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Accuracy
Accuracy describes the agreement between the determined value and the true value.
Example: The true value for the mass of a cube of iron is known to be 7.90 g.
You weigh the same cube of iron and determine it has mass of 7.90 g.
The determined value and the true value are the same, so we can say that we have accurately determined the mass of the iron cube.
Example: You weigh the same cube of iron on a different electronic balance, and this time you determine the mass of the iron cube to be 5.78 g.
The determined value is very different to the true value so we cannot say that we have accurately determined the mass of the iron cube.
We say that the determined value for the mass of the iron cube is inaccurate.
When talking about the accuracy of a measurement, Chemists like to know just how accurate the measurement is.
For this reason, we calculate the percentage relative error as shown below:
percentage relative error 
= 
true value  determined value true value 
× 100 
For an accurate measurement, the pecentage relative error will be very low, close to 0 %
For an inaccurate measurement, the percentage relative error will be high, for example, greater than 5 %.
Example: The true value for the mass of a cube of iron is known to be 7.90 g.
The iron cube is weighed on 4 different electronic balances, and the percentage relative error is calculated for each measurement as shown below:
Balance 
True Value / g 
Determined value / g 
percentage relative error =
true value  determined value true value 
× 100 

1. 
7.90 
7.90 
7.90  7.90 7.90 
× 100 
= 0.00 % 

2. 
7.90 
5.78 
7.90  5.78 7.90 
× 100 
= 26.8 % 

3. 
7.90 
7.62 
7.90  7.62 7.90 
× 100 
= 3.54 % 

4. 
7.90 
7.58 
7.90  7.58 7.90 
× 100 
= 4.05 % 

When balance 1. is used, we can record our determined value as 7.90 g with a percentage relative error of 0.00 %.
When balance 2. is used, we can record our determined value as 5.78 g with a percentage relative error of 26.8 %.
If we know the tolerance of a true measurement, we can decide that the determined value is accurate if it lies within the tolerance levels of the true measurement, and, inaccurate if it lies outside the tolerance levels of the true value.
For example, the cube of iron has a true value of 7.90 ± 0.01 g the the true value lies between 7.90  0.01 = 7.89 g and 7.90 + 0.01 = 7.90 g.
An accurately determined value for the mass of iron would be between 7.89 g and 7.91 g.
An inaccurately determined value for the mass of iron would be less than 7.89 g or greater than 7.91 g.
In order to determine how accurate a measurement is, we need to know the true value.
Often we don't know the true value, so we can't determine how accurate our results are.
Precision
Because it is very rare for Chemists to know how accurate a measurement is, they make a number of measurements under the same conditions until they arrive at a set of measurements that are in good agreement with each other.^{(1)}
The reproducibility of a measurement is known as precision.
When all the measurements are very similar, we say the determined value is known precisely.
If we cannot get measurements that are very similar, we cannot say the value is known precisely, instead we say the measurements are imprecise.
We can make generalisations about the precision of a set of measurements by calculating the range of the experimentally determined values:
range of values = largest value  smallest value
A set of measurements can be described as precise if the range of values is very small, that is, range is close to 0
A set of measurements will be described as imprecise if the range of values is large, that is, range is not close to 0.
Example: The mass of the iron cube was measured 3 times in three separate trials using Balance 1 above.
The results are shown below:
 Trial 1  Trial 2  Trial 3 
Mass /g  7.90  6.14  7.39 

There is a great variety in the values of the mass of the iron cube determined by Balance 1.
range of values = largest value  smallest value
range of values for Balance 1. = 7.90  6.14 = 1.76
1.76 >> 0
Balance 1. is giving imprecise values for the mass of the iron cube.

Using Balance 1. to measure mass does not give reproducible results.
Balance 1. does not give precise measurements of mass.
(Balance 1. should NOT be used until a technician has fixed it!)
Example: The mass of the iron cube was measured 3 times in three separate trials using Balance 2 above.
The results are shown below:
 Trial 1  Trial 2  Trial 3 
Mass /g  5.78  5.77  5.79 

The values of the mass of the iron cube determined by Balance 2. are all very similar.
range of values = largest value  smallest value
range of values for Balance 2. = 5.79  5.77 = 0.02
0.02 ≈ 0
Balance 2. is giving precise values for the mass of the iron cube.

