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Why Does the United States Have Two Different Kinds of Courts?
In the United States, we have two different kinds of courts: federal courts and state courts. It’s been this way since 1788, when a group of nerdy Americans ratified the Constitution. Article III of the Constitution states that the “judicial power of the United States, shall be vested in one Supreme Court, and in such inferior courts as the Congress may from time to time ordain and establish.” With this little sentence, James Madison et al. set the stage for the development of the complex federal judicial system we have today.
Before the ratification of the Constitution, the thirteen original states were governed by the Articles of Confederation, which didn’t provide for a federal judiciary. Instead, legal disputes were largely settled by state courts (the only courts around). This approach had some problems.
First, where cases introduced conflicts between federal and state interests, states had little incentive to enforce federal laws. Second, without a Supreme Court, state courts couldn’t maintain uniformity in their interpretation of federal laws. Third, state courts were too biased to hear disputes between the states themselves. Finally, in the absence of a court with national authority, citizens had no place to file complaints against the national government.
For all of these reasons, we’ve had a federal Supreme Court since the ratification of the Constitution. But wait! You’re a smart cookie. You’ve read in the news that other federal courts exist too. These lower courts were created by Congress in 1789. The number and structure of these lower courts has changed since then, as new laws have been passed and cases decided, but our doubled system of state and federal courts has persisted.
How Is the Federal Court System Structured?
The federal judicial system has three tiers. A litigant in the system starts off in a federal district court. A district court is where the trial actually happens. Where the parties have different stories about what happened, the district court acts as a factfinder, and writes the official summary of events. Based on these findings of fact, the district court renders a legal decision, applying the law to the facts.
If a litigant believes the district court decided the case wrongly, the litigant may ask an appellate court to reverse the decision. In the United States, we have 94 district courts, but only 13 appellate courts. Appellate courts are also known as circuit courts because they preside over “circuits” that govern several district courts within the same geographic region. (An exception is the Federal Circuit, which hears cases that involve specialized subject matter.)
Appellate courts do not have juries, because they do not find facts. Generally, after deferring to the district court’s judgment as to the facts of the case, the appellate court will scrutinize the district court’s judgment for errors of law. If it finds that the district court was right on the facts, but wrong on the law, it will issue a new decision correctly applying the law.
Litigants who are unhappy with the appellate court’s decision can ask the Supreme Court to hear their case. However, the Supreme Court only hears oral arguments in roughly 100 cases every year. As such, decisions of the federal appellate courts are almost always final.
How Are State Court Systems Structured?
State judicial systems are also composed of a state’s trial courts, appellate courts, and a state’s highest court(s). However, each state uses a slightly different naming system for its courts, which can make things confusing. For example, the highest court in New York State is known as the New York State Court of Appeals, while its trial court is known as the New York Supreme Court.
Furthermore, some states have a judicial structure unique to the state. Texas, for example, has two highest courts: the Supreme Court of Texas, which renders decisions in civil cases, and the Court of Criminal Appeals, which renders decisions in criminal cases. Our national highest court, the Supreme Court of the United States, can review the decisions of all state supreme courts.
What Makes Federal Judges Different from State Court Judges?
Rather than being directly elected to office, federal judges are nominated by the president and confirmed by the Senate. Given good behavior, the Constitution grants federal judges the protections of life tenure. Furthermore, Congress cannot reduce their salary. This insulates federal judges from political blowback when they make unpopular decisions.
Generally, state judges must run for office. Furthermore, they rarely enjoy the wide set of career protections granted to members of the federal judiciary. As a result, some people argue that federal courts are better suited to protect constitutional rights, particularly the rights of minorities, and enforce politically unpopular laws.
What Kinds of Cases Are Litigated in Federal vs. State Courts?
The vast majority of cases are heard in state courts. State courts are courts of general jurisdiction, meaning that they can try all cases, except those that Congress has specified should be litigated only in federal courts. Federal courts, on the other hand, are courts of limited jurisdiction: they can only try certain kinds of cases.
Generally, a case can only be heard in federal court if it presents a federal question or involves diversity of citizenship. A case presents a federal question if it involves a violation of federal law. A case involves diversity of citizenship if the opposing parties are citizens of different states (or if one is from a foreign country). Given that so many lawsuits involve diversity of citizenship, federal courts will only hear these cases where the amount of money the parties are arguing over exceeds $75,000.
Can State Courts Decide Issues of Federal Law?
Yes.
