The BriefA Blog about the LSAT, Law School and Beyond
In the next few posts, we will introduce different kinds of graphs. These graphs allow you to visually represent data in ways that emphasize different aspects of the data. In preparation, let's look at some data represented with different graphs:
Tables of Lemonades Sold
Bar graph of lemonades sold on a given day:
Circle graph of lemonades sold:
Box plot of lemonades sold:
Scatterplot of lemonades sold each day:
In subsequent posts, we will talk more about how to read each of these graphs.
Talk of "data" is ubiquitous -- what does that word mean in the context of the GRE? We will think about data as observations of given variables. Now what does that mean?
Well suppose we are trying to help our child increase her earnings from a lemonade stand. Some days, she sells a lot and some days she sells almost nothing. To figure out why, we might start keeping track of how much she sells on a given day. So we would start making a table that looks like:
Now, each of the rows in the table is an observation. And we have our variables at the top of the columns: the date and the number of lemonades sold. And more generally: variables are just the characteristics that we keep track of, while an observation is a set of values for our variables on some particular occasion.
Now, a very simple way to keep track of data is via a frequency distribution. This is a table that records, on the left-hand side, possible values of a given variable, and on the right-hand side, it records how often those values appeared. Applied to the above table, we would get:
|Lemonades Sold||Number of Days (when that many lemonades were sold)|
This table answers questions like "How often did my child sell three lemonades?" To find the answer, we go to the "Lemonades Sold" column and look for the row with three lemonades sold. In that row, the right hand column (corresponding to the number of days) says one. So there was one day where the child sold three lemonades.
And, in addition to a frequency distribution, we can also create a relative frequency distribution which records, on the left-hand side, possible values of a given variable, and on the right hand-side, the percentage of all observations where that value occurred. So, in the above example, we would get:
|Lemonades Sold||Proportion of Days|
since the total number of days is and 20% and 40% and so on through the table.
Now, to really get a handle on what is driving her lemonade sales, we should probably add some more variables (e.g. daily temperature, day of week) and collect some more observations:
|Date||Lemonades Sold||Day of Week||Temperature (Fahrenheit)|
Here are some practice problems on the above concepts:
- Using the above data, construct a frequency table that tells you how often a certain number of lemonades was sold:
Number of Lemonades Sold Frequency 1 1 2 3 3 3 4 1 5 2
In the next few posts, we’ll unpack some of the statistical terms that the GRE throws around. These terms can be seen as ways to answer two main questions:
- What’s Typical
- What’s Possible
In the former category, we have the median, mean, and mode. Essentially, these give you a sense of what the average value is, or what the most common value is. They give you a sense of what a typical value might be.
In the latter category, we have the range, quartiles, and percentiles. These give you a sense of the possibilities and how the values are distributed.
In later posts, we will give actual definitions of these concepts. But first, some overarching advice about learning these concepts.
There are four main ways you’ll get tested on these concepts.
- Given some data and asked to compute the value of one of these terms.
- Given some data and told to alter the data in some way (e.g. by adding 10 to every value) and then compute the value of one of these terms.
- Given some graph/chart and asked to estimate the value of one of these terms.
- Asked a question that requires knowledge of some property of these concepts (e.g. that interquartile range is always less than the range).
So you want to understand the formal definition well enough to do 1 and 2, but also have some intuitive sense for the concepts (which can help in doing problems of type 3 or 4). In what follows, we’ll try to help in building that intuitive sense for these concepts.
Around 20% of GRE questions involve "data analysis" -- essentially reading graphs and tables to determine what they mean, and then applying that knowledge. These questions are generally fairly straightforward, and the key is to be able to read the graphs/tables and to understand concepts like the mean, median, mode, interquartile range, standard deviation, and so on.
We have broken up this guide into five main sections and an appendix:
(Note that posts are still in the process of being written; caveat emptor!)
- What is Data? Variables and frequencies
- Statistical Concepts
- Reading Graphs: An Introduction
- Normal distribution
- Why does it matter?
- Standard normal distribution
- Finding the standard deviation
- Appendix: Understanding All the Graphs
For reasons known only to ETS, number line problems are categorized with geometry. Such problems are basically algebra problems: they present you with some diagram or some facts about certain variables and then ask whether some equations or inequalities could be true, given certain relationships among the variables. Here is a straightforward example:
A. is greater than
B. is greater than
C. The two quantities are equal
D. It cannot be determined which, if any, is greater.
As noted below, number lines have increasing numbers as one goes towards the right. So
Now, a number line simply gives all the numbers from least to greatest. So the numbers to the right of the number line are greater than those to the left. And if there are markings on the number line, then you may (provided the question says nothing to the contrary) assume that the markings are evenly spaced. What does this mean? Well in the following diagram:
you know that the distance between and is only half of the distance between and . Using algebra, we could express this as:
Now, how do we solve number line problems? It will depend on what the question asks for. Some questions ask for which of the following could be true, whereas others ask about which of the following must be true.
