Stat 95 Kinds of Data and Scales of Measurement

Qualitative (categorical) Data

This type of data is all about putting things into groups or categories. It describes qualities or characteristics, not amounts. You can't do math with this type of data, like adding or subtracting it, because the values are labels rather than numbers. This type of data is often used to sort or count items. ex. hair color, favorite sports

Quantitative (Numerical) Data

This is the opposite of qualitative data. This type of data is all about numbers! These are things that you can count or measure, and you can do math with them. Quantitative data can be further broken down into two types

Discrete Data

type of quantitative data. This is data that can only take on specific, countable values. You can't have half a value. For example, the number of goals scored in a game is discrete; you can have 1, 2, or 3 goals, but not 1.5.

Continuous Data

type of quantitative data. This is data that can take on any value within a given range. For example, time, height, and weight are continuous because they can be measured with infinite precision (e.g., 5.2 seconds, 5.21 seconds, 5.215 seconds).

Nominal Scale

used for data that is just names or labels. There is no order or ranking. It's the most basic scale. The only statistical measure you can use on nominal data is the mode (the most frequent category) or a frequency count (how many items are in each category). ex. Jersey numbers: A football player's jersey number, like #10, is a nominal number. It's just a way to identify them. It doesn't mean the player with #10 is better than the player with #34. The number is just a label. This is a qualitative example.

Ordinal Scale

for data that can be put in a clear order or rank, but you can't really say how much "better" or "more" one item is than another. While you know the order, the intervals between each value are not equal. This means you can find the median (the middle value) and the mode, but calculating a mean (average) is not appropriate. ex. Finishing places in a race: You can say that a runner who finished 1st is better than a runner who finished 2nd, who is better than a runner who finished 3rd. There is a clear order. However, you don't know how much time passed between the 1st and 2nd runners, or between the 2nd and 3rd. It's just the rank. This is a qualitative example.

Interval Scale

for data with a clear order, and the distance between each point is equal. The big thing here is that there is no true zero. This means that a value of zero doesn't actually mean "nothing" or "none." Since the intervals are equal, you can do addition and subtraction to find the exact difference between them, and you can calculate the mean. However, you cannot use multiplication or division to create meaningful ratios. ex. Temperature in Celsius or Fahrenheit: If you measure the temperature of a gym, 20 degrees Celsius is 10 degrees hotter than 10 degrees Celsius. The difference is the same no matter where you are on the scale. But 0 degrees Celsius doesn't mean there is "no temperature." It's just a point on the scale. This is a quantitative example.

Ratio Scale

is the most powerful scale. It has a clear order, equal intervals, and it has a true zero. This means that a value of zero means there is none of that thing. Because of the true zero, you can do all kinds of math with it, including multiplication and division, and you can create meaningful ratios. This is the only scale where you can say "this is twice as much as that." Weight: An athlete who weighs 200 pounds is twice as heavy as an athlete who weighs 100 pounds. A weight of 0 pounds means there is no weight at all. This is a quantitative example.

Nominal Scale Limitations

the most restrictive. Its main limitation is that it provides no information beyond categorization. Since the data has no order, you can't perform any mathematical operations like addition, subtraction, or averaging. You can only find the frequency (how many items are in each category) and the mode (the most common category). Trying to rank or compare nominal data is a fundamental mistake.

Ordinal Scale Limitations

primary shortcoming is the lack of equal intervals. You know that one category is "better" or "more" than another, but you can't quantify the difference between them. This means you can't say that the difference between a 1st and 2nd place finish is the same as the difference between a 2nd and 3rd place finish. Therefore, calculating an average (mean) for ordinal data is technically incorrect and can be very misleading. You can, however, find the median and the mode.

Interval Scale Limitations

absence of a true zero point. A value of zero does not signify the complete absence of the property being measured. This means you can't create meaningful ratios or make statements like "A is twice as much as B." For example, while you can say that the difference between 20∘C and 10∘C is 10∘, you cannot say that 20∘C is twice as hot as 10∘C. This is a critical limitation that restricts the types of statistical analyses you can perform, as you cannot use multiplication or division to compare values.

Ratio Scale Limitations

he most powerful, its main limitation is its applicability. It only works for data that has a true zero. You can't force a nominal, ordinal, or interval variable into a ratio scale. Using ratio statistics on data that lacks a true zero will lead to incorrect and nonsensical conclusions, such as claiming a test score of 80 is twice as "smart" as a score of 40, when in fact the test is on an interval scale.

Qualitative Data Nominal Data

Data in the form of categories or labels. ex. types of cars (Sedan, SUV, Truck), Hair Color (blonde, brown, black)

Quantitative Data Nominal Data

Digits are NOT used as quantities, just as labels. ex. Male = 0 Female = 1

Qualitative Data Ordinal Data

Data in the form of categories or labels that have a meaningful order, but the “distances” or differences between categories aren’t necessarily the same size. ex. movie ratings (good, better, best), customer satisfactions (very dissatisfied, neutral, satisfied), colors (red, orange, yellow, green)

Quantitative Data Ordinal Data

Digits used to indicate more or less but not how much more or less. For instance, the difference between 2 and 3 can be different from the difference between 4 and 5. Examples: Ratings like 1, 2, 3, 4, 5; Rankings like 1st, 2nd, 3rd, 4th place.

Quantitative Data Interval Data

Numbers with a consistent, meaningful interval between values, but no true zero point. Temperature of 0 does not mean "no temperature". Examples: Temperature in Celsius or Fahrenheit; Years on a calendar (e.g., 2010, 2015, 2020)

Quantitative Data Ratio Data

Data with a consistent interval and a true zero point, allowing for all mathematical operations including multiplication and division. Examples: Height, weight, age, income.

Simple Random Sampling

“Trust the Math” method. every individual or item in the population (everyone) has an equal chance of being selected for the sample. [A lie in most cases] This is often done using a random number generator.

Stratified Sampling

“Use the Science” Method. the population is divided into subgroups (strata) based on shared characteristics (eg. age, gender, location). a simple random sample is then taken from each subgroup.

Convenience Sampling

“Just for fun” method. The sample is composed of individuals who are readily available and wiling to participate. highly prone to selection bias. the sample is unlikely to be representative of the population, making it difficult to generalize these findings.

Aggregate the values

cells. ex. find the average

Aggregate the attributes

columns. ex. correlation

Aggregate the things

rows. ex. count, compute similarity