Using Balance 2. to measure mass does give reproducible results.
Balance 2. does give precise measurements of mass.
Examples of Accuracy and Precision
Accuracy describes the agreement between the determined value and the true value.
Precision describes the reproducibility of a measurement.
Therefore measurements can be described as one of the following:
 accurate and precise
 accurate but imprecise
 inaccurate but precise
 inaccurate and imprecise
Example 1. The temperature of a laboratory is known to be a constant 25.23°C.
A student uses an alcohol thermometer to measure the temperature of the laboratory three times and records the temperatures in the table below:
 Trial 1  Trial 2  Trial 3 
Recorded Temperature / °C  23.88  23.91  23.24 
How would you describe the accuracy and precision of the student's recorded temperatures?
Solution:
 Deciding if the Student's results are accurate.
To determine accuracy, we need to compare each of the student's results to the true value given as 25.23^{o}C.
Given that we have a number of different measurements using the same thermometer by the student in the same laboratory, we could start by averaging the student's results:
average recorded temperature = 
23.88 + 23.91 + 23.24 3 
= 
71.03 3 
= 23.68 
Next, we can use this average value to calculate the percentage relative error by allowing the average recorded temperature to be the determined value:
percentage relative error = 
true value  determined value true value 
× 100 
percentage relative error = 
25.23  23.68 25.23 
× 100 
percentage relative error = 
6.14 % 

A determined value is accurate if the percentage relative error is low, close to 0 %.
The percentage relative error in the student's results is NOT close to 0 %.
The student's temperature readings are NOT accurate, they are inaccurate.
 Deciding if the Student's results are precise.
A set of measurements are precise if all the measurements are very similar, that is, if there is a small range of values.
Find the range of the student's thermometer readings:
highest recorded temperature is 23.91°C
lowest recorded temperature is 23.24°C
range of values = highest value  lowest value = 23.91  23.24 = 0.67
The range of values is closer to 1 than it is to 0 so we would decide that the student's measurements are NOT precise, they are imprecise.
 Conclusion: the student's temperature readings are neither accurate nor precise.
The student's temperature readings are both inaccurate and imprecise.
Example 2. The volume of water in a volumetric flask is known to be 50.00 ± 0.06 mL at 20°C and 101.3 kPa.
A student uses a 100 mL measuring cylinder to measure the volume of water in the flask three times.
The student's results are recorded in the table below:
 Trial 1  Trial 2  Trial 3 
Recorded Volume / mL  49.84  49.92  49.91 
How would you describe the accuracy and precision of the student's recorded volumes of water?
Solution:
 Deciding if the Student's results are accurate.
To determine accuracy, we need to compare each of the student's results to the true value given as 50.00 ± 0.06 mL
Given that we have a number of different measurements using the same measuring cylinder by the same student in the same laboratory, we could start by averaging the student's results:
average recorded water volume = 
49.84 + 49.92 + 49.91 3 
= 
149.67 3 
= 49.89 mL 
The true value given for the volume of water in the flask is 50.00 ± 0.06 mL.
This means that the true value for the volume of water in the volumetric flask could be as low as 50.00  0.06 = 49.94 mL or it could be as high as 50.00 + 0.06 = 50.06 mL.
In order for the student's measurements to be considered accurate, the determined value needs to be between 49.94 mL and 50.06 mL.
The determined value is the average of the volume measurements recorded by the student which is 49.89 mL.
The student's determined value is below the lowest possible true value for an accurate measurement.
49.89 mL < 49.94
determined value < lowest limit for the true value
The student's volume measurements are NOT accurate, they are inaccurate.
 Deciding if the Student's results are precise.
A set of measurements are precise if all the measurements are very similar, that is, if there is a small range of values.
Find the range of the student's volume measurements:
highest recorded volume is 49.92 mL
lowest recorded volume is 49.84 mL
range of values = highest value  lowest value = 49.92  49.84 = 0.08
The range of values is close to 0 so we would decide that the student's measurements ARE precise.
 Conclusion: the student's volume measurements are not accurate but they are precise.
The student's volume measurements are inaccurate and precise.