Statistics and Uncertainity Lab Report

Statics and Uncertainty

Marilyn Esthappan
Lab Partner: Nisha Sunny
TA: Sajjad Tahir
Physics Lab 106
May 29, 2011
1-3pm

THEORY:
Statistical variation and measurement uncertainty are unavoidable. A theory is consistent if the measurement is 2m +/- 1m. Uncertainty rises from statistical variation, measurement precision, or systematic error.

EXPERIMENTAL OVERVIEW:
Part 1. Coin Toss: Statistical Variation:
Sixteen coins were tossed nine times and the number of heads was counted to determine variation associated with random events.
Part 2. Measurement Precision:
The measurements (length, width, and height) of the wood block were taken with a ruler to estimate the preciseness of using a ruler. The uncertainty of each measurement was measured because this is how much how much the preciseness of the ruler is off. The mass of the block was taken using a scale. All measurements have an uncertainty that effects its calculations.
DATA:
Part 1. Coin Toss: Statistical Variation:

Number of Times for Heads | The Range of Heads Between 6 and 10 | Average for Heads | 7 +/- 8 | 6 | Y | Sigma | 1 +/- 5 | 5 | N | Average per Penny with 16 Pennies | 0 +/- 5 | 7 | Y | Sigma per Penny | 0 +/- 1 | 7 | Y | Percentage Uncertainty | 20% | 8 | Y | Sum of Number of Heads for 144 Pennies | 70 | 9 | Y | Average per Penny for the Whole 144 Set of Coins | 0 +/- 51 | 9 | Y | Sigma per Penny Using Sigma of 6 | 0 +/- .4 | 10 | Y | Percentage Uncertainty | 7.8% | 9 | Y | |

Part 2. Measurement Precision:

| Length (cm) | Width (cm) | Height(cm) | Mass (g) | | Area(cm^2) | Volume(cm^3) | Density(g/cm^3) | Measure | 20 +/- 3 | 8 +/- 5 | 3 +/-6 | 309 +/- 1 | | 172 +/- 5.5 | 621 +/- 1.8 | 0 +/- 4.97601 | Uncertainty | 0 +/- 1 | 0 +/- 1 | 0 +/- .5 | 0 +/-1 | | | | | Maximum | 20 +/- 4 | 8 +/-6 | 3 +/- 6.5 | 309 +/-2 | | 175 +/- 4.4 | 640 +/- 3.56 | 0 +/- 4.82856 | | | | | | Difference | 2 +/- 8.9 | 19 +/- 1.76 | 0 +/- .147 | | | | | | | | | | Percentage | 0.493 | 1.177 | 1.389 | 0 032 | | 1.700 | 3.087 | 2.954 | | | | | | | | | | | | | | | | | | | | | | Expected Percentage | 1+/- 6.69 | 3 +/- 8.9 | 3 +/-1.19 | | Expected Uncertainty (Expected Difference) | 2 +/- 9 | 19 +/- 2 | 0 +/- 1.6 |

DATA, ERROR AND UNCERTAINITY ANALYSIS:
Part 1. Coin Toss: Statistical Variation:
All of the calculations were done on Excel. The data shows 8 out of the 9 tosses, between 6 and 10 coins the coins appeared as heads; giving the average of 7.8 for a coin to be heads. This agrees with the theory of statistical variation since the average was off by 2 in comparison to the actual average of 8. Data is given for 9 tosses with 16 pennies and then with 144 pennies. The data for the 144 pennies wasn’t given by actual tosses, but by calculations. More pennies for tossing gave more variation and less room for uncertainty. The average was calculated by adding the column of heads and dividing by 9. The sigma was calculated by taking the stranded deviation of the column of heads. The percentage uncertainty was calculated by multiplying 100 by the sigma divided by the average.

Part 2. Measurement Precision:
All of the calculations were done on Excel. The data shows the measurements for the wooden block. The maximum calculations are the measured value increased by the uncertainty. The maximum area, volume, and density were calculated by using their formulas (Area=L*W, Volume=L*W*H, Density=M/V). The difference was calculated by subtracting the maximum value from the measure volume. Then the percentage uncertainty was calculated with its formula (100(uncertainty)/(original value)). The expected percentage in area was calculated by the addition of percentage uncertainties in length and width. For volume, the percentage volume was calculated by the sum of the percentage uncertainties in volume. For density, the percentage uncertainty was calculated by the sum of the percentage uncertainties in volume and mass.

CONCLUSIONS:
Part 1:
Statistical variation decreases with the number of pennies. The amount of pennies tossed doesn’t affect the outcome that there is a fifty-fifty chance of heads. As shown in the experiment the average for head was .5 regardless if there were 16 or 144 pennies tossed. The answers to the questions in the instructions follow below:
1.2 Only once was 8 heads were obtained with the average of 7.8 heads.
1.3 8 out of 9 times heads were obtained within the expected range of 6-10. Yes, it was about 2/3 of the time.
1.4 The standard deviation of the data is 1 +/- 5. It is similar to the expected values of sigma expected = 2.
1.5 The number of heads measured varies, but has an average of 7.8 of being heads. Yes, this does agree with theoretical value of 8 because 7.8 is -2 less than 8 which is an acceptable difference.
1.6 Yes, the coins are fair because the expected values is 0.48.
1.7 The percentage uncertainty is 7 +/- 8.
1.8 The total number of heads is 70 out of 144 pennies. No, it does it because the theory says the total number of heads should be 72. The average per penny for the whole 144 pennies is 0 +/- 5. The uncertainty per penny is 0 +/- .4. The percentage uncertainty is 7.8%. It is less than for 16 pennies, because increasing the number of measurements decreases the statistical percentage uncertainty.
Part 2. Measurements Precision:
Measurements have an uncertainty because they are limited by the instruments; therefore affecting calculations. There could also be a possible human error in all the calculations due to measuring or reading a measurement. The answers to the questions in the instructions follow below:
2.1 The smallest increment on the balance scale is grams.
2.5 The computed area is 175 +/- 4.4 cm^2. The computed volume is 640 +/- 1.8 cm^3. The computed density is 0 +/- 4.82856 g/cm^3.