**Explaining Statistics – Levels of Measurement**

STATA procedures for creating Summary Statistics.

The use of statistical reasoning and its application to the processes of justice is a difficult issue. The reality is that most (all?) lawyers, social workers, clients, dental surgeons, and administrators have considerable difficulty in following statistical reasoning. This is consistent with the experience of the DARLInG team who, when preparing documents for the courts have found it necessary to explain the concepts in the most basic terms. This experience is matched by publications specifically aimed at such professional groups – viz.

**Vito GF, Latessa EJ. Statistical Applications in Criminal Justice. Sage Publications 1989. London. ISBN 0-8039-2983-8**

**Kadane JB. Statistics in the Law. Oxford University Press. 2008. Oxford**

**Aitken CGG. Statistics and the evaluation of evidence for forensic scientists.**

** John Wiley & Sons 1995. Chichester.**

For further and more sophisticated statistical explanation the reader is referred to:

**Petrie A & Sabin C. Medical Statistics at a Glance. 2009. Wiley-Blackwell 3rd Edition.**

** Oxford. ISBN 978-1-4051-8051-1**

**Altman DG. Practical Statistics for Medical Research. 1991. Chapman & Hall.**

** London ISBN 0-412-27630-5**

In preparing these explanations it is assumed that the reader is numerate to the level of GCSE mathematics. For non UK readers, this is the level to which 16 year old school children achieve competence before being awarded a ‘General Certificate of Secondary Education’.

Statistics is about numbers and the way these numbers may be handled to summarise the data in a meaningful way. The principles of mathematics and statistics require a very careful consideration of the different types of variables and consequently, the permissable statistical procedures are closely linked to the type or level of measurement.

**Nominal Scale**

This is the** **simplest level and the number classify or categorise the data under consideration. For example. 1 = Male, 2 = Female. Also Racial or Ethnic Group. 1 = UK Caucasian, 77 = Southern Chinese. The TDS used in DAE are usually considered to be of the Nominal Scale, Sometime referred to as Categorical Data.

The type of statistical procedures permissable are comparison of numbers between groups e.g. the number of males and females (counts) in Uk Caucasian and Afro-Trinidadian groups.

An important procedure that is used to test Hypotheses is the χ^{2 test (Chi Squared Test) to test the difference between number of, for example, males and females in two separate groups. }

**Ordinal Scale**

This is a scale where there is a clear hierarchical relationship between the items in the scale. For example, *warm, warmer, warmest. * There is a clear gradient , but the difference between warm an warmer is not the same as the difference between *warmer and warmest*.

This scale concept is of crucial importance in the process of Dental Age Estimation as a group of DAE techniques, based on the concept of Tooth Development Stages (TDS). This concept is central to the DARLInG approach to Dental Age Estimation. This topic is covered in detail in Tooth Development Stages.

There are a whole raft of statistical procedures that relate to the **Ordinal Scale. **These are described in detail in the standard textbooks under the general title of Non-Parametric Statistics. It is paradoxical that these statistical test are occasionally used in DAE. The Tooth Development Stages are on an ordinal scale. But, in the Reference Data Set, the Ordinal scaled items render up numerical data on an Interval Scale. These data are Normally distributed so the more powerful parametric (Normal distribution) tests can be used in calculating Dental Age.

**Interval Scale**

The interval scale has all the characteristics of the ordinal scale with the additional feature that the distance between any two numbers is known. Thus the distance between 2 and 3 is the same as between 34 and 35, and 401 to 402 and so on. This enables conventional mathematical procedures to be used.

The interval scale is particularly useful when the variable of interest Normally distributed. In DAE this is the main focus of statistical procedures. These are known as Parametric tests.

The interval scale is at the core of statistical management of numerical data in relation to DAE.

**Ratio Scale**

All the characteristics of the Interval Scale, but with a true zero. This had no application in DAE.

**Comment**

An understanding of measurement scales is important to the practitioner of Dental Age Estimation whether Dentist, Doctor, Solicitor, Social Worker, Barrister, or Judge.

A particularly well written and comprehensible discourse on Measurement Scales is given in: Nonparametric Statistics for the Social Sciences. Siegel S. 1956 McGraw-Hill. London ISBN 07-085689 3. Pages 21 to 30

~~~~~~~~~~~~~~~~~~~ Page completed 20 June 2014 by Graham Roberts ~~~~~~~