrate each sample to a slightly different endpoint.  These will all be good data points but they will differ slightly due to random or indeterminate errors.  If you average the data points, this random error is represented by the deviation of each individual measurement from the mean.  Random errors are impossible to eliminate, but because they randomly fluctuate around the measurement the average value should reflect a good value for the measurement in the absence of systematic error (see below). 


Systematic error

Systematic error (sometimes referred to as determinate error) is error that is introduced in the lab.  Generally such errors can be avoided or corrected for.  Examples of systematic errors are calibration errors, uncompensated instrumental drift, leakage of material (e.g. gas in a pressure system), incomplete fulfillment of assumed conditions for a measurement (e.g. incomplete reaction in a calorimeter or incomplete drying of a weighed precipitate), or some consistent operational error (parallax, uncompensated human reaction times).  A systematic error is one for which we can definitely predict the direction of its change from the correct value, while random errors can produce a result which is either too high or too low. When determinate errors are made, accuracy rarely matches precision.  Therefore, it is important to calibrate correctly, read instruments correctly, etc. 



Accuracy and Precision


When we speak about experimental values we always want to know the quality of our data.  What exactly do we mean by that and how can we determine the size of our random and systematic errors?  To answer this question we must first define precision and accuracy.


Precision of measurement refers to its reproducibility.  It is a measure of the amount of random error.  If we were to take many measurements how close would they be to each other.  Precision is frequently described by percent deviation. 


% deviation =


Precision is also defined by standard deviation (see below).


Unfortunately, precision cannot give us much information regarding the accuracy of a measurement as may be seen in the figure below:  Although both sets of arrows represent a set of precisely thrown arrows, only picture A would be considered accurate.


So let’s define accuracy-


Accuracy of a measurement tells us how closely a value we measure agrees with what is believed to be true value.  Accuracy is frequently described as a percent difference or percent error between the measured value and the actual value. 


            % error =


The difficulty with determining the error for a measured (or calculated) value is that it is often difficult or impossible to determine the true value.  For measurements taken in the undergraduate laboratory we frequently compare our results with generally accepted results published in the literature as our known or true value.  In such cases we will calculate our percent error.


So how can we perform experiments and get the best results possible?

1.     Perform experiments as precisely as possible to minimize random error.

2.     Analyze each of the measurements to identify possible sources of systematic error and minimize them.

3.     Determine the result using several entirely independent methods of measurement and compare.  If these independent methods give the same final result it is a good indication of accuracy.


What does all this mean and how do we apply it in our laboratory? 


Consider the data acquired for the percent chlorine in the sodium chloride sample (Table 1).  The arithmetic mean (the average) for the data provided is found to be 60.50.



If the true value of the percent chlorine is known to be 60.66%, then we can calculate the absolute error for each measurement and the deviation of each measurement from the average as shown below:


Absolute error             = |(true result – observed result)|

                                                = |(60.66% - 60.50%)|

                                                = 0.16% for trial 1


Deviation        = |(average result – observed result)|

                                    = |(60.50% – 60.50%)|

                                    = 0.00% for trial 1


Table 2


Observed Value




























Note that in this data the average deviation, a measure of the precision, is less than the average error.  This indicates that there is some systematic error in the experiment resulting in poor data.  This example demonstrates how a student might be misled into believing their data was accurate based on good precision.  Unfortunately systematic error can not be described using any clean mathematical theories so it is often ignored although it is frequently found to be orders of magnitude larger than the random errors.  In fact, many published papers have later been shown to be incorrect by amounts far greater that the claimed limits of error.



Given the fact that it is very difficult to identify and quantify all sources of systematic error we frequently fall back on the statistical methods of analyzing random error to get an indication of the error in our results.  An explanation of how we treat random errors follows.


