The BriefA Blog about the LSAT, Law School and Beyond
It’s your first week as an associate at the law firm of your dreams. You’ve passed the bar, received the privilege license that makes you official, and collected your initial research assignments from the partner next door.
Then, the phone rings, and it’s your college friend, Sarah. While you’ve been in law school, Sarah’s been working at Goliath Corp., a huge, nationwide company that recruited her for its management track. You can tell something’s wrong, and soon she tells you: her supervisor has fired her out of nowhere.
Sarah’s performance evaluations were always stellar. She received one promotion and was expecting another. But everything changed when the new boss showed up three months ago, criticizing Sarah for non-existent mistakes and making demeaning, race-tinged comments about her in front of colleagues. She knows she wasn’t fired because she failed to do her job.
You realize Goliath Corp. might have fired Sarah because of her race, which means she might have a claim for race discrimination. You assure her you’ll try to help. Then you hang up and wonder what to do next. Fortunately, you paid enough attention in your 1L Civil Procedure class to know where to look for answers. All you remember are the basics, but that’s enough to get started.
First, you need to know if Sarah’s case is civil or criminal. Those two types of cases involve different disputes, different consequences, and (often) different parties.
In a criminal matter, a prosecutor from the government—often a state, city, or county—charges a defendant with an offense against the community like theft or assault. There may be a single victim of a specific attack, but the prosecutor acts on behalf of all the people. Penalties can range from public service to fines to imprisonment or even death.
Civil matters deal mostly with private disputes, like car accidents or breaches of contract. Instead of bringing charges to protect society, a civil plaintiff brings claims to enforce her own rights. Instead of prison, civil defendants might face money damages or a judge’s order to avoid certain conduct (called an injunction).
Because the disputes and the stakes vary so much, courts apply different rules and standards to criminal and civil matters.
Next, you need to distinguish between the substantive law that will define Sarah’s claims and the procedural rules that will determine how her case makes its way through the court.
Substantive law tells us what legal rights we possess and what duties we have to other people. Laws against fraud, negligence, and breach of contract are prime examples.
Procedural law dictates the steps we take to enforce our rights or defend against a claim that we have violated someone else’s rights. The rules that tell you where to sue and what to file are good examples of procedural law.
With that information, you can already reach two important conclusions:
(1) Sarah should bring a civil claim because her dispute is a private matter between her and Goliath Corp. and
(2) The substantive law Goliath Corp. might have violated is Title VII of the Civil Rights Act. You remember from your Employment Law class that this statute defines and prohibits race discrimination.
Now, you need to turn to the procedural side and consult every litigator’s best friend: the step-by-step manual for handling civil lawsuits. In this case, that manual is the Federal Rules of Civil Procedure (or the “Fed. R. Civ. P.”). (We will explain in a later post why the federal rules—and not state ones—apply!)
Since 1934, the Federal Rules of Civil Procedure have made civil practice uniform across federal courts and helped prevent plaintiffs from “forum shopping”—hand-picking a particular court because some unique rule it followed would give them an unfair advantage. As Rule 1 makes clear, the purpose is to “secure the just, speedy, and inexpensive determination of every action and proceeding.” Along with the federal case law that interprets them, the Fed. R. Civ. P. give lawyers plenty of guidance about what to do—and what not to do—before a federal court.
Next time, we’ll dive into the rules and find out how to launch Sarah’s lawsuit. For now, you’d better get back to those research questions the partner assigned you this morning . . .
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 fact-finder, 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.
Often, you want to use data and graphs to figure out what the relation between two variables is. For example, you may wonder whether your scores improve as you spend more time studying. Or you may wonder how years of education affect one’s lifetime earnings. Scatterplots allow you to plot one variable against another in order to determine what the relationship between them is.
Here’s an example of a scatterplot:
Here, we can see how consuming coffees (on the x-axis, i.e. the bottom axis) affects the number of words one writes (on the y-axis, i.e. the axis on the left-hand side). (You may wonder, sensibly enough, how one can consume non-integer quantities of coffee. We presume that means people drank a partial cup of coffee).
(Part of) the data table corresponding to this graph looks like:
| Coffees | Words written |
| 0 | 290 |
| 1 | 360 |
| 1.8 | 520 |
| 2.6 | 560 |
| 4.4 | 470 |
In the left-hand column, we have the variable for the x-axis (namely the number of coffees one drank) and on the right-hand side, we have the variable for the y-axis (the number of words one writes).
Now, you may be asked to interpret the graph above. So, for example, you may get something like:
Example 1
The above graph comes from a study of how coffee affects literary output. The researchers asked 17 people to drink as much coffee they like and recorded how many words they wrote in the next hour. How many people drank 1 or fewer cups of coffee?
Example 2
Of the people who drank two or more cups of coffee, how many wrote more than 500 words?