State courts can rule on questions of federal law, except where Congress has mandated that a specific kind of case can only be heard in federal court. As the Supreme Court noted in Claflin v. Houseman, federal law is the law of the land––effective in every state. Not only are state courts allowed to rule on federal law, they must enforce federal law to perform their duty of enforcing laws valid within the state.
Can Federal Courts Decide Issues of State Law?
Yes.
The Supreme Court can review decisions of each state’s highest court, but only insofar as a case raises a question of federal law. Decisions of a state’s highest court are final on questions of state law.
The lower federal courts also regularly rule on matters of state law. As we’ve discussed, even a case that exclusively involves state law can enter the federal system if the parties suing have diversity of citizenship. In cases like these, the court must apply state law to decide the issues. Determining which state’s laws to apply is a convoluted process, but the federal courts are theoretically better able to make impartial decisions than the state courts themselves.
Furthermore, cases that do raise federal questions may be tightly bound with matters of state law. In these cases, a federal court may exercise supplemental jurisdiction to decide the state law issues along with the federal.
Of course, these jurisdictional principles are complicated by a variety of exceptions and legal wrinkles—but you'll learn more about those in law school.
Circle graphs (also called pie charts) let you see the relative amounts of different categories. For example, if you run a local grocery store and want to see where your sales are coming from (perhaps because you are considering whether to reallocate floor space), you might look at a chart like the following:
and conclude that as most of your sales come from produce, you may want to allocate more space to new kinds of produce.
Now, when reading a circle graph, the percentages of the different sections will generally be labelled as in the above. So circle graph tells you that in January of 2010, 23% of all sales were from frozen foods, 10% of all sales were from pharmaceuticals, and so on.
Now, if you are given the actual value (as opposed to the proportion) of any category, you can find the value for each category. So, for example:
Example 1
Suppose that in January of 2010, the grocery store sold $23,000 worth of frozen foods. How many dollars worth of canned foods did they sell?
We can also use circle graphs to determine various trends. For example, by comparing the following charts:
we can conclude that the proportion of revenue from canned foods drastically shrunk from January 2010 to 2011.
Example 2
From January 2010 to January 2011, which category grew the most as a proportion of total sales?
Now, if we know the actual value of some category in 2010 and the actual value of some category in 2011, we can calculate the value of each category and 2010 and 2011. Thus, we can calculate the absolute increase in revenue for any particular category as follows:
Example 3
Suppose in January 2010, the total amount of goods sold was $200,000. In January 2011, the amount of canned foods sold was $6,000. Find the absolute change from 2010 to 2011 in the amount of dairy products sold.
Practice Problems
 Suppose produce sales were, in absolute terms, $10,000 greater than pharmaceutical sales in January 2010. What was the total amount of sales for all goods?
 Overall sales in January 2010 were $100,000. How much more revenue was generated by dry foods as compared to canned foods?
 Suppose dry foods in January 2010 were twice as large, in absolute terms, as pharmaceutical sales in January 2011. What is the ratio of overall sales from January 2010 to January 2011?
A box plot is so named for its iconic shape:
(They can also be laid out horizontally). The parts of the graph correspond to:
Looking at a box plot can give you a quick sense of what the distribution of the data looks like. For example:
Example 1
Find the 25th percentile, 50th percentile, 75th percentile, range, and median for the below boxplot:
Example 2
The following chart represents the books read by the 500 fifthgraders in a school over the summer. Suppose no student read exactly 4 books. How many students read more than 4 books?
Example 3
The following chart represents the books read by the 500 fifthgraders in a school over the summer. What is the approximate number of students that have read between 3 and 4 books?
Practice Problems

The following chart represents the books read by the 100 fifthgraders in a school over the summer. What is the approximate number of students that have read more than 5 books? (We assume that people can only read positive integers of books).

The following chart represents the books read by the 500 fifthgraders in a school over the summer. Suppose 7 is the 80th percentile of this data. Approximately how many people read between 6 and 7 books? (We assume that people can only read positive integers of books).
 Researchers measured, over 36 months, how often it rained in a month. About how many months had rain from between 13 and 18 days?
A histogram looks a lot like a bar graph and, indeed, you can think of it as a special kind of bar graph. But instead of having just any kind of category (as a bar graph does), histograms have, as their categories, certain ranges of values. So, for example, suppose the students in your class score the following scores on their test:
Now, those numbers are kind of unwieldy. So to get a sense of how many students are acing your class (getting an A) or failing your class (getting an F), you might categorize their test scores according to certain ranges. So put all the 90100 scores together in one category, all the 80  89 scores in the same category, and so on. Graphing this, we would get the following:
This graph tells us that 4 students scored between 90 and 100; 3 students between 80 and 89; and so on.