Treatment of Random Errors


Although random errors cannot be corrected for, they can be treated statistically in an attempt to establish reliability of the measurement.  The analysis is based on the “normal” error curve (Figure 1).  The curve shows relative frequency of deviation we can expect to find if a large number of measurements are made.  The curve allows us to make determinations about the magnitude of indeterminate errors. 


Figure 1


                                                                                       (a)                                (b)


We can interpret some main points about these curves.

·      The curves are symmetric about the midpoint, which is the arithmetic mean.  Therefore positive and negative deviations are equally likely.

·      The curves rise to a maximum at the midpoint, telling us small deviations occur more often than large deviations.  In fact, if you observe a large deviation a determinate error is most likely involved.

·      The shape of the curve is dependant on inherent precision of the measurements.  Sloppy or crude instrumentation give a high frequency of large deviations (curve b).  Refined measurements with improved precision show large deviations to be improbable (curve ).


Standard Deviation


There is a quantity which is used in common by virtually all scientists and statisticians to measure the size of the random error in a set of data.  This measurement is called the standard deviation statistical significance.  The standard deviation tells us about the shape of the error curve that is associated with a set of experimental measurements.  In order to measure the exact, theoretical value of the standard deviation of a measurement, one must make the measurement many times.  Therefore, the equation of standard deviation below, technically, is only an approximation of the true standard deviation of a particular measurement.  This estimation of the standard deviation is calculated from the relationship:



Here, “σ” represents the standard deviation, d represents the individual deviation and n is the number of trials.


Let’s look at the percent chlorine data in table 2 and compare the standard deviation.


Standard deviation =


Standard deviation tells a lot about the error curve.  A small value for sigma corresponds to a sharp, steeply rising curve, where deviations are close to zero.  On the flip side, a broad, squat curve says large deviations are highly probable.


If we look at the shaded area bound by x = - σ and x = σ, (Figure 2), we see this area is proportional to the probability of observation with a deviation within one unit of σ of the arithmetic mean (located at the midpoint of the curve).   This shaded area represents about 2/3 of the total area, or more exactly, 68.4 % of the total area under the curve. 







Figure 2


What this means to us, is if we make a large number of measurements, we can expect about 2/3 to fall within M – σ to M + σ, or, M ± σ.  Where M is the arithmetic mean and σ is the standard deviation.  Therefore, ~1/3 of the trials would fall outside of these boundaries and hence, would show larger deviations.  Actually, ± 2σ covers most of the area under the curve, about 95%.  This means that 95% of our trials fall within ± 2σ from the arithmetic mean, M - 2σ to M + 2σ or, M ± 2σ.  This leaves about 5% of the measurements out of this range.  In other words, about 1/20 have a deviation of greater than ± 2σ.  Keep in mind the probability of where a measurement falls is valid when dealing with a large number of observations.


For a more complete explanation of this idea view one of the following links




Error analysis will be used to examine the precision and accuracy of useful laboratory equipment.  You will take three sets of measurements regarding volumetric equipment routinely used in the lab.  These pieces of equipment are: the beaker and the graduated cylinder.  In all three sets of data you will determine the volume of the container by weighing the amount of water held by that container and then using the density of water to calculate the volume.  Error analysis will also be used to determine the density of a metal slug using three different methods.  Before proceeding, it would judicious to set up your data table in your lab notebook.  Remember, data is to be recorded directly into your lab notebook, in ink!

Part I - Beaker Calibration

This part of the lab will require a 100 or 150 mL beaker that has a calibration line at the 50 mL point.  (If your beaker is not marked in this way, exchange it with the stockroom for a beaker that is calibrated at the 50 mL point.)  Clean the beaker; making sure the outside is dry before carrying out the following steps.  Record your data in the data table you have prepared in your lab notebook.

a.                    Fill the beaker to the 50 mL mark as carefully as you can.  Adjust the level by using an eyedropper if necessary.

b.                   Weigh the beaker with the water on a balance sensitive to at least .01 grams.  Do not use an analytical balance and don’t forget to zero the balance before you start.   Use the same balance for all your measurements in this set.

c.                    Pour the water out of the beaker until it has stopped draining, and reweigh the now empty beaker.  Do not dry out any remaining drops before this weighing!  This will ensure you are measuring the volume delivered by the beaker, not how much it actually holds.

d.                   Repeat steps a-c two more times, for a total of three trials.

e.                    Using subtraction, determine the mass of the water contained in your beaker for the three individual measurements.

