The nominal, ordinal, interval, and ratio scales are levels of measurement in statistics. These scales are broad classifications describing the type of information recorded within the values of your variables. Variables take on different values in your data set. For example, you can measure height, gender, and class ranking. Each of these variables uses a distinct level of measurement.

Stanley Smith Stevens developed these four scales of measurements in 1946. Analysts continue to use them today because how you record your data affects what you can learn from them and the statistical analyses you can perform.

Nominal, ordinal, interval, and ratio scales are the four primary levels of measurement. These levels are listed in increasing order of the detailed information they provide. Let’s take a look at these measurement scales!

**Related post**: What is a Variable?

## Nominal Scales

A nominal scale simply names categories that values for the variable can fall within. Nominal = name. Analysts also refer to nominal variables as both attribute and categorical data.

Nominal scales have values that you can assign to a countable number of distinct groups based on a characteristic. You can name the categories, but they have no natural order. In some cases, nominal data can define groups in your data that you want to compare, such as the experimental conditions of the treatment and control groups.

Examples of nominal scales include gender, marital status, college major, and blood type.

Binary variables are a type of nominal data. These data can have only two values. Statisticians also refer to binary data as indicator variables and dichotomous data. For example, male/female, pass/fail, and the presence/absence of an attribute are all binary data.

The nominal scale is the lowest level of measurement because you can only group the observations, and you cannot order the groups.

Learn more in-depth about Nominal Data: Definition & Examples.

### Appropriate Calculations for Nominal Scales

You cannot calculate the mean, median, or standard deviation for nominal variables because you only have information about categories. The mode is the proper measure of central tendency for nominal scales because it identifies the most common category.

With nominal data, you frequently need to evaluate the categories’ relative frequencies. Consequently, pie charts and bar charts are conventional methods for graphing these variables because they display counts and relative frequencies for each group. For statistical tests, you can use proportion tests.

**Related posts**: Measures of Central Tendency and Relative Frequencies

## Ordinal Scales

Ordinal scales also name groups in your data, but you can place these groups in a natural order. While the order is crucial, the differences between values might not be consistent. In other words, you can rank the values, but you don’t know the relative degrees of difference between them.

Examples of ordinal scales include:

- Education level (primary, secondary, post-secondary).
- Income (low, middle, and high).
- Overall status (poor to excellent).
- Likert scales (e.g., strongly disagree to strongly agree).
- Rank (such as sporting teams and class standings).

Ordinal data have a combination of properties from nominal scales and quantitative properties. On the one hand, these variables have a limited number of discrete values like nominal data. On the other hand, the differences between values provide order information like quantitative variables. However, the difference between adjacent values might not be consistent. For example, first, second, and third in a race are ordinal data. The difference in time between first and second place might not be the same as between second and third place.

Ordinal variables are a step higher than nominal scales as a level of measurement. These scales group observations, like nominal data, but they also allow you to rank-order the values.

Learn more in-depth about Ordinal Data: Definition, Examples & Analysis.

### Appropriate Calculations for Ordinal Scales

Analysts often represent ordinal scales using numbers, such as a 1-5 Likert scale that measures satisfaction. In number form, you can calculate average scores as with quantitative variables. However, the numbers have limited usefulness because the differences between ranks might not be constant. Consequently, many statisticians consider the mean to be an inappropriate measure for ordinal data.

However, the median is a valid measure of central tendency for ordinal variables because the median refers to the middle-*ranked* value—perfect for rank-order data.

When you need to measure the dispersion of ordinal data, use the range, interquartile range, or the spread between two percentiles.

**Related posts**: Median Definition and Uses, Interquartile Range, and Percentiles: Interpretations and Calculations

## Interval Scales and Ratio Scales for Continuous Data and Integers

Continuous variables can take on all numeric values, and the scale can be meaningfully separated into smaller increments, including fractional and decimal values. There are an infinite number of values between any two values. And differences between any two values are always meaningful. Typically, you measure continuous variables on a scale.

Examples of continuous data include height, weight, and temperature.

Statisticians divide continuous data into two types that you measure using interval and ratio scales. Additionally, these scales can also use integers, such as counts of object or events.

### Interval scales

Interval scales frequently record continuous data, but not always—credit and SAT scores are integers. On these scales, the order of values and the *interval*, or distance, between any two points is meaningful. For example, the 20-degree difference between 10 and 30 Celsius is equivalent to the difference between 50 and 70 degrees. However, these variables don’t have a zero measurement that indicates the lack of the characteristic. For example, zero Celsius represents a temperature rather than a lack of temperature.

Due to this lack of a true zero, measurement ratios are not valid for interval scales. Thirty degrees Celsius is not three times the temperature as 10 degrees Celsius. You can add and subtract values on an interval scale, but you cannot multiply or divide them.

