What is the main difference between a ratio variable and an interval variable?

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While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.

Many statistics books begin by defining the different kinds of variables you might want to analyze. This scheme was developed by S. Stevens and published in 1946.

Definitions

A categorical variable, also called a nominal variable, is for mutually exclusive, but not ordered, categories. For example, your study might compare five different genotypes. You can code the five genotypes with numbers if you want, but the order is arbitrary and any calculations (for example, computing an average) would be meaningless.

An ordinal variable, is one where the order matters but not the difference between values. For example, you might ask patients to express the amount of pain they are feeling on a scale of 1 to 10. A score of 7 means more pain than a score of 5, and that is more than a score of 3. But the difference between the 7 and the 5 may not be the same as that between 5 and 3. The values simply express an order. Another example would be movie ratings, from * to *****.

An interval variable is a one where the difference between two values is meaningful. The difference between a temperature of 100 degrees and 90 degrees is the same difference as between 90 degrees and 80 degrees.

A ratio variable, has all the properties of an interval variable, but also has a clear definition of 0.0. When the variable equals 0.0, there is none of that variable. Variables like height, weight, enzyme activity are ratio variables. Temperature, expressed in F or C, is not a ratio variable. A temperature of 0.0 on either of those scales does not mean 'no heat. However, temperature in Kelvin is a ratio variable, as 0.0 Kelvin really does mean 'no heat'. Another counter example is pH. It is not a ratio variable, as pH=0 just means 1 molar of H+. and the definition of molar is fairly arbitrary. A pH of 0.0 does not mean 'no acidity' (quite the opposite!). When working with ratio variables, but not interval variables, you can look at the ratio of two measurements. A weight of 4 grams is twice a weight of 2 grams, because weight is a ratio variable. A temperature of 100 degrees C is not twice as hot as 50 degrees C, because temperature C is not a ratio variable. A pH of 3 is not twice as acidic as a pH of 6, because pH is not a ratio variable.

The categories are not as clear cut as they sound. What kind of variable is color? In some experiments, different colors would be regarded as nominal. But if color is quantified by wavelength, then color would be considered a ratio variable. The classification scheme really is somewhat fuzzy.

What is OK to compute

OK to compute....

Nominal

Ordinal

Interval

Ratio

frequency distribution

Yes

Yes

Yes

Yes

median and percentiles

No

Yes

Yes

Yes

sum or difference

No

No

Yes

Yes

mean, standard deviation, standard error of the mean

No

No

Yes

Yes

ratio, or coefficient of variation

No

No

No

Yes

Does it matter?

It matters if you are taking an exam in statistics, because this is the kind of concept that is easy to test for.

Does it matter for data analysis? The concepts are mostly pretty obvious, but putting names on different kinds of variables can help prevent mistakes like taking the average of a group of postal (zip) codes, or taking the ratio of two pH values. Beyond that, putting labels on the different kinds of variables really doesn't really help you plan your analyses or interpret the results.

In the 1940s, Stanley Smith Stevens introduced four scales of measurement: nominal, ordinal, interval, and ratio. These are still widely used today as a way to describe the characteristics of a variable. Knowing the scale of measurement for a variable is an important aspect in choosing the right statistical analysis.

Nominal

A nominal scale describes a variable with categories that do not have a natural order or ranking. You can code nominal variables with numbers if you want, but the order is arbitrary and any calculations, such as computing a mean, median, or standard deviation, would be meaningless.

Examples of nominal variables include:

• genotype, blood type, zip code, gender, race, eye color, political party

Ordinal

An ordinal scale is one where the order matters but not the difference between values.

Examples of ordinal variables include:

• socio economic status (“low income”,”middle income”,”high income”), education level (“high school”,”BS”,”MS”,”PhD”), income level (“less than 50K”, “50K-100K”, “over 100K”), satisfaction rating (“extremely dislike”, “dislike”, “neutral”, “like”, “extremely like”).

Note the differences between adjacent categories do not necessarily have the same meaning. For example, the difference between the two income levels “less than 50K” and “50K-100K” does not have the same meaning as the difference between the two income levels “50K-100K” and “over 100K”.

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Interval

An interval scale is one where there is order and the difference between two values is meaningful.

Examples of interval variables include:

• temperature (Farenheit), temperature (Celcius), pH, SAT score (200-800), credit score (300-850).

