Data and error analysis PDF

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17 May, 2021

Data and Error Analysis Purpose: To estimate the uncertainty associated with the measurement of the physical quantities involved in the experiments using simple ideas.

Theory: This lab teaches us how to determine the density of blocks by measuring their length, width, height and mass using data and error analysis. An error here is the uncertainty that occurs in all the measurements regardless of accuracy or precision. For example, while I was measuring the block using a ruler, the block may measure 5.85 cm or 5.86 cm. This means there is a range of possible values for my measurement where the correct value lies. This doubt results to an uncertainty of the measurement. As a result, we can write this as 5.85 cm ± 0.01 cm. To get an accurate answer, it is important to take into account each uncertainty of each physical quantity. The following equations below show uncertainties in length, width, height and mass of the blocks: • Length = Lbest ± δL • Width = Wbest ± δW • Height = Hbest ± δH • Mass = Mbest ± δM

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The following equations below show the fractional uncertainty of length, width, height, mass and density of the blocks: • Fractional Uncertainty in length = δL ————————Lbest

• Fractional Uncertainty in width = δW ————————Wbest

• Fractional Uncertainty in height = δH ————————Hbest

• Fractional Uncertainty in mass =

δM ————————Mbest

• Fractional Uncertainty in Density = Massbest ————————- x √ (δL/Lbest)2 + (δW/Wbest)2 + (δH/Hbest)2 + (δM/Mbest)2 Lbest x Wbest x Hbest

Fractional uncertainty is used when we want to compare the precision of different measurements. It is calculated by the uncertainty of the measurement (for example ± 0.01cm from the above example) divided by the measurement of the physical quantity (for example the length of the block from the above example 5.85cm). This equation is used to determine the uncertainty of length, width, height and mass of the block. Fractional Uncertainty in Density is calculated differently. To calculate density we divide Mass by volume. Volume is calculated by multiplying

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the length, width and height of the block. Therefore from the above equation, we dived the best Mass by the Volume of the block (length x width x height). To obtain the fractional uncertainty we multiply this value with square root of each fractional uncertainty in length, width, height and mass and square each of the values as shown above in the equation. The following equation below shows the standard deviation formula:

σ3 = √ ∑(ρ - ρ1)2 / N - 1

Standard deviation is used to tell how measurements in a data set are spread out from the average value. It shows the variability in data. A low standard deviation would mean that the values are close to the mean. Whereas a high standard deviation would mean that the values are spread out. The above equation is used to calculate the standard deviation. It is calculated by subtracting the sum of each block’s density value from the average standard deviation. N here means 6 - 1 since we have 6 blocks.

Apparatus: • Six rectangular plastic blocks, • A millimeter ruler and • Digital scale

17 May, 2021 Data table:

Table 1.1: Data collection

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17 May, 2021 Calculations:

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Graph: Density can be determined from the slope of the line. This can be obtained by using two points on the best fit line and calculating the change in mass divided by the change in volume.

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Results: • Average density from the data table = 0.9166 gram/cm3 • Standard deviation of density = 0.0228 gram/cm3 • Density from the graph = 0.9535 ram/cm3 • Uncertainty in density: I.

Block 1: 0.9128 ± 0.0121

II. Block 2: 0.8869 ± 0.0082 III. Block 3: 0.9270 ± 0.0165 IV. Block 4: 0.9259 ± 0.0147 V. Block 5: 0.9418 ± 0.0160 VI. Block 6: 0.9055 ± 0.0079

Discussion: The slope of the graph of Mass/Volume shows the rate of change of how much mass changes in comparison to volume. The graph shows a straight line. This is because both the variables mass and volume are proportional. The graph has a linear line because for every unit that mass increases, the volume increases as well. Both these variables have a fixed relationship. In this lab, I learnt how to determine the density of the blocks and estimating uncertainty associated with the measurements of length, width, height and mass. I obtained six values of density of six blocks. These values are shown in the data table. Based on these six values of density, I can see that these values are close to each other. Using these values, we were able to calculate the average density and the standard deviation. The average density from the data table is 0.9166

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gram/cm3 and the density from the slope of the straight line is 0.9535 gram/cm3. Both these methods have produced close results.

Uncertainty is an estimation of error present in data. All measurements have some degree of uncertainty because of systematic error or random error. No matter how careful we are with measuring, there can always be uncertainty regardless of accuracy and precision. It is important to take into account the uncertainty component while measurement of physical quantities. It helps make the data results stronger. This experiment has helped me understand the importance of uncertainty while taking measurements. In future experiments, I can use the concept of data and error analysis and estimate the uncertainty while taking measurements. Taking the source of the error into account will help me not further magnify that error....