Example 1
In the following histogram, approximately how many students scored a 170 or higher?
Example 2
In the below histogram, what if anything can we conclude about the median of the data?
Example 3
In the following chart, what can we conclude about the range of the data?
Practice Problems
 In the following histogram, approximately how many students scored a 170 or lower?

In the below histogram, what if anything can we conclude about the range of the data?
 In the below histogram, what is the minimum possible number of students who scored higher than 75? What about the maximum?
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 some particular occasion when we record the values of our variables.
Now, a very simple way to keep track of data is via a frequency distribution. This is a table that records, on the lefthand side, possible values of a given variable, and on the righthand side, it records how often those values appeared. Applied to the above data, we would get:
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 lefthand side, possible values of a given variable, and on the right handside, the percentage of all observations where that value occurred. So, in the above example, we would get:
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:
Here are some practice problems on the above concepts:
Practice Problems:
 Using the above data, construct a frequency table that tells you how often a certain number of lemonades was sold:
 Using the above data, how many lemonades were sold on Tuesdays?
 Using the above data, how many days had more than 3 lemonades sales?
 On days where the temperature was at least 73 degrees, how many lemonades did she sell on average?
 On days where the temperature was less than 73 degrees, how many lemonades did she sell on average?
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 central questions:
 What’s Typical?
 What’s Possible?
In response to the question of "What's Typical?" we can find the median, mean, or mode of the data. These give you a sense of what the average value is, or what the most common value is; in short, they describe what a typical value might be.
And in answering "What's Possible?" we can find the range, quartiles, or percentiles of the data. These give you a sense of the possibilities and how the possibilities 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, find the value of one of these concepts (e.g. the mean, median, range, etc.).
 Given some data, alter the data in some way (e.g. by adding 10 to every value), and then find the value of one of these concepts.
 Given some graph/chart, estimate the value of one of these concepts.
 Answer some question about some concept's general properties (e.g. that the interquartile range is always less than or equal to the range).
Thus, you want to understand the formal definition well enough to do questions of the first and second types. But you also want to have some intuitive sense for the concepts, so that you can do questions that fall in the third and fourth categories.
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:
Example 1
x y
A. is greater than
B. is greater than
C. The two quantities are equal
D. It cannot be determined which, if any, is greater.
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:
Example 2
Let and . Must the following be true?
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 twofold: 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.
Practice Problems:
1. The markings on the below number line are evenly spaced:
Which of the following must be true (select all that apply):
A.
B.
C.
2. Suppose we know and Which of the following must be true (select all that apply):
A.
B.
C.
3. Suppose we know Which of the following must be true (select all that apply):
A.
B.
C.
D.
E.
4. The markings in the below number line are equally spaced. Which of the following could be true:
A.
B.
C.
5. The markings in the below number line are equally spaced. Which of the following must be true:
A.
B.
C.
Geometry questions sometimes involve various graphs. For example:
Example 1
Find the area of the following square:
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:
Example 2
A rectangle has a side whose vertices are at (1, 3) and (2, 2). Find the length of that side.
In this vein, a more realistic problem might look like:
Example 3
A triangle has vertices at (1, 4), (2,3), and (0,0). Find its perimeter.
Similarly, graphing problems can involve angles:
Example 4
A triangle has vertices at and . Find the degree of angle .
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.
Practice Problems:
1. A rectangle (shown below) has vertices at (0,1), (2, 0), and (2,5). Where is its fourth vertex?
2. A circle has its center at point and its circumference includes the point . Find its radius.
3. A triangle has vertices at (1, 4), (2,3), and (0,0). Find its area.
4. The square in the following diagram has points at (2, 1) and (1, 4) as labelled. What is the area of the square?
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.
Example 1
The circle below has a radius of 2 and has a length of . What is the degree of angle ?
Example 2
The circle below has a radius of 6 and the degree of is 47 degrees. What is the length of ?
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 onehalf 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.
Practice Problems
1. The below circle has a center at . Find :
2. Find the value of
3. In the following diagram, Find
4. The radius of the following circle is Find
5. The radius of the following circle is 1. Find
6. The radius of the below circle is 1. Find
7. In the below circle, which is larger?
A.
B.
C. They are equal.
D. It cannot be determined.
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.
Practice Problems
1. What is the sum of the interior angles for a 13sided polygon?
2. Find the value of any interior angle of a regular 14sided polygon.
3. Find the angles of the below polygon: