Statistical Analysis of Facial Proportions

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.

Males

Females

Mouth

Nose

Ratio

Mouth

Nose

Ratio

5.1

3.4

.5

4.5

3.2

.406

5

3

.667

6

4

.5

5

3.5

.429

6.5

3

2.167

3.5

2.7

.296

4.5

3

.5

3.5

2.8

.25

5

2.8

.786

5.2

2.9

.793

4.5

3.2

.406

5.5

3.3

.667

6

4.2

.429

4.5

3.7

.216

4.5

3.1

.452

4.8

2.5

.92

5

2.5

2

4.5

3.4

.324

4.5

3.3

.364

7.5

2.8

2.679

4.5

3.5

.286

5

3.5

.429

5.1

3.2

.594

5.4

3.5

.543

7

4.5

.556

5.4

2.8

.929

6

2.5

2.4

5.2

3.3

.576

4.3

2.7

.593

4.5

3

.5

5

3.3

.515

4

2.5

.6

5

3.8

.316

5.7

3

.9

5.5

3.6

.528

4.7

2.8

.679

5.2

3

.733

5

3

.667

5

2.8

.786

4.6

3.1

.484

5.4

3.5

.543

4.5

3.3

.364

4.7

3

.567

4.8

3.5

.371

4.8

3

.6

4.2

2.8

.5

5

3.3

.515

5.2

2.8

.857

5.4

3.5

.543

Males and Females measurements of the Proximal and Central Phalanges.

Proximal

Central

Ratio

Proximal

Central

Ratio

5

3.1

.613

4.5

3.4

.324

5.5

3.2

.719

5.2

3.1

.677

3.6

2.4

.5

4.5

3

.5

3.4

2.9

.172

4.5

2.5

.8

3.2

2.4

.333

3.5

3

.167

5

3

.667

5.3

0.2

.656

4.4

3

.467

4

2.8

.429

4

2.5

.6

4

2.5

.6

3.7

2

.85

3.5

2.8

.25

3.4

2.8

.214

3.8

3

.267

4.1

2.8

.464

4.8

3.1

.548

3.7

3

.233

4.5

3

.5

5.1

2.9

.759

3.5

2.5

.4

4

2

2

4.1

3.2

.281

3.5

2

.75

4.6

2.7

.704

4.5

2.1

2.143

5

32

.563

4.5

2

2.25

4.8

3.1

.548

4.8

2.5

.92

4.7

3.2

.469

3.7

2.5

.48

4.2

3

.4

4.2

3.8

.105

4.4

3.2

.375

4.6

3.3

.394

5

2.9

.724

4.2

3

.4

5

3.6

.389

4.6

3

.533

4.5

2.7

.667

4.7

2.9

.621

4.8

3

.6

5.8

3.3

.758

4.7

2.9

.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. Therefore can be used to estimate the parent population.

Males

Unbiased estimate of the

Population mean = 1.606

Females

Unbiased estimate of the

Population mean = 1.603

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 0.093

24

= 0.097

Females

Unbiased estimate of

The population variance = 25 x 0.067

24

= 0.071

Therefore 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.

Males

Females

This 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.

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