# Statistical Analysis of Facial Proportions

The aim of my investigation is to find how beautiful the year seven pupils are in my school, according to Pythagoras’ Theory that the more beautiful a person is, the closer the measurements of certain features of the body, to the ratio of 1:1.

618.The samples that I will take will be random samples of males and females in year seven.The samples will be taken in the following way.Each of the pupils in year seven has a number next to their name in the teachers’ register. I will select random numbers using the random number generator facility on my graphic calculator, and the numbers that match up with the names on the register will be selected from the population and the submitted to the analysis.I will select 25 people from the register at random and submit the same people to both measurements.

If the same number is selected again, I will take the next random number as the next person for analysis.The measurements that I will be taking will be:1. The width of the mouth and the width of the bottom section of the nose.2. The length of the proximal phalanges and the length of the central phalanges of the hand.

These measurements will then be expressed as a ratio of the smaller to the larger in the form 1:n.Methods of measurements.* Measurements of the Proximal Phalanges and Central Phalanges were taken using a 30-cm ruler. These were measured to the nearest mm.* Measurements of the mouth (from crease of bottom and top lips, on either side of the mouth) and nose (from widest points of either side of the nose) were taken using a 30-cm ruler.

These were measured to the nearest mm.To keep the test fair the same rulers were used throughout the investigation.Results.Males and Females measurements of the Mouth and Nose.MalesFemalesMouthNoseRatioMouthNoseRatio5.

13.41.54.53.21.406531.

667641.553.51.4296.532.1673.

52.71.2964.531.53.

52.81.2552.81.7865.

22.91.7934.53.21.

4065.53.31.66764.21.

4294.53.71.2164.53.11.

4524.82.51.9252.524.53.

41.3244.53.31.3647.

52.82.6794.53.51.

28653.51.4295.13.21.

5945.43.51.54374.51.

5565.42.81.92962.52.45.

23.31.5764.32.71.5934.

531.553.31.51542.51.653.

81.3165.731.95.53.

61.5284.72.81.6795.

231.733531.66752.81.7864.63.

11.4845.43.51.5434.53.

31.3644.731.5674.83.

51.3714.831.64.22.

81.553.31.5155.22.81.

8575.43.51.543Males and Females measurements of the Proximal and Central Phalanges.ProximalCentralRatioProximalCentralRatio53.11.

6134.53.41.3245.53.

21.7195.23.11.6773.

62.41.54.531.53.

42.91.1724.52.51.

83.22.41.3333.531.167531.

6675.30.21.6564.431.46742.

81.42942.51.642.51.

63.721.853.52.81.253.

42.81.2143.831.2674.

12.81.4644.83.11.

5483.731.2334.531.55.12.

91.7593.52.51.44224.

13.21.2813.521.754.

62.71.7044.52.12.1435321.

5634.522.254.83.11.5484.

82.51.924.73.21.4693.

72.51.484.231.44.

23.81.1054.43.21.

3754.63.31.39452.91.

7244.231.453.61.3894.631.

5334.52.71.6674.72.91.

6214.831.65.83.31.

7584.72.91.621Measurements of the Mouth and Nose Ratios.MalesThe sample mean = 1.606The sample variance = 0.

093FemalesThe sample mean = 1.603The sample variance = 0.067Since the samples are random, the distribution of the sample means are equals to the actual populations being estimated. Therefore can be used to estimate the parent population.MalesUnbiased estimate of thePopulation mean = 1.606FemalesUnbiased estimate of thePopulation mean = 1.

603However, the sample variance is a biased estimator of the population variance.To convert this to an unbiased estimator, this method was used;If S squared is the sample variance of a sample size n then,x S squared is an unbiased estimator of the population variance.MalesUnbiased estimate ofthe population variance = 25 x 0.09324= 0.097FemalesUnbiased estimate ofThe population variance = 25 x 0.

06724= 0.071Therefore for the whole population:Males X ~ N (1.606,0.097)Females X ~ N (1.603,0.071)To compare these results, I will calculate 95% Confidence intervals for males and females, and compare the size of the interval.

I will also compare where the intervals lie in relation to each other.Males:X ~ N (1.606,0.097)Females:X~ N (1.603,0.071)Placing of confidence intervals.

MalesFemalesThis shows a similarity between the sexes Males and Females. As you can see from the diagram the confidence intervals are very close in size. The size of the male interval is 0.244 and the size of the female interval is 0.209.Also these intervals are closely aligned showing a further similarity.

This shows a similarity between males and females.I can be 95% certain that the mean will fall within this interval.Measurements of proximal and central phalanges ratio.MalesThe sample mean = 1.598The sample variance = 0.086FemalesThe sample mean = 1.498The sample variance = 0.028Since the samples are random, the distribution of the sample means are equals to the actual populations being estimated. Therefore can be used to estimate the parent population.MalesUnbiased estimate of thePopulation mean = 1.598FemalesUnbiased estimate of thePopulation mean = 1.498However, the sample variance is a biased estimator of the population variance.To convert this to an unbiased estimator, this method was used;If S squared is the sample variance of a sample size n then,x S squared is an unbiased estimator of the population variance.MalesUnbiased estimate ofthe population variance = 25 x 0.08624= 0.090FemalesUnbiased estimate ofThe population variance = 25 x 0.02824= 0.029Therefore for the whole population:Males X ~ N (1.598,0.086)Females X ~ N (1.498,0.029)To compare these results, I will calculate 95% Confidence intervals for males and females, and compare the size of the interval. I will also compare where the intervals lie in relation to each other.Males:X ~ N (1.598,0.086)Females:X ~ N (1.498,0.029)Placing of confidence intervals.MalesFemales