# Statistical Analysis of Facial Proportions

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Introduction

Maths coursework Statistical Analysis of Facial Proportions The coursework task that I am following is S1 Task A. 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.

Middle

Proximal Central Ratio Proximal Central Ratio 5 3.1 1.613 4.5 3.4 1.324 5.5 3.2 1.719 5.2 3.1 1.677 3.6 2.4 1.5 4.5 3 1.5 3.4 2.9 1.172 4.5 2.5 1.8 3.2 2.4 1.333 3.5 3 1.167 5 3 1.667 5.3 0.2 1.656 4.4 3 1.467 4 2.8 1.429 4 2.5 1.6 4 2.5 1.6 3.7 2 1.85 3.5 2.8 1.25 3.4 2.8 1.214 3.8 3 1.267 4.1 2.8 1.464 4.8 3.1 1.548 3.7 3 1.233 4.5 3 1.5 5.1 2.9 1.759 3.5 2.5 1.4 4 2 2 4.1 3.2 1.281 3.5 2 1.75 4.6 2.7 1.704 4.5 2.1 2.143 5 32 1.563 4.5 2 2.25 4.8 3.1 1.548 4.8 2.5 1.92 4.7 3.2 1.469 3.7 2.5 1.48 4.2 3 1.4 4.2 3.8 1.105 4.4 3.2 1.375 4.6 3.3 1.394 5 2.9 1.724 4.2 3 1.4 5 3.6 1.389 4.6 3 1.533 4.5 2.7 1.667 4.7 2.9 1.621 4.8 3 1.6 5.8 3.3 1.758 4.7 2.9 1.621 Measurements of the Mouth and Nose Ratios. Males The sample mean = 1.606 The sample variance = 0.093 Females The sample mean = 1.603 The sample variance = 0.067 Since the samples are random, the distribution of the sample means are equals to the actual populations being estimated.

Conclusion

Males The sample mean = 1.598 The sample variance = 0.086 Females The sample mean = 1.498 The sample variance = 0.028 Since 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. Males Unbiased estimate of the Population mean = 1.598 Females Unbiased estimate of the Population mean = 1.498 However, 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. Males Unbiased estimate of the population variance = 25 x 0.086 24 = 0.090 Females Unbiased estimate of The population variance = 25 x 0.028 24 = 0.029 Therefore 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. Males Females

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