Two Kinds of Numerical Outcomes
In What Is a Random Variable, you learned that a random variable assigns a number to each possible outcome of a chance process. The next useful question is: what kinds of numbers can it take? A variable that counts separate items behaves differently from one that measures a quantity that can vary across a range. This distinction helps us choose and interpret an appropriate probability model.
The number of siblings a student has is a count: 0, 1, 2, 3, and so on. A student cannot have 2.4 siblings. A student’s height, on the other hand, is a measurement. Between two different heights, there are many other possible heights. These are typical examples of discrete and continuous random variables.
“Countably infinite” means that the values can be listed in sequence, even if the list does not end. For example, a count might take the values 0, 1, 2, 3, and so on. The possible values are separate; there are no possible count values between 2 and 3.
For a continuous variable, the possible values are not separated by fixed jumps. Between 170 centimeters and 171 centimeters, for example, there are possible heights such as 170.5 centimeters, 170.25 centimeters, and many others. A continuous model treats measurement values as varying along an interval, not as restricted to the handful of values a measuring device displays.
Start with the Variable’s Definition
The same general subject can lead to different kinds of variables, depending on what is defined. “Number of siblings” is a count and therefore discrete. “Height in centimeters” is a measurement and is modeled as continuous. Likewise, the number of minutes a person waits when recorded as whole minutes is discrete as recorded, while the actual duration of the wait is modeled as continuous.
The important question is not whether a number appears in the data. Both kinds of variables use numbers. Instead, ask whether the variable counts separate items or measures a quantity that can vary across an interval. Then check its possible values. This definition-first approach extends the advice from What Is a Random Variable: name what the variable counts or measures before classifying it.
Worked Example: Number of Siblings or Height?
A school counselor defines \(S\) as the number of siblings a randomly selected student has and \(H\) as that student’s height in centimeters. Classify each random variable and explain the difference.
State. \(S\) counts siblings. \(H\) measures height in centimeters. Both variables describe numerical features of the selected student, but one is a count and the other is a measurement.
Plan. For each variable, consider the values it can take. Separate count values have no possible values between consecutive integers. Measurement values can vary between nearby points in an interval.
Do. The possible values of \(S\) are nonnegative whole numbers: 0, 1, 2, and so on. A value such as \(S=2.4\) would not make sense because a count of siblings cannot be fractional. Thus, \(S\) is discrete.
Height can be 165 centimeters, 165.4 centimeters, or a value between those examples. The measuring instrument may display only a rounded value, but the student’s height is a measurement that can vary across a range. Thus, \(H\) is modeled as continuous.
Conclude. \(S\) is discrete because it counts siblings and takes separate whole-number values. \(H\) is continuous because it measures height, which can vary across an interval. The distinction comes from what each variable represents, not from the fact that both are recorded with numbers.
Recording Precision Does Not Always Determine the Model
Measurements are often rounded because instruments and records have limited precision. A height might be reported as 165 centimeters, a race time as 18.6 seconds, or a temperature as 21 degrees Celsius. The recorded values may look like a finite set of separate numbers. But that does not necessarily mean the underlying variable is discrete.
It helps to distinguish the underlying quantity from the recorded value. If the quantity is a measurement that can vary across an interval, it is often modeled as continuous, even when each observation is rounded. If the variable is explicitly defined as the rounded record, then that recorded variable takes separate values and is discrete. State clearly which version the problem asks about.
A count is different. Reporting a count as a whole number is not merely rounding a more precise count: the number of broken parts, siblings, or customers is already a whole-number count. By contrast, rounding a measured length to the nearest centimeter changes how the measurement is recorded.
Worked Example: Measured Height and Rounded Height
A garden project records the height of each seedling in centimeters, rounded to the nearest whole centimeter. Let \(H\) be a seedling’s underlying height and \(R\) be its recorded, rounded height. Classify \(H\) and \(R\).
State. \(H\) represents the actual height measurement in centimeters. \(R\) represents the whole-number value written in the project’s record after rounding.
Plan. Classify each variable according to its own definition. Do not assume that the record and the underlying measurement have the same possible values.
Do. The underlying height \(H\) can vary between, for example, 12 centimeters and 13 centimeters. It could be 12.3 centimeters or another value in that interval. Therefore, \(H\) is modeled as continuous.
The recorded variable \(R\) can take whole-number values such as 12 or 13 centimeters, but not 12.3 centimeters as recorded. Its possible values are separate, so \(R\) is discrete. For instance, an underlying height of 12.3 centimeters would be recorded as 12 centimeters when rounded to the nearest whole centimeter.
Conclude. \(H\) is continuous because it represents the underlying measurement. \(R\) is discrete because it represents a rounded record that takes separate whole-number values. The classification depends on exactly what the random variable is defined to represent.
Why the Distinction Matters for Probability Models
A probability model describes the possible values of a random variable and how probability is associated with them. For a discrete random variable, it is natural to discuss the probability of each possible value, such as the probability that a randomly selected household has exactly two pets. The possible values can be listed or described as separate count values.
