Percentage Error In Measurements
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The difference between two measurements is called a variation in the measurements. Another word for this variation - or uncertainty in measurement - is
Absolute Error Formula
"error." This "error" is not the same as a "mistake." It does percentage error definition not mean that you got the wrong answer. The error in measurement is a mathematical way to show the relative error uncertainty in the measurement. It is the difference between the result of the measurement and the true value of what you were measuring. The precision of a measuring instrument is determined by
Relative Error Formula
the smallest unit to which it can measure. The precision is said to be the same as the smallest fractional or decimal division on the scale of the measuring instrument. Ways of Expressing Error in Measurement: 1. Greatest Possible Error: Because no measurement is exact, measurements are always made to the "nearest something", whether it is stated or not. The greatest possible error
Types Of Errors In Measurement
when measuring is considered to be one half of that measuring unit. For example, you measure a length to be 3.4 cm. Since the measurement was made to the nearest tenth, the greatest possible error will be half of one tenth, or 0.05. 2. Tolerance intervals: Error in measurement may be represented by a tolerance interval (margin of error). Machines used in manufacturing often set tolerance intervals, or ranges in which product measurements will be tolerated or accepted before they are considered flawed. To determine the tolerance interval in a measurement, add and subtract one-half of the precision of the measuring instrument to the measurement. For example, if a measurement made with a metric ruler is 5.6 cm and the ruler has a precision of 0.1 cm, then the tolerance interval in this measurement is 5.6 0.05 cm, or from 5.55 cm to 5.65 cm. Any measurements within this range are "tolerated" or perceived as correct. Accuracy is a measure of how close the result of the measurement comes to the "true", "actual", or "accepted" value. (How close is your answer to the accepted value?)
of Accuracy Accuracy depends on the instrument you are measuring with. But as a general rule: The degree of accuracy is half a unit each side of the unit of measure Examples: When your instrument measures in "1"s then any value between relative error calculator 6½ and 7½ is measured as "7" When your instrument measures in "2"s then any
Absolute Error Calculator
value between 7 and 9 is measured as "8" Plus or Minus We can show the error using the "Plus or Minus" sign: absolute error example ± When the value could be between 6½ and 7½ 7 ±0.5 The error is ±0.5 When the value could be between 7 and 9 8 ±1 The error is ±1 Example: a fence is measured as 12.5 http://www.regentsprep.org/regents/math/algebra/am3/LError.htm meters long, accurate to 0.1 of a meter Accurate to 0.1 m means it could be up to 0.05 m either way: Length = 12.5 ±0.05 m So it could really be anywhere between 12.45 m and 12.55 m long. Absolute, Relative and Percentage Error The Absolute Error is the difference between the actual and measured value But ... when measuring we don't know the actual value! So we use the maximum possible error. In the example above https://www.mathsisfun.com/measure/error-measurement.html the Absolute Error is 0.05 m What happened to the ± ... ? Well, we just want the size (the absolute value) of the difference. The Relative Error is the Absolute Error divided by the actual measurement. We don't know the actual measurement, so the best we can do is use the measured value: Relative Error = Absolute Error Measured Value The Percentage Error is the Relative Error shown as a percentage (see Percentage Error). Let us see them in an example: Example: fence (continued) Length = 12.5 ±0.05 m So: Absolute Error = 0.05 m And: Relative Error = 0.05 m = 0.004 12.5 m And: Percentage Error = 0.4% More examples: Example: The thermometer measures to the nearest 2 degrees. The temperature was measured as 38° C The temperature could be up to 1° either side of 38° (i.e. between 37° and 39°) Temperature = 38 ±1° So: Absolute Error = 1° And: Relative Error = 1° = 0.0263... 38° And: Percentage Error = 2.63...% Example: You measure the plant to be 80 cm high (to the nearest cm) This means you could be up to 0.5 cm wrong (the plant could be between 79.5 and 80.5 cm high) Height = 80 ±0.5 cm So: Absolute Error = 0.5 cm And: Relative Error = 0.5 cm = 0.00625 80 cm And: Percentage