Sometimes, you will see a scatterplot that also has a “trend line” like so:
The trend line is an attempt to infer, from the available data, what the general pattern looks like. Generally, it will be a straight line chosen (by some algorithm) to be an optimal fit for the data.
Example 3
According to the trend line in our graph, approximately how many words will someone who drank 2 cups of coffee write?
Finally, scatterplots will often use time as the variable on the x-axis. This is because we are often interested in knowing how some variable (e.g. value of a share, net worth, world record for running a marathon) changes with time. We say that time plots are the scatterplots that use time as a variable. Here is an example:
The S&P 500 is an index of, roughly speaking, the share value of the 500 largest publicly traded companies. Here, we can see that it has grown considerably over the past 20 or so years.
Example 4
By (approximately) how much has the S&P 500 increased from 1/1/1999 to 1/1/12?
Example 5
By (approximately) what percentage has the S&P 500 increased from 1/1/95 to 1/1/18?
![\[\frac{2700 - 500}{500} = \frac{22}{5} = 440\%\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-765fc71f5abcdb8e360bdbcb736b14b2_l3.png "Rendered by QuickLaTeX.com")
Univariate vs. Bivariate
Some graphs only use one variable. Those graphs are called univariate. Other graphs use two variables; they are called bivariate.
How can a graph use only one variable? Well consider the histogram. It tells you how many observations fall into certain brackets. So, for example:
tells you that 4 students scored between 90 and 100; 3 between 80 and 89; and so on. The raw data for this kind of graph just looks like:
| Score |
| 92 |
| 94 |
| 93 |
| 100 |
| 82 |
| 83 |
| 85 |
| 72 |
| 62 |
Where we just record different observations of a single variable. Thus, we can call such graphs univariate. Other examples of univariate graphs include circle graphs and bar graphs.
By contrast, scatterplots involve two variables. See, for example:
Whose data look like:
| Coffees | Words written |
| 0 | 290 |
| 1 | 360 |
| 1.8 | 520 |
| 2.6 | 560 |
| 4.4 | 470 |
| 4.2 | 492 |
| 5.3 | 455 |
| 4.9 | 462 |
| 0.8 | 310 |
| 0.1 | 270 |
| 1.3 | 280 |
| 0.5 | 255 |
| 2.1 | 540 |
| 2.4 | 510 |
| 3.2 | 572 |
| 2.9 | 580 |
| 4.7 | 400 |
Thus we see that for scatterplots, we need two variables, one for the x-axis and another for the y-axis. Thus, we call these bivariate. And since time plots are just a special kind of scatterplot (namely one that uses time as a variable), we get that time plots are also bivariate.
We have already talked about what the mean, median, and mode are. But in the examples we discussed previously, we gave you the data and asked you to find the mean/median/mode. But you can also sometimes, to a limited degree, get information about the mean/median/mode just from the graph of the data. For example:
Example 1
Which of the following two quantities is larger?
Average number of words written 
A. The average number of words written is larger
B. is larger
C. They are equal
D. It cannot be determined from the information given
. But 
which is less than 200. Thus, A is correct.Now, sometimes you will be asked to manipulate the data/graph given before estimating the mean/median/mode. So, for example:
Example 2
Suppose the researchers for the above graph miscounted the number of words each participant wrote. They accidentally multiplied the number of words written by each participant by 2. Fix this error in their data by dividing the observations in the above graph by 2 to get the new average. Now, which of the following two quantities is larger?
New average of words written 
A. The average number of words written is larger
B. is larger
C. They are equal
D. It cannot be determined from the information given
we get that the new average must be 
. And again, 
so A is the correct answer.
Now, we can generally find the median from a graph as well:
Example 3
Find the median number of words written in the above scatterplot.
And again, we can modify the given data and find the new median:
Example 4
Suppose that, this time, the researchers under-counted the number of words each participant wrote. Find the median of the corrected data, multiplying the number of words written by each participant by 2.
And finally, we could try to find the mode of the above graph. It is somewhat hard to tell whether some of the points are the same on the above graph, so I will cheat and just tell you that there is no mode; every value occurs just once. But on the GRE, rest assured that if the question asks you to find the mode from the graph, the relevant points will be fairly clearly marked. Then, it is just a matter of counting up how many observations each value has (e.g. how many people wrote 500 words; wrote 550; etc.).
Finding quartiles/percentiles
Now, it will not always be possible to find the quartile of a graph. For example, if you are given a circle graph:
and asked to find the various quartiles, the question simply makes no sense. But of course, in a box plot:
the quartiles just correspond to where the lines of the box are.
