Accuracy, Precision, Average Deviation, Error

          

Mathematics deals with pure numbers, and pure numbers are precise, or exact.  In science, getting precise numbers is difficult unless, for example, you divide a number by 2, which is exact, or multiply by a whole number.  However, most data is not exact.  It might be possible to get a precise value for the number of people in a room, bit if there are 50 people moving about it might be difficult.  You might count them 3 or 4 times and get a different value each time.  Your value might not be very accurate or precise.  In science, there is often a wide range of acceptable values, and few numbers are exact.

 

Accuracy is a measure of how close a number is to the actual value (how far from the actual value a particular measurement falls).  In many experiments, accuracy cannot be determined; only an estimate of the accuracy can be found, because the actual value is not known.  Precision, on the other hand, refers to how close repeated measurements are to each other (a measure of the variation present in a set of readings).

 see drawing on board here

Although standard deviation is a more accurate method of finding the error margin we will use the average deviation method because it is relatively easy to calculate.

To Find Average Deviation (precision of measurement)

1. Find the average value of your measurements.
2. Find the difference between your first value and the average value.  This is called the deviation.
3. Take the absolute value of this deviation.
4. Repeat steps 2 and 3 for your other values.
5. Find the average of the deviations.  This is the average deviation

 The average deviation is an estimate of how far off the actual values are from the average value, assuming that your measuring device is accurate.  You can use this as the estimated error.  Sometimes it is given as a number (numerical form) or as a percentage.

 To Find Percent Error

1. Divide the average deviation by the average value.
2. Multiply this value by 100.
3. Add the % symbol.

Sample Problem: 

A man wants to see if his car gets the number of miles per gallon (mpg) claimed by the dealer.  he took data for five fill ups and found that he got 27, 33, 28, and 32 mpg.  The car manufacturer stated that he should get 32 mpg.  Find if the stated mpg agrees with the advertised value.

Solution:  The average value is found to be 31 mpg.  (27 + 33 + 35 + 28 + 32) / 5 = 31 mpg.  Precision is found to be plus or minus 3, because (4 + 2 + 4 + 3 + 1) / 5 = 2.8, which is rounded to 3, as this is just an estimate of how far off the answer is.  This give a value of 31, plus or minus 3 mpg – or between 28 and 34 miles per gallon.  It shows the car is doing OK.

 

Average Deviation Problems

 

1.  On five different tankfuls of gas a pickup got 12, 15, 16, 12, and 15 mpg.  Find the average mpg of this truck.

2.  Find a value that gives an estimate of the error in gas mileage of the truck in problem 1.

3.  Find the percent error for mileage given the numerical error that you found in problem 2.

 4.  Suppose that instead of using a truck, the person in the problem had used a car that got 39, 45, 47, and 41 mpg on four tanks of gasoline.  Find the average value for the mpg, and estimate how much range in value is expected.

5.  Find the estimated percent error in problem 4.

6.  You walk down a football field and find that in 100 yards, you have taken 150 steps.  You then measure a distance by walking 240 steps.  How far have you gone?

 

Once the estimated error has been found for a measurement two rules can be used in the calculation of an error estimate when doing mathematical operations:

1. When adding or subtracting, add the numerical error.
2. When multiplying or dividing, add the percent error.