Example: I estimated 260 people, but 325 came. 260 − 325 = −65, ignore the "−" sign, so my error is 65 "Percentage Error": show the error as a percent of the exact value ... so divide by the exact value and make https://www.mathsisfun.com/numbers/percentage-error.html it a percentage: 65/325 = 0.2 = 20% Percentage Error is all about comparing a guess or estimate to an exact value. See percentage change, difference and error for other options. How to Calculate Here is the way to calculate a http://www.staff.vu.edu.au/mcaonline/units/percent/pererr.html percentage error: Step 1: Calculate the error (subtract one value form the other) ignore any minus sign. Step 2: Divide the error by the exact value (we get a decimal number) Step 3: Convert that to a percentage (by absolute error multiplying by 100 and adding a "%" sign) As A Formula This is the formula for "Percentage Error": |Approximate Value − Exact Value| × 100% |Exact Value| (The "|" symbols mean absolute value, so negatives become positive) Example: I thought 70 people would turn up to the concert, but in fact 80 did! |70 − 80| |80| × 100% = 10 80 × 100% = 12.5% I was in error by 12.5% Example: The report said the percentage error in carpark held 240 cars, but we counted only 200 parking spaces. |240 − 200| |200| × 100% = 40 200 × 100% = 20% The report had a 20% error. We can also use a theoretical value (when it is well known) instead of an exact value. Example: Sam does an experiment to find how long it takes an apple to drop 2 meters. The theoreticalvalue (using physics formulas)is 0.64 seconds. But Sam measures 0.62 seconds, which is an approximate value. |0.62 − 0.64| |0.64| × 100% = 0.02 0.64 × 100% = 3% (to nearest 1%) So Sam was only 3% off. Without "Absolute Value" We can also use the formula without "Absolute Value". This can give a positive or negative result, which may be useful to know. Approximate Value − Exact Value × 100% Exact Value Example: They forecast 20 mm of rain, but we really got 25 mm. 20 − 25 25 × 100% = −5 25 × 100% = −20% They were in error by −20% (their estimate was too low) InMeasurementMeasuring instruments are not exact! And we can use Percentage Error to estimate the possible error when measuring. Example: You measure the plant to be 80 cm high (to the nearest cm) This means you could be up to 0.5 cm wrong (the plant could be between 79.5 and 80.5 cm high) So your perce
found by measurement and the "true value' of the quantity. eg an object that has a mass of 120 g may be shown to weigh 130 g on an imperfect weighing machine. True weight: 120 g Measured weight: 130 g Error: +10 g Measurement errors arise because of inevitable imperfections in the measuring instrument and limitations of the human eye. Errors come in all sizes, and sometimes you need to decide if the error in your measurement is so big that it makes the measurement useless. (see examples below) Errors can be positive or negative. An electric current might be measured as Examples The effective size of the error depends on the actual size of the error the size of the measurement itself Example 1 Measuring a Line Actual length of line: 11 cm Length of line when measured: 12 cm Error is (Measured Length - Actual Length) Error is (12 cm - 11 cm) = 1 cm. The error expressed as a fraction of the actual size is Example 2 Measuring the height of a person Actual height is 1.72 cm = 1270 mm If the error in measurement is only 1 mm, then expressing this as a fraction of the actual size Have a Go Problem 1 Voltage is measured with a multimeter. A particular multimeter is being tested. True voltage of the multimeter: 224 V. Measured voltage: 220 V. Calculate the actual error and the percentage error. You will notice that in this example the error is a negative value Problem 2 Another multimeter is being tested. True voltage of the multimeter: 150 V Measured voltage: 153 V Calculate the actual error and the percentage error. In this case the error has a positive value. Practice Questions Question 1 Answer 1.3 hectares Question 2 answer + or - 0.2% Question 3 answer + or - 0.2 cm Question 4 answer + or - 32.2 sec Question 5 answer + or - 0.2% Question 6 answer + or - 1.2% Solution 1 Actual Error = Measured Voltage -.True Voltage = 220 - 224 V = (-) 4 V back to Have a Go Solution 2 Actual Error = Measured Voltage- True Voltage = 153-150 V = (+) 3 V The multimeter is slightly less accurate than the one in the