Uncertainty+and+error+in+measurement

// This part deals mostly with experimental practices and making measurements. It is particularly useful for your practical work and lab reports but is also assessed in papers 1 and 2. //

There are two types of errors that can occur when doing an experiment.
 * Uncertainty and error in measurement **

These occur due to variations in the working condition or the experimenter’s performance. Supposing you are measuring temperature and there is a wind blowing, thus affecting the temperature readings- so every time you measure the temperature you get a different value like once you get 20 o C, another time 21 o C, and other times 25 o C and 29 o C. //These random errors affect the **precision** of the experiment. // However, repetition can be used to reduce the effect of random errors. Repeat the experiment several times and calculate an average of the value you get.
 * //__Random uncertainties/errors __//**

These occur due to some problems with the system that can cause the values you get to differ from the actual value. One simple example can be when you use a ruler with a broken end to measure lengths. Supposing the ruler starts from the 2 cm mark. So everything you measure will be 2 cm bigger than its actual length, if you forget to deduct the 2cm from the values. Thus you have a systematic error. //Systematic errors affect the **accuracy** of the experiment. //
 * //__Systematic errors __//**

Precision refers to the agreement between the values you get. For example if you titrate an acid and a base and do the experiment 3 times and get the value of acid required to neutralise the base to be 30 cm 3 , 31 cm 3  and 29 cm 3 , then your results are precise; they are close to each other. Thus there are lower chances of you having random errors in your experiment. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">However, if you had 25 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">, 31 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> and 28 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">, these values do not agree with each other (they aren’t close to each other). Thus you can see that the readings in this case are no precise.
 * //<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">PRECISION //**

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Accuracy refers to how close your values are to the theoretical value. For example if you are titrating an acid with a base in order to find the concentration of the acid and you get 1.2 mol dm -3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> while the actual concentration is 1.3 mol dm -3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">, then your experiment is very accurate since the value you got is very close to the actual/theoretical value. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">__//Therefore, if your experiment is accurate it means that the systematic errors are less//__.
 * //<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">ACCURACY //**

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">As we saw earlier the readings 30 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">, 31 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> and 29 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> quite agree with each other, they are within a range of ±1 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> from each other. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> Let’s take an example: Whenever you are taking a reading, let’s say from a thermometer you can’t say that you know exactly where the graduation lies for example when measuring temperature, the liquid may lie between two graduation marks. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> In the figure it is between 29 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> and 30 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">. So you take the reading to be 30 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">. But the reading isn’t exactly 30 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">, it is slightly lower. Your reading is accurate to 1 o <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">C. Thus your reading can be expressed as 30 ± 1 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> (you reading can lie anywhere in that range from 29 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> to 31 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">). Therefore whenever you take a reading with that thermometer you have to add a ±1 cm 3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> to show the range since your reading isn’t exact, it can be higher or lower.
 * Uncertainty range **

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Another very important thing when making measurements and calculations is to consider the number of significant figures. You know that any number from 1 to 9 is significant so if you have 99 there are 2 significant figures, if you have 1123478911 there are 10 significant figures. So it is basically a matter of counting how many digits the number contains However, finding the number of significant figures can become when there are zeros in the number. Some simple zeros can help in such cases. Basically, zeros are significant when they come after a non-zero digit. Look at these examples: <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">00153.5 – the number has 4 significant figures. The two zeros are not significant since they come before a non-zero digit. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">0.01535 – again this is 4 significant figures. The zero is not significant. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">1.01535 – this number has 6 significant figures. The zero is now a significant digit since it comes after a non-zero digit. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">153.50 – in this case again the zero is significant since it comes after a number. So the number has 5 significant figures. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">However, sometimes zeros before the decimal point may not be significant. For example 153000, the zeros here may/may not be significant. This is because we don’t know if the figure is exactly 153000 or rounded off. To solve such problems the number is expressed in the scientific notation. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Thus if you have 1.53000 X 10 5 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> then the value has 6 significant figures <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">But if you had 1.53 X 10 5 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> then the zeros aren’t significant so the value has only 3 significant figures
 * Significant figures **

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">When making calculations such as addition/subtraction/multiplication/division it is important to know how many significant figures you answer should have: <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">For example if you add: 0.234 + 1.9 + 6.75 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">If you add these in the calculator you get 8.884, but should you leave the answer as 8.884??? **//NO//**. It is wrong- you have to give your answer as 8.9 why?? (Read on J <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">)
 * Significant figures in calculated results **

//<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">The rule is simple: //

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Let’s look at the previous example: in the previous example of adding 0.234 + 1.9 + 6.75, you can see that <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 0.234 has 3 decimal places <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 1.9 has 1 decimal places <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 6.75 has 2 decimal places <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">The number with the lowest number of decimal places is 1.9, so your answer should also have only 1 d.p (decimal place). So you give your answer as 8.9
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">When adding/subtracting numbers, your answer should have the same number of decimal places as that value in your calculation that has the lowest number of decimal places (confused??? What did I just say. It doesn’t make sense).

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">For example: 2.0 X 965 X 2.3, the calculator answer is 4439 but you can’t leave the answer with 4 significant figures. Look at the figures in the calculation <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 2.0 has 2 significant figures <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 965 has 3 significant figures <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">- 2.3 has 2 significant figures <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">The lowest number of significant figures is two. So your answer should also have two significant figures. So your answer should be 4400.
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">When multiplying/dividing numbers your answer should have the same number of significant figures as that value in your calculation that has the lowest number of significant figures.

//<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">(These two rules might be difficult to understand at first, but keep in mind that from now on whenever you make a calculation you have to bear these in mind and make sure you give your answer to an appropriate number of significant figures or decimal places.) //

By the end of this lesson you should be able to: media type="googleplusone" key="" width="360" height="18" media type="facebooklike" key="http%3A%2F%2Fibchem4u.wikispaces.com%2FUncertainty%20and%20error%20in%20measurement" width="360" height="74" **<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">NEXT UP- UNCERTAINTIES IN CALCULATED RESULTS **
 * Describe what random and systematic uncertainties are
 * Distinguish between precision and accuracy
 * State uncertainties as an uncertainty range
 * State the results of your calculations to an appropriate number of significant figures