For a continuous random variable, probability questions usually concern ranges of values, such as the probability that a randomly selected seedling is between 12 and 14 centimeters tall. A continuous model describes probability across intervals of measurement values. The exact mathematical tools for probability distributions come next; for now, the key point is that counts and measurements have different structures of possible values.
Using the wrong classification can make a model unsuitable or lead to unclear probability statements. A model built for separate count values may not represent the possible values of an underlying measurement. Likewise, treating a count as if it could take any decimal value would introduce impossible outcomes. Correct classification is an early modeling decision, made before calculating probabilities.
Worked Example: Defects per Package or Package Weight?
A quality-control team randomly selects a sealed package. Let \(D\) be the number of damaged pieces inside and \(W\) be the package’s weight in grams. Explain how the variables should be classified and how that distinction affects the kind of probability question that makes sense.
State. \(D\) counts damaged pieces in one package. \(W\) measures the package’s weight in grams.
Plan. Examine the possible values of each variable, then give an example of a probability question suited to its count or measurement scale.
Do. \(D\) can be 0, 1, 2, and so on, up to the number of pieces in the package. It cannot be 1.5 damaged pieces. The possible values are separate counts, so \(D\) is discrete. A suitable question is, “What is the probability that the package contains exactly one damaged piece?”
\(W\) is a measurement. The underlying weight can vary across a range and can have values between, for example, 250 grams and 251 grams. It is modeled as continuous, even if a scale records it to the nearest gram. A suitable question is, “What is the probability that the package weighs between 250 and 251 grams?” If the variable were instead defined as the scale’s rounded whole-gram reading, that recorded variable would be discrete.
Conclude. \(D\) is discrete because it counts damaged pieces; \(W\) is continuous because it measures weight. A probability model for \(D\) focuses on separate count values, while a model for \(W\) focuses on ranges of measurement values. The package and selection process are the same, but the variables call for different ways of describing possible results.
A Practical Classification Routine
When a problem asks you to classify a random variable, use the variable’s definition rather than relying on a quick impression. This short routine helps identify both the type and the reason.
Determine whether the variable counts items, measures a quantity, or describes a recorded version of a quantity.
For a count, identify the separate possible integer values. For a measurement, consider whether values can vary throughout an interval.
Decide whether rounding is only a way to record an underlying measurement or is part of the variable’s definition.
Use a complete sentence that connects the variable’s meaning to its possible values.
This routine also helps with less obvious cases. The number of seconds a runner takes to finish is a measurement and is modeled as continuous, even if a results sheet reports whole seconds. The number of seconds shown on that sheet is a different, rounded recorded variable. The number of text messages received in a day is a count, so it is discrete. “Age” depends on the definition: exact age measured in years is a continuous quantity, while age in completed whole years is a discrete recorded count of years.
Common Mistakes and AP Exam Tips
- Calling every variable with decimal values continuous. A decimal may be a code or a rounded record. Explain what the variable represents and examine its possible values.
- Calling a measurement discrete just because it is rounded. If the variable is the underlying height, time, or weight, rounding for a report does not change the nature of that measurement. If the variable is specifically the rounded report, classify that recorded variable separately.
- Assuming every discrete variable has only a few possible values. A count can have many possible values. Discrete means the values are separate and countable, not necessarily that there are only two or three.
- Using “it is a number” as the explanation. Continuous and discrete variables are both numerical. A useful explanation names the count or measurement and describes the possible values.
- Confusing the observation with the variable. A recorded height of 165 centimeters is one value. The random variable \(H\) describes the height measurement that could take other values.
- Forgetting units or context. “Weight is continuous” is less clear than “\(W\), the package’s weight in grams, is modeled as continuous because it is a measurement that can vary across an interval.”
For a full-credit response, name the variable, classify it, and justify the classification from its definition. For example: “\(D\), the number of damaged pieces in a package, is discrete because it is a count and can take only separate whole-number values.” Or: “\(W\), the underlying package weight in grams, is modeled as continuous because it is a measurement that can vary across an interval.” If rounding matters, say whether you mean the underlying measurement or the rounded record.
Key Takeaway
Counts such as number of siblings and number of damaged pieces take separate, countable values, so they are discrete. Measurements such as height, time, and weight can vary across intervals, so they are modeled as continuous. A rounded record can be discrete even when the underlying measurement is continuous.
Check Your Understanding
Classify each variable as discrete or continuous, and explain your reasoning. When rounding is involved, state whether the variable is the underlying measurement or the recorded value.
- Let \(X\) be the number of siblings reported by a randomly selected student. What values can \(X\) take, and why is it discrete?
- A device measures the temperature of a greenhouse to the nearest degree Celsius. Compare the classification of the underlying temperature with the classification of the rounded reading.
- Let \(T\) be the actual time, in seconds, it takes a cyclist to complete a course. Is \(T\) a count or a measurement? Explain its likely classification.
- A warehouse records the number of boxes loaded onto each truck. Why is this variable discrete even if a truck can carry a large number of boxes?
- A scale reports a parcel’s weight to the nearest tenth of a kilogram. Explain the difference between the parcel’s underlying weight and the recorded reading.