Lab report rewrite

Introduction

This experiment goes through the statistical methods of a given data which will be used to analyze the experiment. These statistical methods are used to calculate important values such as: the mean, median, mode, stander deviation, and confidence level of the mean. Other than that one will be able eliminate a measurement based on Chauvenant’s criterionIn this experiment the data specified with 50 cylinders with unknown material. Each cylinder has it own mass, diameter, and height. Based on that one will be able to apply all of the statistical methods required in this lab. The determination of the nominal density, and uncertainty of density of cylinder an unknown material is also required in this lab.

Experimental Methods

Since the experiment was theoretical, the values of 50 cylinder was given to the student. Most of the calculations and findings was done using Excel.

1- From the given data the histogram was plotted for mass, diameter and height.

2- After that the centural tendency and dispersion was measured.

3- Assuming the distribution was Gaussian, using the Chuvenat’s criterion the data was studied to check if any of the measurement could be discarded.

4- In addition, 90% and 98% confidence interval was found for each of the mean values.

5- From the available information volume and density was both found.

6- Step 2 and 4 was done for the volume and dencity except 95% confidence interval.

7- An equaiton was driven and uncertainty was calculated for volume and density based on 95% confidence level, and compared to the previous step.

8- The mass as a function of volume was plotted and the least fit square curve was shown.

= for the mean

for Standard deviation equation (S.D)

Z = to get the Chauvenet’s criterion (T)

CI = for confidence interval.

Analysis and Results

You need to answer the following questions in the memo report and show any sample calculations in the attachments:

1. Plot a histogram of each of the three measurements: mass, diameter, and height of the cylinders. Each histogram must contain at least 5 bins.

Figure 1 Mass Histogram

Figure 2 Diameter Histogram

Figure 3 Height Histogram

2. Determine the central tendency of each measurement using the mean, the median, and the mode.

Table 1 Central Tendency of Mass

Table 2 Central Tendency of Diameter

Table 3 Central Tendency of Height

3. Determine the dispersion measure of each measurement using the standard deviation.

4. Assume that the distribution of each measurement is Gaussian or normal error distribution. Using Chauvenant’s criterion, can any of these measurements be discarded? If yes, determine the new central tendency and dispersion measure (i.e., repeat steps 2 and 3).

Chauvenant’scriterion values for n=50 is 2.576

Calculating values using

Sample calculation

Table 4Chauvenant’scriterion Table

Based on calculation we can reject 24mm value for diameter as outlier

Central tendency and dispersion after removing outlier

Table 5 Central tendency and dispersion after removing outlier

5. Determine the 90% and 98% confidence interval for each of the mean values.

Table 6 90%, 98% confidence interval for Mass

Table 7 90%, 98% confidence interval for Diameter

Table 8 90%, 98% confidence interval for Height

6. Determine the volume and density of each cylinder in cm3 and kg/m3 respectively.

Table 9 Volume and Density Table

7. Determine the following statistics of the calculated values of the volume and density of the cylinders:a) Mean, b) Median, c) Mode, d) Standard Deviation, e) 95% confidence level of the mean.

Table 10 Mean, Median, Mode and S.D for Volume

Table 11 Mean, Median, Mode and S.D for Density

8. Derive an equation and calculate the uncertainty in the volume and density calculations based on the 95% confidence level.

The volume and density of cylinders is calculated by measuring dimensions D, and H.

It is assumed that the measurements will follow Gaussian distribution with standard deviation , which is same for all the measurements.

Therefore uncertainties at 95% confidence levels are given by

Or in other words

The error formula is

Table 12 Uncertainty in Volume and Density

9. Compare the uncertainties calculated in Step 8 with the statistics found in Step 7.

While comparing uncertainties in step 8 with the confidence interval we found out that these uncertainties fall within the range 95% confidence interval.

10. Plot the mass of each cylinder as a function of the volume of each cylinder. Fit a leastsquares curve to this plot. Physically, what does the slope of the least squares linerepresent? Compare, if possible.

Figure 4 Mass as a function of Volume

The slope of the curve means how mass is changing with respect to volume. As in this case slope is -0.0001 means mass and volume are not dependent on each other.

Conclusions and Recommendations

In this lab we have used statistical methods to analyze experimental data. Also determined the nominal density, and uncertainty of density, of cylinders of an unknown material. Using the raw data of 50 measurements (diameter, height, and mass) of cylinders of an unknown material we first made the histogram of mass, diameter and height of cylinder. The volume and density of each of the cylinder is found using the raw data. We also determined the uncertainty in volume and density of the cylinder and found out that uncertainty lie within the range of 95% confidence interval.

References