For example, suppose you are testing the precision of a scale, and you observe five measurements: 11, 13, 12, 14, 12. After sorting, these values are listed as 11, 12, 12, 13, 14. The highest measurement is 14.
For the scale measurement data, the lowest value is 11.
Range=x(max)−x(min){\displaystyle {\text{Range}}=x(max)-x(min)} For the sample data, the range is: Range=x(max)−x(min)=14−11=3{\displaystyle {\text{Range}}=x(max)-x(min)=14-11=3}
The mean is not actually part of calculating the range or precision, but it is generally the primary calculation for reporting the measured value. The mean is found by adding up the sum of the measured values and then dividing by the number of items in the group. For this set of data, the mean is (11+13+12+14+12)/5=12. 4.
For this example, use the same sample data as before. Assume that five measurements have been taken, 11, 13, 12, 14, and 12. The mean of these values is (11+13+12+14+12)/5=12. 4.
Algebraically, the absolute value is shown by placing two vertical bars around the calculation, as follows: Absolute deviation=|x−μ|{\displaystyle {\text{Absolute deviation}}=|x-\mu |} For this calculation, x{\displaystyle x} represents each of the experimental values, and μ{\displaystyle \mu } is the calculated mean. For the values of this sample data set, the absolute deviations are: |12−12. 4|=0. 4{\displaystyle |12-12. 4|=0. 4} |11−12. 4|=1. 4{\displaystyle |11-12. 4|=1. 4} |14−12. 4|=1. 6{\displaystyle |14-12. 4|=1. 6} |13−12. 4|=0. 6{\displaystyle |13-12. 4|=0. 6} |12−12. 4|=0. 4{\displaystyle |12-12. 4|=0. 4}
Average deviation=Σ|x−μ|n{\displaystyle {\text{Average deviation}}={\frac {\Sigma |x-\mu |}{n}}} For this sample data, the calculation is: Average deviation=0. 4+1. 4+1. 6+0. 6+0. 45{\displaystyle {\text{Average deviation}}={\frac {0. 4+1. 4+1. 6+0. 6+0. 4}{5}}} Average deviation=4. 45{\displaystyle {\text{Average deviation}}={\frac {4. 4}{5}}} Average deviation=0. 88{\displaystyle {\text{Average deviation}}=0. 88}
Your data represents an entire population if you have collected all the measurements possible from all possible subjects. For example, if you are conducting tests on people with some very rare disease, and you believe that you have tested everyone with that disease, then you have the entire population. The standard deviation formula in this case is: σ=Σ(x−μ)2n{\displaystyle \sigma ={\sqrt {\frac {\Sigma (x-\mu )^{2}}{n}}}} A sample set is any group of data less than an entire population. This is actually going to be used more often. The standard deviation formula for a sample set is: σ=Σ(x−μ)2n−1{\displaystyle \sigma ={\sqrt {\frac {\Sigma (x-\mu )^{2}}{n-1}}}} Notice that the only difference is in the denominator of the fraction. For an entire population, you will divide by n{\displaystyle n}. For a sample set, you will divide by n−1{\displaystyle n-1}.
Using the same set of measurements as above, the mean is 12. 4.
For the five data values in this sample, these calculations are as follows: (12−12. 4)2=(−0. 4)2=0. 16{\displaystyle (12-12. 4)^{2}=(-0. 4)^{2}=0. 16} (11−12. 4)2=(−1. 4)2=1. 96{\displaystyle (11-12. 4)^{2}=(-1. 4)^{2}=1. 96} (14−12. 4)2=1. 62=2. 56{\displaystyle (14-12. 4)^{2}=1. 6^{2}=2. 56} (13−12. 4)2=0. 62=0. 36{\displaystyle (13-12. 4)^{2}=0. 6^{2}=0. 36} (12−12. 4)2=(−0. 4)2=0. 16{\displaystyle (12-12. 4)^{2}=(-0. 4)^{2}=0. 16}
For the sample data set, these are: 0. 16+1. 96+2. 56+0. 36+0. 16=5. 2{\displaystyle 0. 16+1. 96+2. 56+0. 36+0. 16=5. 2}
This example has only five measurements and is therefore only a sample set. Thus, for the five values being used, divide by (5-1) or 4. The result is 5. 2/4=1. 3{\displaystyle 5. 2/4=1. 3}.
σ=1. 3=1. 14{\displaystyle \sigma ={\sqrt {1. 3}}=1. 14}
The standard deviation is perhaps the most common measurement of precision. Nevertheless, for clarity, it is still a good idea to use a footnote or parentheses to note that the precision value represents the standard deviation.
Precision is not the same as accuracy. Accuracy measures how close experimental values come to the true or theoretical value, while precision measures how close the measured values are to each other. It is possible for data to be accurate but not precise or to be precise but not accurate. Accurate measurements are close to the target value but may not be close to each other. Precise measurements are close to each other, whether or not they are close to the target.
Range. For small data sets with about ten or fewer measurements, the range of values is a good measure of precision. [13] X Research source This is particularly true if the values appear reasonably closely grouped. If you see one or two values that appear far from the others, you may wish to use a different calculation. Average deviation. The average deviation is a more accurate measure of precision for a small set of data values. [14] X Research source Standard deviation. The standard deviation is perhaps the most recognized measure of precision. Standard deviation may be used to calculate the precision of measurements for an entire population or a sample of the population. [15] X Research source
For example, for one series of data, the result could be reported as 12. 4±3. However, a more explanatory way to report the same data would be to say “Mean=12. 4, Range=3. ”
