Accuracy and Precision
How to Calculate Precision
Four Methods:
Precision means that a measurement using a particular tool or implement produces similar results every single time it is used. For example, if you step on a scale five times in a row, a precise scale would give you the same weight each time. In math and science, calculating precision is essential to determine if your tools and measurements work well enough to get good data. You can report precision of any data set using the range of values, the average deviation, or the standard deviation.
Steps
Calculating the Range

Determine the highest measured value.It helps to begin by sorting your data in numerical order, from lowest to highest. This will ensure that you do not miss any values. Then select the value at the end of the list.
 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.

Find the lowest measured value.Once your data has been sorted, finding the lowest value is as simple as looking at the beginning of the list.
 For the scale measurement data, the lowest value is 11.

Subtract the lowest value from the highest.The range of a set of data is the difference between the highest and lowest measurements. Just subtract one from the other. Algebraically, the range can be expressed as:
 Average deviation=Σx−μn{\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}}}
 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}}{n1}}}}
 Notice that the only difference is in the denominator of the fraction. For an entire population, you will divide byn{\displaystyle n}. For a sample set, you will divide byn−1{\displaystyle n1}.

Find the mean of the data values.As with calculating the average deviation, you will begin by finding the mean of the data values.
 Using the same set of measurements as above, the mean is 12.4.

Find the square of each variation.For each data point, subtract the data value from the mean, and square that result. Because you are squaring these variations, whether the difference is positive or negative does not matter. The square of the difference will always be positive.
 For the five data values in this sample, these calculations are as follows:
 (12−12.4)2=(−0.4)2=0.16{\displaystyle (1212.4)^{2}=(0.4)^{2}=0.16}
 (11−12.4)2=(−1.4)2=1.96{\displaystyle (1112.4)^{2}=(1.4)^{2}=1.96}
 (14−12.4)2=1.62=2.56{\displaystyle (1412.4)^{2}=1.6^{2}=2.56}
 (13−12.4)2=0.62=0.36{\displaystyle (1312.4)^{2}=0.6^{2}=0.36}
 (12−12.4)2=(−0.4)2=0.16{\displaystyle (1212.4)^{2}=(0.4)^{2}=0.16}
 For the five data values in this sample, these calculations are as follows:

Calculate the sum of the squared differences.The numerator of the standard deviation fraction is the sum of the squared differences between each value and the mean. To find this sum, add together the figures from the previous calculation.
 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}
 For the sample data set, these are:

Divide by the data size.This is the one step that will differ for either a population calculation or a sample set calculation. For a full population, you will divide byn{\displaystyle n}, the number of values. For a sample set, you will divide byn−1{\displaystyle n1}.
 This example has only five measurements and is therefore only a sample set. Thus, for the five values being used, divide by (51) or 4. The result is5.2/4=1.3{\displaystyle 5.2/4=1.3}.

Find the square root of the result.At this point, the calculation represents what is called the variance of the data set. The standard deviation is the square root of the variance. Use a calculator to find the square root, and the result is the standard deviation.
 σ=1.3=1.14{\displaystyle \sigma ={\sqrt {1.3}}=1.14}

Report your result.Using this calculation, the precision of the scale can be represented by giving the mean, plus or minus the standard deviation. For this data, this will be 12.4±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.
Deciding How to Report Precision

Use the word precision correctly.Precision is a term that describes the level of repeatability of measurements. When collecting a group of data, either by measurement or through an experiment of some kind, the precision describes how close together the results of each measurement or experiment are going to be.
 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.

Choose the best measure of precision.The word “precision” does not have a single meaning. You can represent precision using several different measurements. You need to decide the best one.
 Range. For small data sets with about ten or fewer measurements, the range of values is a good measure of precision.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.
 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.

Report your results clearly.Very often, investigators will report data by giving the mean of the measured value, followed by a statement of the precision. The precision is shown with a “±” symbol. This provides an indication of precision, but it does not clearly explain to the reader if the number following the “±” symbol is a range, standard deviation, or some other measurement. To be very clear, you should define what measure of precision you are using, either in a footnote or parenthetical note.
 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.”
Community Q&A

QuestionHow do you measure accuracy?wikiHow ContributorCommunity AnswerAccuracy is a measure of how close you are to the known, expected value of what you are measuring. If you have a known weight of 10 kg, for example, and you put it on a scale and the scale says "9.2," then your scale is accurate within 0.8 kg.Thanks!

QuestionHow do I calculate the level of precision of an equipment? It's an electrolyte analyserwikiHow ContributorCommunity AnswerUse it to take several measurements and then follow the directions in this article.Thanks!

QuestionHow do you know if a measurement is precise?wikiHow ContributorCommunity AnswerWhen the mean absolute deviation or the standard range is as close to zero as possible.Thanks!

QuestionIf the examples are like these 1s, 2s, 2s, 1s, 0s, 3s, 2s, 2s, 1s, 1s and average is 1.5, so is this precise?wikiHow ContributorCommunity AnswerThere is never a direct answer to the question "is this precise?" The question is, "Is it precise enough?" Your data has a range of plus or minus 3 seconds, an average deviation of 0.7 seconds. Do you think that is precise enough for what you're using the data for?Thanks!

QuestionHow do I find the accuracy and precision of the number 750.5?wikiHow ContributorCommunity AnswerYour accuracy is plusorminus your rounding factor, which is 0.01. There is no precision involved, because you do not have replicates if this number.Thanks!

QuestionWhat's the conclusion made after calculating the standard deviation as a way of determining the precision?wikiHow ContributorCommunity AnswerEvery single measurement you make could be ± 0.88lbs different from what you see.Thanks!

QuestionWhat is the precision of 14050?wikiHow ContributorCommunity AnswerThis is a use of the word "precise" that really means how specific the measurement is. It's like the old "rounding off" that you probably learned in third grade. This number is rounded to the nearest 10, so that is its level of precision.Thanks!

QuestionWhat is the more precise measurement from the pair 54.1cm and 54.16?wikiHow ContributorCommunity AnswerThis is a different use of the word "precise" than used in this article. For your question, "precise" means the smallest measure possible, so 54.16 is a more precise measure than 54.1.Thanks!

Question51.03 is more accurate than 51.032. True or false?Top AnswererIf you're comparing 51.03 to 51.032, the latter has been measured to the nearest onethousandth of a unit, while the former has been measured only to the nearest onehundredth of a unit. So 51.032 is more accurate.Thanks!
 If one of your trial values is much higher or lower than the rest of your values, do not exclude this number from your calculations. Even if it was a mistake, it is data and should be utilized for a proper calculation.
 In this article, only five values were used for mathematical simplicity. In an actual experiment, you should perform more than five trials to achieve a more accurate calculation. The more trials you run, the closer you will get to a clear precision value.
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