ACTIVIDADES PARA CAPTACIÓN DE COLABORADORES, CLIENTES Y

2.0 0.0 13.5 12.5 13.0 11.5 12.0 10.5 11.0

Observed Value

The third method used in the examination o f the data to establish whether the sample could be representative o f a normally distributed population was the

Kolm ogorov-Sm irnov test: The particular version used was the one-sample

Kolmorgorov-Smirnov test with the Lilliefors correction. The value calculated is the probability that the amount o f deviation from normality shown in the sample distribution could arise from random sample factors. The null hypothesis being tested is that the sample under investigation has been taken from a population in which the measurement has a normal distribution. In this project measurements with significance values o f 0.05 or less have been taken as evidence that the sample is not what would be reasonably expected from a normally distributed measurement in the parent population. An example o f the presentation o f the results o f the Kolm ogorov-Sm irnov (Lilliefors) test is shown below. It is not the statistic itself but the significance level which is quoted, as this is the value used to assess the normality o f the sample under investigation. SPSS calculates the values to four decimal places. In the presentation o f the results in Chapter Six the values relating to measurement in millimetres are quoted to 0.01 mm (tw o decimal places) which is the level o f precision to which the callipers measured.

Table 4.1 Example of Kolmogorov-Smirnov (Lilliefors) test results

Length of the first permanent molar for females from age group 7-9 months from the Hakel sample (left side) n=8.

Measurement Kolmogorov-SnfrrnQy Q-Q graph Stem and leaf Normal? (Lilliefors)signîficancë '

value

form plot form

M ,L > 0.2000 straight line normal yes

The results o f the stem and leaf, Q-Q plots and K olm orgorov-Sm im ov test are presented in tables in Chapter Six (see Tables 6 .1-6.18), in the final column an

assessment o f whether the sample can be said to com e from a population in which the measurement is normally distributed is made. The form o f the stem and leaf plots and Q-Q plots are described in the tables in Chapters Six. The format used is shown above (Table 4.1) the value for significance from the K-S Lilliefors test is given then the q-q plot and the stem and leaf plot are described. In the final column an assessment o f the normality o f the measurement is given if none or only one test indicates a deviation from normality the distribution is classed as normal ‘y es’ in the table, if tw o o f the tests/graphs indicate a slight deviation from the normal distribution this is denoted by if tw o or more tests/graphs indicate a stronger deviation from the normal distribution then ‘n o ’ is entered in the table.

Once the normality o f the sample data had been examined, an investigation o f factors which could cause variation within the samples was carried out. These factors were asymmetry, sexual dimorphism and age-related variation

To assess the degree o f variation due to asymmetry scatter plots o f pairs o f measurements (e.g. length against width) were done with the right and left sides labelled separately. A more precise means used to test the null hypothesis that ‘there is no significant difference between the measurements o f the right and left sides’ is to use Student’s t-test. This test measures the difference between the means o f the tw o groups and assesses whether the difference between the means is significant, or whether it could arise by random sampling factors or sample size. If the difference between the means is small, the test will return a low t-value, and if the significance values are higher than 0.05 then the null hypothesis can be retained. I f the difference between the means o f the tw o samples is large then a t-value larger than 2 (depending on sample size) and a significance value below 0.05 result, indicates that the null hypothesis should be rejected. The values included in presentation o f results were:

• the means for each o f the tw o groups • the mean difference (A -B )

• the t-value

• the degrees o f freedom • the significance

An example o f this format is shown below in Table 4 .2.

Table 4.2 Example of reported results for a t-te

Measurement M ean A Right Mean B Left Mean diff. A-B t-value degrees o f freedom significance M iL 18.5 18.5 0.0 0.21 316 0.832

The t-test was also used to test the null hypothesis that ‘there is no difference between measurements o f males and fem ales’. In the test the males were group A and the females group B , the results were reported in Chapter Six using the same format as w as used for the results o f the asymmetry t-test (see Table 4 .2 above).

The third factor considered to contribute to variation within samples was that o f age related variation due to eruption and wear o f the teeth. This was investigated using analysis o f variance (A N O V A ) with the Bonferroni test. In this method more than tw o groups can be simultaneously examined for differences between groups In the t-test the group means were compared in A N O V A mean squares are used instead. The A N O V A identifies any tw o groups that are different at a pre-set level (in this project the 0.05 significance level was used). The A N O V A procedure plots a diagram in which the groups entered are listed along the left side and the top then any significant differences between groups are marked by an asterisk in the lower triangle. In Chapter Six, the presentation o f results for analysis o f variance includes:

• the means for each group (1-4) • total degrees o f freedom • the F-ratio

• F-probability

• description o f the differences between groups.

Table 4.3 shows an example o f the format in which the A N O V A results are presented in chapters Six and Seven.

Table 4.3 Example of reported results for analysis of variance

Measurement total degrees of freedom F ratio probability Patterns of differences between groups at j 0.05 level

Means for groups

1 2 3 i 4 M,L 117 3.3498 0.0216 2 from 4 18.5 19.1 18.9 18.6

Figure 4.13 Example of ANOVA result chart

The group numbers refer to the four age groups described above.

e g Grp Grp Grp Grp 1 4 3 2 Grpl Grp4 Grp3 Grp2 *

Once the possible causes for variation within the samples had been investigated, the tw o samples were examined together to find out if there were any differences

between them. Scatter plots o f pairs o f measurements in which the different groups were separately labelled were produced (an example is shown in Figure 4.14). A selection o f scatter plots are presented in Chapter Six.

Figure 4.14 Example of scatter plot for two dimensions of one tooth comparing the data from Hakel and the Domestic rare breeds.