Part II - Graduated Cylinder Calibration


For calibration of the graduated cylinder you will need your 10 mL graduated cylinder.  You will also need a plastic bottle and lid, capable of holding at least 60 mL.  The plastic bottles will be provided for you.  You will also use a balance that is sensitive to at least .01 grams.  Your instructor will tell you which balance type to use.  Wash the graduated cylinder with soap and water and thoroughly rinse with deionized water.  (Remember, dirty glassware is evident by water droplets sticking to the sides.)  Do not use heat to dry your graduated cylinder!  (Why should this be avoided?)  Do not dry the graduated cylinder; just let all the water drain out of it before you begin.


a.     Weigh the plastic bottle and its lid to a precision of at least 0.01 g on the appropriate balance.  This bottle should be empty but it does not have to be dry on the inside. Record the weight in the data table that you have prepared for part II.


b.     Fill a clean 250 mL beaker with about 200 mL of deionized water.  Record the temperature of the water.  Use this water for the ten fillings of the graduated cylinder. 


c.     Fill the graduated cylinder to the 10.0 mL mark.  Make sure the bottom of the meniscus is just about even with the 10.0 mL mark on the cylinder.  You may find it helpful to use an eyedropper to make the final adjustment.


d.     Now pour the contents of the cylinder into the plastic bottle.  Allow about 10 seconds for the cylinder to completely drain into the bottle.  Place the lid on the plastic bottle, weigh it and record the weight directly into your data table.


e.     Again, as in part (c), transfer another 10.0 mL of the D.I. water into the same sample bottle and reweigh it.  Do not empty the bottle from the first transfer.  Just keep adding the water to the bottle; it will be fairly full at the end of this exercise.


f.      Repeat step (c) until you have placed a total of six samples of water into the plastic bottle.  This means you will have a total of 7 measurements.


g.     Add another column on the right of your data table and fill it with the mass of the water delivered in each transfer.  You will need this value later when interpreting your results.





For each of the following, determine the mean and standard deviation for the masses of water contained in the beaker and the graduated cylinder.  Show your calculations.


            Part I – Beaker Calibration

a.     Calculate the mean and standard deviation for the mass of water contained in the beaker.

b.     Assuming the density of water is 1.0 g/mL, the volume in the beaker will have the same numerical value in mL as its mass in grams.  Based on your results, what is the most likely volume contained in your beaker when filled to the 50 ml mark?


Part II - Graduated Cylinder Calibration

a.     Calculate the mean and standard deviation for the mass of water contained in the graduated cylinder.

b.     Determine the density of water used in your measurements from a table in the handbook of Chemistry and Physics.  Use this density to convert the mean value of the mass held in the cylinder and its standard deviation to the corresponding volumes.



·      Make a table that shows the mean volume for each device in Parts I, and II.  Show what limits (i.e., ± mL) you would expect about 95% of the measurements to fall.

·      For each piece of equipment in parts I, and II classify the precision and accuracy of the volume contained or delivered as good, fair or poor.  Briefly explain why you rated them the way you did.



Masterton and Slowinski: Elementary mathematical preparation for General Chemistry. Saunders Publishing, 1974

Shoemaker, Garland, and Steinfeld. Experiments in Physical Chemistry. McGraw Hill, 1974.

Kratchvil, H.: Chemical Analysis.  Barnes and Noble, 1969

Handout: Experiment E1A and E1B, Grossmont College, 2002


Check for some interesting overheads regarding statistical data treatment.