Ratio

A ratio variable, has all the properties of an interval variable, and also has a clear definition of 0.0. When the variable equals 0.0, there is none of that variable.

Examples of ratio variables include:

• enzyme activity, dose amount, reaction rate, flow rate, concentration, pulse, weight, length, temperature in Kelvin (0.0 Kelvin really does mean “no heat”), survival time.

When working with ratio variables, but not interval variables, the ratio of two measurements has a meaningful interpretation. For example, because weight is a ratio variable, a weight of 4 grams is twice as heavy as a weight of 2 grams. However, a temperature of 10 degrees C should not be considered twice as hot as 5 degrees C. If it were, a conflict would be created because 10 degrees C is 50 degrees F and 5 degrees C is 41 degrees F. Clearly, 50 degrees is not twice 41 degrees.  Another example, a pH of 3 is not twice as acidic as a pH of 6, because pH is not a ratio variable.

Learn more about the difference between nominal, ordinal, interval and ratio data with this video by NurseKillam

 OK to compute....

Nominal

Ordinal

Interval

Ratio

 Frequency distribution

Yes

Yes

Yes

Yes

 Median and percentiles

No

Yes

Yes

Yes

 Add or subtract

No

No

Yes

Yes

 Mean, standard deviation, standard error of the mean 

No

No

Yes

Yes

 Ratios, coefficient of variation

No

No

No

Yes

Knowing the measurement scale for your variables can help prevent mistakes like taking the average of a group of zip (postal) codes, or taking the ratio of two pH values. Beyond that, knowing the measurement scale for your variables doesn’t really help you plan your analyses or interpret the results.

Note that sometimes, the measurement scale for a variable is not clear cut. What kind of variable is color? In a psychological study of perception, different colors would be regarded as nominal. In a physics study, color is quantified by wavelength, so color would be considered a ratio variable. What about counts? 

There are occasions when you will have some control over the measurement scale. For example, with temperature, you can choose degrees C or F and have an interval scale or choose degrees Kelvin and have a ratio scale. With income level, instead of offering categories and having an ordinal scale, you can try to get the actual income and have a ratio scale. Generally speaking, you want to strive to have a scale towards the ratio end as opposed to the nominal end.

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Each scale is represented once in the list below. 

• Favorite candy bar
• Weight of luggage
• Year of your birth
• Egg size (small, medium, large, extra large, jumbo)

Each scale is represented once in the list below. 

• Military rank
• Number of children in a family
• Jersey numbers for a football team
• Shoe size

Answers: N,R,I,O and O,R,N,I

There are other ways of classifying variables that are common in statistics. One is qualitative vs. quantitative. Qualitative variables are descriptive/categorical. Many statistics, such as mean and standard deviation, do not make sense to compute with qualitative variables. Quantitative variables have numeric meaning, so statistics like means and standard deviations make sense. 

This type of classification can be important to know in order to choose the correct type of statistical analysis. For example, the choice between regression (quantitative X) and ANOVA (qualitative X) is based on knowing this type of classification for the X variable(s) in your analysis.

Quantitative variables can be further classified into Discrete and Continuous. Discrete variables can take on either a finite number of values, or an infinite, but countable number of values. The number of patients that have a reduced tumor size in response to a treatment is an example of a discrete random variable that can take on a finite number of values. The number of car accidents at an intersection is an example of a discrete random variable that can take on a countable infinite number of values (there is no fixed upper limit to the count).

Continuous variables can take on infinitely many values, such as blood pressure or body temperature. Even though the actual measurements might be rounded to the nearest whole number, in theory, there is some exact body temperature going out many decimal places That is what makes variables such as blood pressure and body temperature continuous. 

It is important to know whether you have a discrete or continuous variable when selecting a distribution to model your data. The Binomial and Poisson distributions are popular choices for discrete data while the Gaussian and Lognormal are popular choices for continuous data.

The list below contains 3 discrete variables and 3 continuous variables:

• Number of emergency room patients
• Blood pressure of a patient
• Weight of a patient
• Pulse for a patient
• Emergency room wait time rounded to the nearest minute
• Tumor size

Answers: d,c,c,d,d,c

Note, even though a variable may discrete, if the variable takes on enough different values, it is often treated as continuous. For example, most analysts would treat the number of heart beats per minute as continuous even though it is a count. The main benefit of treating a discrete variable with many different unique values as continuous is to assume the Gaussian distribution in an analysis. 

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