If the questions asks which equations could be true, then we are looking for either a set of numbers that makes the equation true and is compatible with the diagram/given information, or we want to show that the equation is somehow inconsistent with our diagram/given information.
If the questions asks which equations must be true, then we are looking for either a set of numbers that is compatible with the diagram/given information but makes our equation false, or we want to show that the equation somehow follows from the diagram/given information.
Whether you should look for a specific example or some proof of the consistency/inconsistency of the given equation is a judgment call that you need to make in the moment. It's hard to give general principles about when to do which, but after doing some practice problems and just thinking about the equation at hand, you should develop some sense of how to decide if an equation is consistent or not. For example:
Let and . Must the following be true?
It might seem like, well, if is a bigger number then of course 1 divided by a bigger number must be smaller than 1 divided by a smaller number! If you keep the numerator the same, then the bigger the denominator, the smaller the whole fraction. So, for example, let and Of course in that case.
But now consider this possibility: what if is negative? So let be some positive number and be some negative number. It's easiest to think about this with concrete numbers, so let's try Then, we get that So the equation doesn't have to be true and the answer is no.
After doing some practice problems, you get a sense of which numbers to consider when you come to one of these problems. And by running through those numbers quickly, you can often show that some equation either can be satisfied (which answers whether the equation could be true) or can be violated (which answers whether the equation must be true).
The only real piece of advice I have here is two-fold: First, always try easy, concrete numbers. Don't just think about "one negative and one positive value." It's a lot more general to think that way, but also a lot harder to evaluate. Try, instead, or and . Nice, easy numbers to calculate. And if you want a small number, try or Second, it is easy to think about what happens if both are normal, natural numbers (e.g. 1 and 2). But also think about what happens if they are small, positive numbers (e.g. and ) or only one is positive (e.g. 1 and -1) or both are negative (e.g. -1 and -2). If the equation works (or is always violated) in all of those cases, then probably it's always true (or always false).
When it comes to proving that some equation is either always true/false given the diagram/information in the question, look for easy inferences. And if you can't see a way to prove it quickly, consider just flagging it and coming back. If you've gone through a few diverse examples and they all come out the same way, probably the equation actually is always true (or always false) and the time spent confirming that might be better spent on other problems.
1. The markings on the below number line are evenly spaced:
Which of the following must be true (select all that apply):
Note that we can express and in terms of :
And so, so that A must be false.
And as for B, which is greater than 0 since So B must be true.
And as for C, is a negative number multiplied by some positive number. So it is negative and C must be false.
2. Suppose we know and Which of the following must be true (select all that apply):
For A: consider what happens if is a very large negative number while are very small positive numbers. So try and Then, the equation is false. So A can be false and hence it is not the case that it must be true.
For B: let's try to find a counter-example. It's easiest to first think of some cases where So suppose and are both some normal positive number, like 2. Then, we would get:
But of course, we could pick a negative value of for which this would not be true. So let , for example. So B can be false as well.
For C: what if are extremely small? Suppose but Then, we get:
which is not true. So C can also be false.
3. Suppose we know Which of the following must be true (select all that apply):
A: Doesn't have to be true since all three numbers could be negative.
B. Doesn't have to be true since could be barely bigger than . For example:
C: Doesn't have to be true since all three numbers could be positive.
D. Doesn't have to be true since could be very small. For example: Or, let be positive and have be negative.
E. Doesn't have to be true since could be almost the size of . For example:
4. The markings in the below number line are equally spaced. Which of the following could be true:
Again, let's put all of our values in terms of . So we get:
Then, for A: and as is positive, we get that So A must be false (i.e. cannot be true).
B: Let's substitute for to get:
which, as must be positive, simply cannot be true.
C: and this cannot be less than since is positive.
Thus none of the above could have been true.
5. The markings in the below number line are equally spaced. Which of the following must be true:
This time, let's suppose that Then, we get:
Now, let's evaluate the above:
A: since So A must be true.
B: so B must be true.
C: so C must be true as well.
Geometry questions sometimes involve various graphs. For example:
Find the area of the following square:
We see that the square has points at (3, 1) and (1, 2). Using the Pythagorean Theorem, we can find the length of that side as Thus, the side is units long and so the square has an area of 5.
Now, these problems do not differ substantially from what we have before. Essentially, graphing problems give you the same information in a slightly different way. So instead of telling you how long the sides of a particular shape are, they might give you the coordinates of the endpoints. But of course, we know how to infer the length of that side, for we know the Pythagorean Theorem:
A rectangle has a side whose vertices are at (1, 3) and (2, 2). Find the length of that side.
