Introduction to blog

At the moment I'm slowly building a website for Statistics. The idea is that the site describes the various terms in a clear and efficient way, including references and examples.

On this blog I'll upload some parts of it already so others can already make use of the content and comment and give suggestions for further improvements.

1.1. Decimal marker and rounding

Dot or comma
Belief it or not, a whole range of symbols is used as a decimal marker and thousands separator. The dot under (.) and the comma (,) are probably the most widely known, but also the dot middle (∙), dot above (˙) or comma above (') are used in various countries.

The International Bureau of Weight & Measures, has noted that only the dot or comma should be used as a decimal separator (Bureau International des Poids et Mesures, 2003, p. 169). To separate thousands only a space should be used and not a dot or comma (Bureau International des Poids et Mesures, 1948, p. 70, 2003, p. 169).

So the only two ways to write down seven million and a half are 7 000 000,5 or 7 000 000.5 (or without the spaces).

Rounding
The rounding used here is the ‘round half away from zero’. If we want to round to two decimals, this method means we look at the third decimal. If the third decimal is 5 or higher (i.e. 5, 6, 7, 8 or 9) the second decimal gets increased by one. If the third decimal is lower than 5 (i.e. 0, 1, 2, 3, or 4) the second decimal remains unchanged.

1.375999 ≈ 1.38
1.374999 ≈ 1.37
-1.375999 ≈ -1.38
-1.374999 ≈ -1.37

This is the method also described by the National Center for Education Statistics (NCES, 2012).

>>Next section: Cases, questions, variables and values

References
Bureau International des Poids et Mesures. (1948). Résolution 7 (pp. 70–71). Presented at the 9e Conférence Générale des Poids et Mesures, Paris: BIPM.
Bureau International des Poids et Mesures. (2003). Résolution 10. Presented at the 22e Conférence Générale des Poids et Mesures, Paris: BIPM. Retrieved from http://www.bipm.org/fr/convention/cgpm/comptes_rendus.html
NCES. (2012). Analysis of Data / Production of Estimates or Projections. In 2012 NCES Statistical Standards. Retrieved from http://nces.ed.gov/statprog/2012/pdf/Chapter5.pdf

1.2. Cases, variables, values and scores

When collecting data, you collect it from something. This something could be people, things, days, etc. and are called the cases. From each case we like to know different things that can vary per case, the things we want to know from each case are called the variables. Each variable can vary, the possible variations are the values. Finally the specific value of a variable for a case is called a score.

A survey for example has as cases people (or respondents) and the questions on the survey can be considered the variables. A small difference between a survey question and a variable, is that the name of the variable is often shorter. The question ‘What is your gender?’ will have as a variable simply ‘Gender’ and the possible variations are Male and Female. My score for the variable gender would be Male.

In Table 1 a classification of types of questions is shown.

Table 1
Types of questions

Type	Description	Example
1	Open questions
1a	Open questions asking for a number	What is your age? ___
1b	Open questions asking for text	What is your name? ____
2	Closed questions
2a	Closed question, single answer	What is your gender? O Male  O Female
2b	Closed question, multiple answer	Which TV series do you watch? □ Got  □ Walk. Dead  □ Southpark □ Other
3	Semi-closed
3a	Semi-closed, single answer	What is your favourite brand? O Nike O Adidas O Puma O Other, please specify:____
3b	Semi-closed, multiple answer	Which brands do you like? □  Nike □  Adidas □  Puma □  Other, please specify:____
4	List of questions on same scale	Please indicate the level you agree or disagree with the following statements. Strongly disagree Disagree Agree Strongly agree I enjoy this book O O O O The book is clear O O O O The layout is nice O O O O The style is good O O O O

If type 4 uses some form of scale for each of its questions (as in the example), then this type is also known as a Likert [1] scale (Likert, 1932). Note that each question from the Likert scale is called a Likert item, the combined set of the questions is then the Likert scale.

