Triangles and Quadrilaterals

When you know the lengths of the three sides of the triangle you know all need to know to make the triangle, that is, there is nothing more needed to define that exact triangle you ‘re talking about. Quadrilateralsare different, we need more than just the four lengths because they can swing all over the place and change shape. We examined one particular family of four sided figures, ones with two pairs of equal lengths. These make parallelograms if the same lengths don’t meet at any corner (vertex) and they make kites otherwise. Parallelograms have sides which are parallel and have opposite angles equal but their diagonals are not always perpendicular. Kites have one pair of angles the same, where two different lengths meet, and their diagonals meet at right angles (perpendicular). If we know the sizes of the sides of a parallelogram or kite we can work out the distance round the shape (perimeter) but we cannot work out the area. For parallelograms we need to know the perpendicular distance between a pair of sides (ph) and the length of that side (s) to get the area A = ph * s. For kites we need to know the length of the two diagonals r and t, so that the area A = ½ * r * t.

You can work out the area if you know the angles but we’ll save that till later.

What Kind of Data?

Categorical nominal/ordered Numerical discrete/continuous.

It is very difficult to decide which kind of data we’re dealing with at times. Sometimes it’s very clear for example when the data is the name of the country you were born in (categorical nominal) when the answer is basically a name, or if you ask someone to tick how they feel when they pick from: bad, ok, good (categorical ordered) when there’s a name but a sensible sequence to the possible answers. The difference between these two sorts of data is pretty clear in these cases.

Sometimes numbers are used as categorical data, eg What is the number of your door? There is clearly a sequence to the answers and they may indicate how far a house is along a street (categorical ordered), however this may not make sense if the street has a mix of odd and even numbers on different sides and they don’t match up, or if the houses are built in a circle as there will be a gap of one between neighbouring houses except for the first and last which will seem to have an enormous gap but are beside each other. Another data which looks like numbers but are categorical, is the identification number on a footballer’s shirt.

Numerical Data

Numerical data is always written as a number and is the result of counting things or measuring things. If it’s counted it’s discrete if it’s measured it’s continuous. For example the amount of sugar you buy is measured, (we generally don’t count the number of grains). The number of apples you buy is counted, but the weight of apples you buy is measured, so the number of apples is numerical discrete but the weight is numerical continuous. This seems easy until you measure something and write down the answer. If you measure the weight of a bag of sugar you will get a number very close to 1kg, perhaps 1.003kg or 0.996kg, so have you counted how many grammes (a thousandth of a kilo) in the bag or measured it? When you measure the height of a tree is it counting the number of centimetres? This rather silly question determines whether the answer is discrete or continuous. One way to look at it, is that a measurement (continuous) can always be more accurate, but counting is always exact. The best guide is to ask if there are more than 20 possible answers. If there are more than 20 then it’s close enough to continuous if there are fewer it’s discrete. Another good guide is to ask whether a result between two that you already have could actually happen; for example you may measure potato plants to be 38cm and 39cm and it’s perfectly plausible that another one could be 38.4cm, these are continuous, but if you count elephants in a field you won’t get two and a half an elephants! So the number of elephants is numerical discrete.

This is a difficult area of statistics to understand, so if you’re confused then you are getting to grips with the problem. It’s like looking at a picture on a screen and asking if it’s a collection of dots (discrete) or a real picture (continuous), it‘s usually a collection of dots if you get a big magnifying glass, but if the dots are small enough they look continuous so it’s perfectly ok to say it’s a continuous picture.

The Example of Telephone Numbers

The 7 digit phone number is very interesting.

It looks like discrete numerical data as it only comes in whole numbers, and just like counting sheep,there’s no meaning to the gaps between the numbers. So it might be numerical discrete.

It might be considered to be continuous as there are lots and lots of possible results and the likelihood of two people getting the same number is zero (if you measure people’s height in cm to 4decimal places it’s unlikely that two people will be have the same number).

The problem with looking at phone numbers as numerical data is that you can’t really add the numbers and get another number. If you add numbers of sheep together you get an answer which can be a quantity of sheep, if you add heights of trees together you get another height even if it is a very tall tree. If you add two phone numbers together you probably won’t have a phone number at all. So phone numbers are actually a categorical.

Is it categorical ordered? Is there a sequence? Clearly they can be put in increasing numerical order, but what does this order mean? Nothing really for the 7 digit number, perhaps the area code means something? Or is it just like a name? It’s just like a name.

The test is that you can use a phone system perfectly well using the letters associated with each digit 3def,4ghi.... instead of the digits.So the phone numbers are categorical nominal.