Examples of interval scales include temperature in Celsius and Fahrenheit, credit scores (300-850), SAT scores (200-800), and dates on a calendar.

Interval scales are a step higher than ordinal data as a level of measurement. This type allows you to order the values, like ordinal data, but it also allows you to assess the degree of difference between any two values.

With interval scales, you can calculate the mean and standard deviation for the central tendency and variability measures, respectively. However, the median and interquartile range can be more appropriate for skewed distributions.

The coefficient of variation is not valid for this type of measurement because it incorporates a ratio, which leads us to the next section!

**Related post**: Skewed Distributions

### Ratio Scales

Ratio scales can be continuous or discrete data. When it is discrete, it’s usually a count. For this level of measurement, intervals are still meaningful. Additionally, these variables have zero measurements representing a lack of the attribute. For example, zero kilograms indicates a lack of weight. Consequently, measurements *ratios* are valid for these scales. 30 kg is three times the weight of 10 kg. You can add, subtract, multiply, and divide values on a ratio scale.

Examples of ratio scales include temperature in Kelvin (with its absolute zero that represents no temperature), height, weight, speed, and time periods.

Ratio scales are the top level of measurement. Like interval scales, they let you order observations and know the difference between any two values. Additionally, they allow you to assess ratios. A height of 4m is twice as tall as 2m. A period of 10 minutes is twice as long as 5 minutes.

Ratio scales allow you to calculate the same statistics as interval scales plus ratios. Consequently, a measurement like the coefficient of variation is valid for this type of measurement.

**Related post**: Coefficient of Variation

## Summary of Nominal, Ordinal, Interval, and Ratio Scales

Knowing whether your data use the nominal, ordinal, interval, or ratio level of measurement can help you avoid analysis mistakes. Consider the following table that summarizes the capabilities of the various levels.

Valid |
Nominal |
Ordinal |
Interval |
Ratio |

Frequency Distributions | Yes | Yes | Yes | Yes |

Median, percentile ranges | No | Yes | Yes | Yes |

Addition, subtraction, mean, standard deviation | No | No | Yes | Yes |

Multiplication, division, ratios, coefficient of variation | No | No | No | Yes |

**Related post**: Frequency Distributions

## Choosing Your Levels of Measurement

Frequently, the data you need to collect dictates your choice of measurement scale. For instance, if you’re recording gender, eye color, and marital status, those are definitely nominal data. However, if you’re recording time and temperature values, those are either interval or ratio scales. If you’re ranking sports teams, it’ll be an ordinal data.

Occasionally, you’ll need to decide which measurement scale to use. For example, you might want to compare ordinal income groups (low, middle, high). You could also measure income using their currency values, which is a ratio scale.

In these situations, you want to record the data using the level of measurement closest to the ratio end of the spectrum whenever possible. If you wish, you can always recode higher levels down to lower levels, such as converting ratio scales to ordinal and nominal data. However, suppose you only record the nominal or ordinal level of measurement. In that case, you can’t later recode it up to a higher level because you won’t have the necessary detail.

For example, you can recode income in currency to low-, middle-, and high-income groups. However, if you have only the subjects’ income groups, you can’t recover their income in currency.

Color is an interesting variable. If you’re recording eye, hair, and clothes color, it’ll always be a nominal variable. However, in physics, color is more appropriately a ratio scale variable of wavelength.

Occasionally, you can convert an interval to a ratio scale. For example, you can convert temperature in Celsius (Interval) to Kelvin (ratio).

For more information about graphing and analyzing data by their levels of measurement, read the following posts:

- Qualitative vs. Quantitative Data
- Data Types and How to Graph Them
- Hypothesis Testing by Data Types
- Choosing the Correct Type of Regression Analysis

## Reference

Stevens, S.S., On the Theory of Scales of Measurement, Science, 1946

cindy preston says

i’m conducting a study using an ordinal scale 1-5. I’m not sure what the best way is to analyze. there are 40 questions. each question is important to determine how often a specific answer was reported, this is the mode. How do I compare question to question in order to determined specific patterns. Can i do a regression analysis on a nonparametic measure? Or how do i go about this. It is only to investigate preliminary information, not truths.

Jim Frost says

Hi Cindy,

Working with ordinal data can be difficult. It’s often not valid to calculate the mean for ordinal data. However, you can use the median. So, consider comparing the median for different questions. As for regression analysis, you can ordinal logistic regression. This analysis is designed for ordinal outcomes (dependent variables). Although, if you need to use ordinal independent variables, you’ll need to enter them as either continuous or categorical variables depending on which produces the best fit.

I hope that helps!