We can apply the Pythagorean Theorem to get that which is the length of that side.
In this vein, a more realistic problem might look like:
A triangle has vertices at (1, 4), (2,3), and (0,0). Find its perimeter.
This problem simply involves repeated uses of the Pythagorean Theorem. Draw in the following additional triangles:
And we can apply the Pythagorean Theorem to get the following lengths:
So the answer is
Similarly, graphing problems can involve angles:
A triangle has vertices at and . Find the degree of angle .
We know that is a right isosceles triangle since and Thus, we know that
In short, if you have a geometry problem that involves certain points or a diagram, you can generally use those points or that diagram to infer aspects like the lengths of certain sides or the degrees of various angles, and then use that information as you normally would.
1. A rectangle (shown below) has vertices at (0,1), (2, 0), and (2,5). Where is its fourth vertex?
To answer this question, we need to remember that the opposite sides of a rectangle are the same length and point in the same direction. So the bottom side of the rectangle, going from (0, 1) to (2, 0) travels in the same direction and for the same distance as the top side of the rectangle going from (2, 5) to our mystery point.
And we know how far the bottom side changes: it gains 2 units in the x-direction and loses 1 unit in the y-direction. So too for the top side. So (2,5) must become (4, 4) and that is the coordinate of our missing point.
A similar route would be to note that these are congruent triangles:
and so, again, the change over the course of the longer side of the rectangle must be the same. So in going from (0, 1) to (2, 5), there is a gain of 2 units in the x-direction and 4 units in the y-direction. So (2,0) must undergo the same change, giving us (4,4).
2. A circle has its center at point and its circumference includes the point . Find its radius.
Recall that the radius of a circle is just the distance from the center to the circumference (at any point on the circumference, since they are all equally far from the center). So we just need to find:
which, by the Pythagorean Theorem, is just
3. A triangle has vertices at (1, 4), (2,3), and (0,0). Find its area.
The key observation is that we can take the area of the surrounding rectangle and subtract the area of bits outside of our desired triangle, like so:
Thus, we get that the surrounding rectangle has an area of 8 and that the triangles in red have an area of Thus, our triangle has an area of
4. The square in the following diagram has points at (2, 1) and (1, 4) as labelled. What is the area of the square?
To find the area of the square, we just need to find the length of any side. We are given, in effect, a diagonal of the square. We know that if a square has sides of length , then its diagonal will be units long. And we can use the Pythagorean Theorem for the above to find the length of the diagonal:
so we get that the diagonal is units long. Dividing by we get that the sides must be units long and so the area of the square is 5.
There are also two relationships between angles and circles which are crucial. First, let's define the term "central angle":
Central Angle: A central angle is an angle located at the center of a circle with endpoints on the circumference of that circle
Now, the crucial fact to learn is:
Central Angles and Arcs: In the following diagram:
The degree of angle AOB =
Thus, we can determine the degree of any central angle by looking at the length of its arc, and vice versa.
The circle below has a radius of 2 and has a length of . What is the degree of angle ?
The total circumference is so takes up of the total circumference, so the angle must be degrees.
The circle below has a radius of 6 and the degree of is 47 degrees. What is the length of ?
The total circumference is and the arc takes up of the total circumference. Thus, its length is .
A related fact to the above is:
Circumference Angle Theorem: In the following circle with radius :
The degree of angle ACB = .
This theorem is also called the Inscribed Angle Theorem.
In essence, if you have an angle on the circumference of a circle, its degree will be one-half the degree of a central angle with the same endpoints. In pictures:
Now, what may be surprising about this is that it does matter where on the circumference the angle begins from. In other words:
the angles and above have the same degree.
Finally, you must be careful to make sure that you are comparing the angle on the circumference to the correct central angle. For example:
and do not have the same degree. That is because they give rise to different central angles:
Thus, it is not enough to look at the endpoints of your angle. You also need to look at where on the circumference the angle is, so you can be sure to compare it to the relevant central angle.
1. The below circle has a center at . Find :
Since intersects a diameter of the circle, we know that the central angle there will be 180 degrees, and so by our Circumference Angle Theorem.
2. Find the value of
This is a bit of a trick question. Even though it looks like a right angle, there is no way to know that based solely on the above (see here). So the right answer would be that we cannot determine the value of
3. In the following diagram, Find
The total circumference is . Thus, our arc takes up of the total circumference. So
4. The radius of the following circle is Find
The total circumference of the circle is Thus, our arc takes up of the total circumference and thus
5. The radius of the following circle is 1. Find
Note that we have a right isosceles triangle, since the legs of the triangle are also radii of the circle and thus have a length of 1. So by the Reverse Pythagorean Theorem,
6. The radius of the below circle is 1. Find
Again, from the center, we would have a right isosceles triangle so the central angle of that arc must be 90 degrees. Thus,
7. In the below circle, which is larger?
C. They are equal.
D. It cannot be determined.
As noted above, it does not matter where on the circumference your angle is located; it just matters which central angle it corresponds to. In this case, both and have the same central angle. If the center of the circle is at point , then they both correspond to the central angle Thus, they are equal.
