# Fundamentals

## Measurement levels

**(if you prefer to watch a video on this than read, click here)**

Depending on how something is measured, we can ask different questions. In the example survey I could ask myself how many people were at least interested in statistics, but I couldn’t really ask myself how many people were at least male. How a variable is measured has a consequence on the operations that can be performed on them, and which statistical tools can used. For example we can calculate an average age but not an average gender.

Stevens (1946) classified variables based on which basic operations can be performed on them, and created four so-called measurement levels: nominal, ordinal, interval and ratio. I will combine interval and ratio into one category called scale (which is something SPSS also does). If you are curious about the difference between those two please see the appendix at the bottom of this page.

A variable is said to have a **nominal** measurement level, if the values are non-numeric and have no logical order (besides perhaps alphabetical). The variable *gender* for example has a nominal measurement level, since the order of the possible values can be in any way you want (although there are many discussions sometimes about it :-)). Also open questions that ask for text are nominal, for example *first name*.

If the values are non-numerical, but do have a logical order to them, the measurement level is **ordinal**. For example a variable that uses the options 'fully disagree', 'disagree', 'agree', 'fully agree', is ordinal. Note that it is about the original values being non-numerical. Often with ordinal variables the values are coded with numbers (e.g. 1 = fully disagree to 5 = fully agree), but this would still be ordinal, since the original values are non-numerical. Also numbers that have been categorized create an ordinal measurement level, so for example age category (e.g. 0 < 20, 20 < 30, etc.) would be ordinal.

In case you do have real numbers the measurement level will be **scale** (interval and ratio). So for example when asked to fill out your age, the variable *age* will have a scale measurement level. With 'real' numbers I mean that a phone number for example is not really a number on which you can perform calculations (adding the digits of a phone number might be fun but there it has no meaningful interpretation), so a phone number would be considered nominal since there is not really a logical order.

For the analysis of the data this course will do this by measurement level. It is therefore important to be able to determine the correct measurement level. To help with this the flow chart in Figure 1 can be helpful.

**Figure 1**

*Decision flow chart for measurement level*

For example if we ask people their phone number, then although we could calculate an average phone number, it would not make any sense. There is also not any particular order in these phone numbers, so the measurement level would be nominal.

A special type of variable is a **binary variable**, also known as dichotomous (Warner, 2013; Weinberg & Abramowitz, 2008), or Bernoulli (Nelson, Coffin, & Copeland, 2003) . A variable is binary if it only has two possible outcomes (e.g. pass/fail, on/off, male/female, right/wrong, etc.). Note that strictly speaking a binary variable can be considered nominal, ordinal, interval, and ratio. Since there are only two options to choose from. In most cases it is often considered as a nominal variable.

Although the classification from Stevens (nominal, ordinal, interval and ratio) works well for many variables, there are some that are difficult to determine. Especially cyclic measurements, like for example weekdays. Weekdays (Monday to Sunday) are not numbers, so it is not interval or ratio. There seems to be a logical ordering, so ordinal might actually be a good first guess. However, what is this order then? Some will argue that a week starts on Monday, while others might say on Sunday. So, although there is an order, the starting point is not clear. Unfortunately, there is no consensus on what these cyclical types of variables should be. It can depend on the actual analysis that you want to perform with it.

One more fundamental term to be discussed before we can start analysing data, it's the most important but also most difficult one to understand; significance. This will be the topic for the next section.

**What about interval and ratio levels? (click to expand)**

As mentioned in this crashcourse we will not go into the difference between interval and ratio level, but call them scale. However for those interested the difference between interval variables and ratio variables have to do with the 0 (zero) and how this was chosen. If the zero was chosen so it means the absence of something, then the measurement level will be ratio, if the zero was chosen somewhat arbitrary then it will be interval. This might sound a bit vague, but hopefully some examples make things clearer. With money for example the 0 really means the absence of money (no matter which currency), but 0 degrees in temperature does not mean the absence of temperature (at least not in Fahrenheit and Celsius). Money is therefor an example of a ratio variable, and temperature (in Fahrenheit and Celsius) of interval.

Examples of ratio variables are many: money, weight, length, number of children, number of cars, number of..., miles per gallon, etc. However, examples for truely ordinal variables are hard to find. In my search I always find temperature (in Fahrenheit or Celsius) and sometimes Year, but that is about it (also often IQ but that is not truely an interval scale).

The names of these two scales actually come from a mathematical consequence of their attributes. With an interval scale, the intervals between the values are the same. Take for example 10 ^{o}C = 50 F, 15 ^{o}C = 59 F, 20 ^{o}C = 69 F, so going up 5 ^{o}C is always going also up 9 F. The intervals remain the same. However 20 ^{o}C is twice as much as 10 ^{o}C, but in Fahrenheit is 69 F not twice as much as 50 F. The ratios do therefor not remain the same.

With ratio scales, you might guess it, the ratios do remain the same. If we would convert for example $ to € using $5 = €4.07, $10 =
€8.14, $15 = €12.21, you can see that again for every $5 we go up, we also go up €4.07, but now also that indeed $10 is twice as much as $5, and also in € we then have doubled it.

Note that for the interval scale, the intervals have to be equal. To make things even more complex, there are scales that might have an absolute zero, but no equal intervals. For example the Richter scale to measure earthquakes. At zero there is no movement of the earth at all (an absolute zero), but a two on the Richter scale is actually ten times as much movement than a one. The Richter scale is therefor ordinal. One thing to help with this is to plot a graph of various results of the scale used (for example the Richter scales) against what you actually try to measure (earth movement). If this forms a straight line, then you have at least an interval level, otherwise ordinal.

With the temperature I used Fahrenheit and Celsius. There are temperature scales that use a so-called absolute zero temperature. They set as zero the absolute zero. At this point molecules don't move at all (and you can't go slower than not moving) (strickly speaking from quantum mechanics this is not fully true, but serves the point for now). Scales that use the absolute zero will be ratio, for example Kelvin and Rankine. Consequently we will now for sure then that a doubling in Kelvin will also be a doubling in Rankine.

As a final note on this, we can add and subtract scores from an interval scale, but will need a ratio scale if we also want to multiply and divide. There not many analyses I know of that are allowed for ratio variables, and not for interval (but I might be wrong). One such analyses that is only allowed on ratio level would be the geometric mean.

**Fundamentals**

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