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Absolute error is a measure of the difference between a measured value and the true value of a quantity. It is a fundamental concept in mathematics and is used in a variety of fields, including engineering, physics, and finance. In this article, we will define absolute error, and provide five examples of its use.

Definition:

Absolute error is the absolute difference between the measured value of a quantity and its true value. It is represented by the symbol “|E|” and is calculated using the following formula:

|E| = |measured value – true value|

For example, if the true value of a quantity is 10 and the measured value is 12, the absolute error would be |12-10| = 2.

It’s important to note that absolute error does not take into account the direction of the error (i.e. whether the measured value is greater or less than the true value). This is in contrast to relative error, which does take the direction of the error into account.

Examples:

1. Measuring the volume of a liquid: Suppose you are tasked with measuring the volume of a liquid in a graduated cylinder. You read the scale and determine that the volume is 25 mL. However, due to the inherent imprecision of the instrument and your own ability to read the scale accurately, the true volume of the liquid may be slightly different. If the true volume is actually 26 mL, the absolute error in your measurement would be |25-26| = 1 mL.
2. Calculating the distance between two points: Suppose you are using a map to calculate the distance between two points. You measure the distance on the map using a ruler, and determine that it is 10 cm. However, due to the scale of the map and the inherent imprecision of the ruler, the true distance may be slightly different. If the true distance is actually 11 cm, the absolute error in your measurement would be |10-11| = 1 cm.
3. Estimating the weight of an object: Suppose you are trying to estimate the weight of an object using a bathroom scale. You step on the scale and read the display, which shows a weight of 150 lbs. However, due to the imprecision of the scale and your own ability to read the display accurately, the true weight of the object may be slightly different. If the true weight is actually 151 lbs, the absolute error in your measurement would be |150-151| = 1 lb.
4. Measuring the temperature of a substance: Suppose you are trying to measure the temperature of a substance using a thermometer. You read the thermometer and determine that the temperature is 100°F. However, due to the imprecision of the thermometer and your own ability to read it accurately, the true temperature of the substance may be slightly different. If the true temperature is actually 101°F, the absolute error in your measurement would be |100-101| = 1°F.
5. Calculating the slope of a line: Suppose you are trying to calculate the slope of a line using the formula “m = (y2-y1)/(x2-x1)”. You measure the coordinates of the two points and plug them into the formula, and determine that the slope is 2. However, due to the imprecision of your measurements and the inherent imprecision of the formula, the true slope may be slightly different. If the true slope is actually 2.1, the absolute error in your calculation would be |2-2.1| = 0.