# Uncertainty and Error

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## Contents

##### Uncertainty and Error

In 1st year Lab courses you will encounter the difficult task of error analysis. Here is a summary of all the different ways of calculating it, from the simple to the complex.

Whenever you make a measurement, there is an error in your measurement. Error here does not mean that your measurement is wrong, but that you are **uncertain** about its exact value. So, another term for "error" is the measurement's "uncertainty". When you report the mass to be 5.2kg +/- 0.5kg, you are saying you are certain that the actual mass lies between 4.7kg and 5.7kg.

The notation used for an error of a value called **x** is **dx**. ie. If you are measuring a mass **m** of an object, its error would be **dm**.

### Measuring Errors

There are many ways to measure an uncertainty/error. The most common way is to take the error to be half of the smallest division of your measuring device. For example, if your metre stick has tick-marks every 0.01m, then your error is +/- 0.005m. The reasoning is that if, for example, you see that the length lies between the 0.75m and 0.76m tick marks, you cannot confidently say that the length is exactly 0.755m. You can't say it's 0.756m, 0.754m, or 0.758m either. It is good practice to take the middle number, 0.755m and report an error of half the smallest division: 0.005m. Now when you report the length to be 0.755m +/- 0.005m, you are saying the length is somewhere between 0.750m and 0.760m, which is what you observed.

### Absolute Errors

Absolute errors refer to the actual value of the error. In the above example, 0.005m was the absolute error of that 0.755m measurement. It gives the actual value of the error (as opposed to fractional error, below, which only give an error relative to the measurement). Note that absolute errors have units because they correspond to actual lengths.

### Percent Error, Fractional, or Relative Error

If you measured a value **x** and have an absolute error **dx**, then the fractional/percent error in your measurement is calculated by **dx/x**. Some Examples: - You measure a rod to be **1.03m +/- 0.05m**. The value is 1.03m and the error is 0.05m. So, the fractional error, **dx/x**, is 0.0485, or 4.85%. With proper rounding, the fractional error is 0.05 , or 5%. Note that there are **no units** given in fractional errors. - You measure that a laser is deflected by an angle **b = 15.3 degrees** with an error **db = 0.2 degrees**. The relative error, **db/b** would be 0.2/15.3 = 0.01, or 1%.

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