Is Exact Height Discrete or Continuous

Perhaps one of the simplest but toughest questions for my intro (and graduate) stats students seems to be those asking to classify a variable as discrete or continuous.

My quick rule of thumb (heuristic) is to think about whether the variable is countable or whether it must be measured. I tried to come up with a mnemonic like "population-parameter; sample: statistic" but the best I could do is "finger : digit: discrete" since you have to count your fingers.

Dogs, cats, people, houses, touchdowns, are countable, so they are discrete variables. And we do not often think of dividing a dog or house into parts, e.g. 1.6 dogs, so again that sounds like they are discrete.

Things we measure are Continuous

A person's weight, gallons of water, the length of a football field, the speed of a car, the temperature of the ocean, price of gas, all must be measured, so they are continuous variables. Another clue is that continuous variables are often stated as fractions or decimals, as in 2.5 gallons of gas.

But then there are some things that are not so obvious. Take the height of buildings. Most commonly, we in America tend to think of buildings in terms of the number of floors. I can remember when the Empire State Building was the tallest building in the world and us kids could quickly tell you it had 102 floors. Floors are countable, so that would make the height of a building discrete, right?

Well, in many other places, buildings are measured. In fact, if you Google the height of the Empire State Building, you would find it is 443.2 meters tall if you include the antenna, which most building owners do so they can brag a bit more. Well, that sounds like someone had to actually measure the height, doesn't it, so that means building height is really a continuous variable. And, of course, we all have to measure our own height as well.

Time is Continuous

Another confusing variable is time. We tend to think of time as the number of years we have lived, or the days until Christmas, which we can count. But when I was a little kid, I always answered "six and a half" when asked how old I was. So, that was a clue that time could be divided into smaller pieces, which would mean time is not really a discrete variable.

And today, we no longer count the number of moons that have passed. We measure time using very accurate devices. If you want a bit of distraction, check out this page showing a timeline of our universe.

I do hope your statistics instructors do not put purposefully confusing questions on quizzes and exams about time. But if they do, you are now armed to be able to persuade them your answer that time is a continuous variable is correct.

Unless your instructor is also a particle physicist. Then, they are likely to tell you time is both discrete and continuous.

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Data can be Descriptive (like "high" or "fast") or Numerical (numbers).

And Numerical Data can be Discrete or Continuous:

Discrete data is counted,
Continuous data is measured

Discrete Data

Discrete Data can only take certain values.

Example: the number of students in a class

We can't have half a student!

Example: the results of rolling 2 dice

Only has the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12

Continuous Data

Continuous Data can take any value (within a range)

Examples:

• A person's height: could be any value (within the range of human heights), not just certain fixed heights,
• Time in a race: you could even measure it to fractions of a second,
• A dog's weight,
• The length of a leaf,
• Lots more!

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ORGANISING DATA

After collection and processing, data need to be organised to produce useful information. It helps to be familiar with some definitions when organising data. This section outlines those definitions and provides some simple techniques for organising and presenting data.

Definition

The word variable is often used in the study of statistics and so it is important to understand its meaning. A variable is:

ANY TRAIT THAT IDENTIFIES DIFFERENT VALUES FOR DIFFERENT PEOPLE OR ITEMS.

Height, age, amount of income, country of birth, grades obtained at school and type of housing are examples of variables. Variables may be classified into various categories, some of which are outlined in the following pages.

NOMINAL VARIABLES

A nominal (also called categorical) variable is one that describes a name or category.

EXAMPLE

1. The method of travel to work by people in Darwin at the time of the 1996 Census was:

Method of travel to work	Number of people
CAR AS DRIVER	23,617
CAR AS PASSENGER	3,699
BICYCLE	1,335
WALKED	1,703
BUS	1,335
WORKED AT HOME	1,012
MOTOR BIKE/SCOOTER	577
TAXI	284
TRAIN	25
FERRY/TRAM	11

In this case the variable 'method of travel to work' is nominal because it describes a name.

NUMERIC VARIABLE

A numeric variable is one that describes a numerically measured value. However, not all variables described by numbers are numeric. For example, the age of a person is a numeric variable, but their year of birth, despite being a number, is a nominal variable.

Numeric variables may be either continuous or discrete:

CONTINUOUS VARIABLE

A variable is said to be continuous if it can take any value within a certain range. Examples of continuous variables may be distance, age or temperature.

The measurement of a continuous variable is restricted by the methods used, or by the accuracy of the measuring instruments. For example, the height of a student is a continuous variable because a student may be 1.6321748755... metres tall.

However, when the height of a person is measured, it is usually only measured to the nearest centimetre. Thus, this student's height would be recorded as 1.63m.

Note that continuous variables are usually grouped using class intervals (explained shortly). They are grouped to make them easier to handle as part of the general process of organising data into information.

DISCRETE VARIABLE

Any variable that is not continuous is discrete. A discrete variable can only take a finite number of values within a certain range. An example of a discrete variable would be a score given by a judge to a gymnast in competition: the range is 0-10 and the score is always given to one decimal place.

Discrete variables may also be grouped. Again, this is done to make them easier to handle.

NOTE: measurement of a continuous variable is always a discrete approximation.

ORDINAL VARIABLE

An ordinal variable is one that can be placed in order. Numeric variables are always ordinal, while only some nominal variables are ordinal.

1. A teacher may rank a class of students in order according to their behaviour:

Behaviour	Number of Students
Excellent	5
Very Good	12
Good	10
Bad	2
Very bad	1

In this case the variable 'behaviour' is nominal and ordinal.

Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only take $N = 3$ distinct values it is a discrete random variable with a probability mass function.

For example, we could have $N = 3$ people with heights $150$ cm, $160$ cm, and $170$ cm, and thus the probability of any particular height occuring is $1/3$.

Now, consider the case of $X^N$ when $N = 6$ billion, i.e. the heights of the real-world human population. We now have $6$ billion distinct values for $X^N$. Although there is now a very large range of values $X^N$ can take, it is still a discrete random variable as those $6$ billion discrete values are the only values of $X^N$ that can occur.

Therefore, heights of humans is actually a discrete random variable and not a continuous random variable? Everywhere I look it says that human heights is a continuous random variable with a pdf, but it seems from the above that it is actually a discrete random variable with a pmf?

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