Denise Martin says

Hi Jim, I got another question for you. Why when calculating standard deviation for a sample n-1 is used instead of N (for a population)? Why is this so? Can you explain this in dummy terms for a high school student and myself for that matter to understand? THanks,

Denise Martin

Jim Frost says

Hi Denise,

When you calculate the variability (or standard deviation) for the entire population, you use the total size in the denominator (N).

But when you’re working with a sample to estimate the whole population, there’s a chance you might not get a perfect representation. Statisticians have found that by luck of the draw, samples tend to be more similar than the population, making the variability seem smaller than it really is for the entire population. We say that the sample standard deviation is biased estimate of the population standard deviation because it tends to be too low.

To correct for this underestimation, statisticians use n – 1 instead of n when calculating the standard deviation for a sample. This tweak makes the estimated variability a bit larger because you’re dividing by a smaller number. Using n – 1 in the denominator makes the sample standard deviation estimate an unbiased estimate of the population standard deviation.

There is a more complicated proof for this, but that’s the gist.

I hope that helps clarify it!

Al Antelman says

Hi Jim;

I have six levels of effort:

0 – Zero effort is require

1 – Minimum effort required

2 – Moderate effort required

3 – Extreme effort required

4 – Maximum effort required

5 – Impossible effort required

I have a detailed description for each level of effort.

Are my level of efforts interval or ratio? If they are interval, is there a way to convert them to ratio?

Jim Frost says

Hi Al,

Those are ordinal data. The only way to make them ratio level is if you can actually quantify the amount of effort using some sort of measuring. Because these levels use a qualitative assessment, they can’t be ratio or interval, only ordinal. I know you’re using numbers to code the levels, but given the qualitative nature of the data, they’re ordinal.

Marty Shudak says

Thank you, Jim. Excellent review of such important concepts! I was wondering if you can explain the 3rd to last paragraph I copied below about hair color being ordinal? I thought hair color should be nominal??

Color is an interesting variable. If you’re recording eye, hair, and clothes color, it’ll always be an ordinal variable. However, in physics, color is more appropriately a ratio scale variable of wavelength.

Jim Frost says

Hi Marty,

That’s a great catch! You’re correct that it should be nominal. My brain must have slipped a gear there! I’m off to fix that now.

Denise Martin says

High School Test scores (such as 78, 80, or 95 percent correct).

Interval or ratio?

Jim Frost says

Hi Denise,

Those scores would be ratio scale because there is a true zero. It’s possible to get 0% scores correct, which is the absences of any correct answers. Additionally, if one student got 80% correct and another got 40%, it’s accurate to say that the first student got twice as many correct as the second student. That’s a 2:1 ratio. Hence, ratio scale.

Andrew says

Hi Jim. So, if I asked “What is your weight in kilograms? and gave a choice: <50 kgs, 51-70kgs, 71-90kgs, more than 110kgs, these have to be interval scales because you cannot be zero kgs?

Jim Frost says

Hi Andrew,

That would be an ordinal scale. You can place the observations in a natural order, but you don’t know the precise difference between observations. Additionally, because you don’t know the precise difference (i.e., the interval between values), it can’t be interval scale.

Melissa Menier says

Hi, Jim! Looking for your insight on this: I see many sites claiming that age is a ratio level of measurement, but I don’t envision “birth” (zero) as “Absence of Age”, and I have analysts calculating a mean using both positive (post-natal diagnosis age) and negative (pre-natal diagnosis age) which renders the interpretation useless. Thoughts?

Jim Frost says

Hi Melissa,

You have an interesting question!

Yes, I would definitely say that human age is a ratio scale variable. More generally, time periods (i.e., durations) are ratio scale. They have a natural zero point and cannot be negative. You can a time period of zero, which has an absence of a duration. Age of a human is the duration since birth. Hence, it is ratio scale. To test this, ask yourself whether it makes sense to say that a baby who is two years old is twice as old as a one-year old baby. It does make sense because the duration is twice as long. Only ratio scales allow you to assess the ratio like that. On this scale, zero represents the instant of birth, or zero duration since birth. So, I’d think of it less as an absence of age but an absence of duration since birth.

However, you bring up an interesting point with the negative ages for a scale with both pre-natal and post-natal ages. You’re correct that is an interval scale. It doesn’t make sense to use ratios of positive and negative ages. For interval scales, it is appropriate to calculate the means. So, it’s not causing any problems there.

If you lop off the pre-natal ages, it becomes a ratio scale. So, when considering only post-natal ages, you can treat it as ratio scale, as discussed above.

Bob E. says

When you say,” Examples of ratio scales include temperature in Kelvin (with its absolute zero that represents no temperature)…” did you mean,” …absolute zero that represents no heat….”

Jim Frost says

Yep! That’s it exactly!