Uncertainty and Error in Measurement

Let’s Think… You measure the temperature of a glass of water 10 times, using 10 different thermometers. Results in Celsius : 19.9, 20.3, 20.2, 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2 How would you accurately report the final temperature?

Let’s Think… Answer the following questions related to the data collected. 1.Is it reasonable to assume that the actual temperature is about 20ᵒC? Why or why not? 2. If not, what would the reasonable result be? Actual result…  20.1  /- 0.2 ᵒC This result is both reasonable And precise!

Precision Refers to how close several experimental measurements of the same quantity are to each other In previous example: – If one student took all the temp. readings, they would be said to be repeatable, whereas if several students had taken the readings they would be said to be reproducible.

Accuracy How close a precise reading is to the true reading (literature value) – If all thermometers are reading 20ᵒC when the true value of the temp is 19ᵒC, then they are giving precise but inaccurate readings – This is due to systematic errors Due to the apparatus itself, or due to the way in which the readings are taken

Systematic error

Random Error Occur if there is an equal probability of the reading being high or low from one measurement to the next Might include variations in the volume of the glassware due to temp fluctuations, or the decision on ten an indicator changes color during and acid base titration

Significant Figures Significant figures are critical when reporting scientific data because they give the reader an idea of how well you could actually measure/report your data.

Significant Figures Four Main Rules 1)All non zero numbers are ALWAYS significant ex) 125, 34, 971 2)Leading zeros are NOT significant. They indicate the position of the decimal ex) , 0.01,

3) Zeros that are sandwiched between non zero numbers are ALWAYS significant. ex) 103, 4009, ) Trailing zeros at the right end of a number are significant ONLY if the number contains a decimal point ex) 100, 20 (1 sig fig) ex) 100., 20.0 (three sig figs)

Practice... NumberAmount of Sig FigsRule 48, , , , ,3,4 3,000,00011, ,4

Addition and Subtraction When adding or subtracting, it is the number of decimal places that is important. When using a balance that measures +/-0.01 g, the answer can be quoted to two decimal places, which may increase or decrease the number of significant figures.

Addition and Subtraction Example: 7.10 g+3.10 g=10.20 g 3 sig figs3 sig figs4 sig figs g+5.16 g=7.20 g 4 sig figs4 sig figs3 sig figs

Multiplication and Division When multiplying or dividing, it is the number of significant figures that is important. The number with the fewest significant figures used in the calculation determines how many significant figures should be used when writing the answer

Multiplication and Division Examples 4.56x1.4=6.38  sig figs2 sig figs3 sig figs2 sig figs 3.217x2.64=  sig figs3 sig figs6 sig figs 3 sig figs

Practice… 1) = 3.6 2)32.0 – 27.0 = 5.0 3)( ) – 3.9 = 4.6 4)6.2 x 7 = 43.4  40 5)46.12 / 13 = 3.54  3.5 6)(2.3 x 4.00) / 1.7 = 5.41

Uncertainties It the degree of tolerance is not stated on an apparatus, you will have to estimate Degree of tolerance, is where the manufacturer lists the uncertainty of the apparatus

If you are making a single measurement with a piece of apparatus, the absolute uncertainty & the percentage uncertainty can both be easily stated. For example, Consider measuring 25.0 cm 3 with a 25 cm 3 pipette that measures to +/- 0.1 cm 3. The absolute uncertainty is 0.1 cm 3 and the percentage uncertainty is equal to… 0.1 / 25.0 x 100 = 0.4%