Now, with respect to triangles, there are two main facts you need to know:
Triangles have 180 degrees: The angles of a triangle add up to 180 degrees.
Reverse Pythagorean Theorem: Triangles whose side lengths obey the Pythagorean Theorem (i.e. the triangle has side lengths and ) must have a right angle.
Sometimes, a problem will be somewhat sneaky: instead of directly telling you that a triangle is a right triangle, it will give you the side lengths and leave you to make that inference. (Recall here our Reverse Pythagorean Theorem)
The final fact which can help in finding an unknown angle is the following formula for the sum of all the angles of a polygon:
Polygon Angle Sum Formula: For an -sided 2D shape, the sum of all the interior angles is equal to
What is an interior angle? Well on a polygon, we have some sides and some vertices (points where the sides meet). For example:
At each vertex, there is are two angles:
The interior angle, sensibly enough, is the angle that is insidethe polygon. In the above diagram, it is the blue angle.
Thus, our formula lets us calculate, for any polygon, the sum of all of its interior angles. This can be helpful when you are given almost all of the angles of a polygon and you need to find the final angle.
1. What is the sum of the interior angles for a 13-sided polygon?
We simply apply our above formula to get:
2. Find the value of any interior angle of a regular 14-sided polygon.
By our above formula, the sum of all interior angles of our polygon must be: Recall that a regular polygon has angles and sides of equal degree and length, respectively. Thus, each angle must be degrees.
3. Find the angles of the below polygon:
Note that by our above formula, the sum of all these interior angles must be degrees. Thus, we get:
and solving, we get
Given parallel lines, there are four main relationships that angles can have to one another: vertical angles, interior angles relationships, exterior angle relationships, and corresponding angles. These relationships can all be derived from one another, and in what follows, we include some of those derivations so that, if you forget one of them on exam day, you can recover it from knowledge of the others.
Vertical Angles: Two angles formed by the same lines are vertical angles if they are on opposite sides of the point of intersection between the two lines.
This is a fundamental fact that is often critical in finding an unknown angle. For example:
We know that since they are vertical angles. Then, we get: so each angle must be 90 degrees.
In the above diagram,
In the following diagram, let and let What is the value of ?
We know that so we simply plug in to get and hence
We know that so we simply plug in to get and hence
In the above diagram,
In the below diagram, let and What is the value of ?
We know that so we can plug in to get: which implies and so
Corresponding Angles: Two angles (which are in the same relative position) formed by a single line cutting across two parallel lines. Such angles are equal.
In this diagram, and are corresponding angles and thus equal in degree.
1. Find the value of .
As a corresponding angle, we get:
2. Find the value of .
First, as a corresponding angle, we get:
and thus as an opposite interior angle.
3. Find the value of
As a vertical angle, we get:
and then as a same-side interior angle, we get that and thus that
4. Find the value of
Since the triangle above is an isosceles right triangle, we know:
and as a corresponding angle, we get:
This problem exhibits a trait that is common to many problems about angles: we need to use several different facts about angles in combination with one another. So when solving these problems, once you make some inference about the value of a previously unknown angle, you should see what further inferences can be made about still more angles. You may not immediately see how this information could get you to the final answer, but you should still keep it in mind when considering your next move.
Many geometry problems either directly ask about the degree of certain angles or will require you to figure out the degree of an angle in the course of solving some problem. In successive posts, we will cover the basics of what an angle is, how we can use parallel lines to find unknown angles, and how we can use triangles/circles/polygons to find unknown angles.
Angles: Some Preliminaries
To understand angles, we need to begin with the idea of a line:
Line: a straight line that continues in both directions without end.
Now, an angle appears where two lines intersect. More formally,
Angle: a measure of how much you would need to turn one line to make them part of the same line
And as you can see in the above diagram, there are actually four angles formed when two lines intersect. So there is a convention for naming each of the four angles:
[Practice problems with matching angles: label points on the lines and degree measures and have them match the two]
Now, here are two important facts about angles:
Lines Have 180 Degrees: The angle formed by a single line has 180 degrees.
Circles Have 360 Degrees: The angle formed by a circle is 360 degrees.
Now, we also have two relationships among lines which are very important:
Parallel Lines: Parallel lines go in the same direction and therefore never intersect. We often denote that lines and are parallel by writing , as in the below diagram:
Perpendicular Lines: Perpendicular lines intersect at a 90 degree angle. We denote two lines as perpendicular by writing :
In our next post, we will talk about some of the relationships that exist between angles and lines and shapes.