Questions with multiple answers (type 2b and 3b) are often not considered as a variable, but a collection of variables. Each option is considered a yes/no question. In the example of the TV shows the multiple answer question, could also have been rephrased using the following four single answer questions:

• Do you watch Got? O Yes O No
• Do you watch Walk. Dead? O Yes O No
• Do you watch Southpark? O Yes O No
• Do you watch Other? O Yes O No

How to enter questions in SPSS can be found in Appendix A.

>>Next section: Measurement levels

References
Likert, R. (1932). A technique for the measurement of attitudes (Vol. 22). New York: The Science Press.

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[1] Pronounced with a short i as in sick, and not as in lie.

1.3. Measurement level

Each variable has also a measurement level. These are used to classify the variables and depending on the type of measurement level different statistical techniques can be used. The classification described by Stevens (1946) is the one most frequently used and also described here.

Nominal has ‘no’ in it. It is used for variables that have no logical order (besides perhaps alphabetical). An example of a variable on a nominal level is TV shows. The order we put the TV shows in, does not really matter.
For variables on nominal level we can only compare groups based on the quantity.

As the name implies, for ordinal variables there is a logical order, but note that numbers are not yet being used. An example is Educational Level. There is a clear order in this.
Besides comparing the different groups, we can now also make statements about greater or less (e.g. 45 respondents had secondary school or less).

Interval is for variables that are using numbers, but the zero of the scale was somewhat arbitrary chosen. For example temperature in degrees Celsius (or Fahrenheit) is at an interval level. The zero was somewhat arbitrary chosen as the boiling point of water (and later flipped to be the freezing point).
This level is called ‘interval’ because there are equal intervals between values. The difference between 5 degrees Celsius and 10 degrees, is the same as the difference between 20 and 25. Note that for ordinal variables this is not the case, the difference between Strongly disagree and Agree, might be different than the difference between Agree and Neutral.
At an interval level the zero does not really mean nothing. For example at 0 degrees Celsius there is still a temperature, and also someone with an IQ of 0, still has intelligence (but nothing much). This means we can also add and subtract the values on an interval level.

The last level is ratio, for variables that are using number, and have a true zero. This means that zero really means nothing. If your income is zero, you are really not earning anything. Ratio variables are also relative speaking equal (hence the name ratio). If you earn 200 one day, and 400 the next than this is really twice as much (no matter which currency), while for example 10 degrees Celsius is not twice as much as 5 if you would convert them to Fahrenheit. There is one temperature scale that is considered ratio, which is Kelvin who uses the absolute zero as a starting point.
At ratio level we can now also divide and multiply.

Sometimes no distinction is made between nominal and ordinal, and the term categorical is used. Similar SPSS does not make a distinction between interval and ratio and simply calls these scale.

Another way to classify variables is into discrete and continuous. A discrete variable is a variable where the values are distinct from each other. This means that both nominal and ordinal variables will also be discrete. However interval and ratio variables could either be continuous or discrete. A continuous variable is if there are no interruptions. The number of people is a discrete variable, but also interval. Money, weight and length are examples of continuous variables.

Note that the measurement level classification from Stevens is not without criticism. Velleman & Wilkinson (1993) give a nice overview of various problems with Stevens classification. For those interested in reading more on measurement levels the article from Sarle (1997) might be a nice starting point.

>>Next chapter: Descriptive Statistics

References
Sarle, W. S. (1997, September 14). Measurement theory: Frequently asked questions. Retrieved May 3, 2015, from ftp://ftp.sas.com/pub/neural/measurement.html
Stevens, S. S. (1946). On the Theory of Scales of Measurement. Science, 103(2684), 677–680. doi:10.1126/science.103.2684.677
Velleman, P. F., & Wilkinson, L. (1993). Nominal, Ordinal, Interval, and Ratio Typologies are Misleading. The American Statistician, 47(1), 65–72. http://doi.org/10.1080/00031305.1993.10475938 

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