Now, it is not really feasible to read off, say, what the 96th percentile looks like, just based off of a graph. But some important facts to keep in mind are that: the highest value in your graph will be greater than the 99th percentile (since all of the observations will be less than or equal to that observation’s value). Similarly, the 1st percentile will be greater than the lowest value of the graph. Thus, knowing the maximum and minimum (which you can read off of a graph) can give you some idea of the limits of your percentiles.
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 re-allocate 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?</span> <span class="s1">[ss_toggleable_text]</span> <span class="s1">We know from the above diagram that 23% of all sales in January of 2010 were of frozen foods. Thus, let](https://7sage.com/wp-content/ql-cache/quicklatex.com-f99902444a2da1cdb834377bc52c7e4a_l3.png "Rendered by QuickLaTeX.com")
x
![be the total amount of sales made in January 2010. We get that <span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="https://7sage.com/wp-content/ql-cache/quicklatex.com-b6dcd5d410df414731b2d745fb5f7108_l3.png" height="12" width="101" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[.23x = 23000\]" title="Rendered by QuickLaTeX.com"/> so](https://7sage.com/wp-content/ql-cache/quicklatex.com-8b0c249921056e68956e284b63b110ed_l3.png "Rendered by QuickLaTeX.com")
x = 100000.
7000 worth of canned food.
[/ss_toggleable_text]
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 
6,000. Find the absolute change from 2010 to 2011 in the amount of dairy products sold.
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
[think of another example]
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:
$$100, 98, 68, 88, 79, 92, 90, 85, 86$$
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 90-100 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
[think of another example]
A bar graph tells you how many objects in a given category you have. For example, in the following bar graph:
Each bar tells you how many children fall into that height range. So, for example, there are 2 children between 0 and 4 feet; there are 7 between 4 feet 1 inch and 4 feet 6 inches, and so on.
Example 1
How many children are between 4 feet 7 inches and 6 feet? How many children are taller than 6 feet?
We can also have segmented bar graphs like the following:
These bar graphs allow us to compare different groups, in this case the group of 1 year olds against the group of 2 year olds, 3 year olds, and so on. We can thus see how their preferences shift over time.
To read the graph, the horizontal signs tell us which categories we have (here, the different ages 1 to 5) and then the vertical component tell us how many members of that category (e.g. how many one year-olds) in accordance with the key to the right. Thus, the left-most blue bar tells us how many 1 year-olds have "Mighty Man" as their favorite superhero. The right-most yellow bar tells us how many 5 year-olds have "Valiant Vanessa" as their favorite hero and so on.
Example 2
Which superhero gains the most fans as the children mature from being 1 years-old to 5 years-old?
Example 3
Which superhero has the most supporters overall (i.e. across all the age groups)?
Example 4
[Give a match the table to the possible bar graph question; simply 5 category bar graph]
In this post, we’ll define the mean, median, and mode. Along the way, we’ll work through an example of how to find each of them. We’ll also talk about why the median is responds less to outliers than the mean does, a fact which can sometimes be crucial in solving a GRE problem.
First, some definitions. The mean (or average) is found by adding up all of the values of some variable and then dividing by the number of observations. So to go back to our lemonade example:
| Date | Lemonades Sold |
| 7/2/19 | 1 |
| 7/3/19 | 2 |
| 7/4/19 | 5 |
| 7/5/19 | 2 |
| 7/6/19 | 3 |
Now, finding the median is a little trickier. Imagine lining up all of the values for your variable from least to greatest. Then, we find the one in the middle. So for example, if we have the data:
![\[1, 4, 9, 5, 3\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-e0f74009ac0a6f848810cc8693312e67_l3.png "Rendered by QuickLaTeX.com")
Then, we order it from least to greatest to get:
![\[1, 3, 4, 5, 9\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-2cfa0b4d558ca1cd33f321d23562c6b8_l3.png "Rendered by QuickLaTeX.com")
And then we pick the middle value, namely 
.
Sometimes, if there is a long string of values, it will be hard to see which value is the middle one. So take the data:
![\[1, 9, 2, 3, 3, 3, 4, 6, 7, 8, 1, 2, 3, 1, 1\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-1cdefe4712bba2b2ea841f497bef2aae_l3.png "Rendered by QuickLaTeX.com")
We order it from least to greatest:
![\[1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-fa23f9744889c2f748f3f9534a16996d_l3.png "Rendered by QuickLaTeX.com")
And then we simply cross off the numbers at the end, each pair at a time, to get:
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
...
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
And thus the median is 3.
Now, sometimes we will have an even number of observations. So suppose we have the data:
![\[1, 2, 3, 4\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-c6855fd47da5c12506bd2d90c2d8c31c_l3.png "Rendered by QuickLaTeX.com")
Crossing through the numbers on the end, we get:
1, 2, 3, 4
But now, it seems, we are stuck. For the median is a single number - not two numbers! And yet if we cross out any more, we will eliminate all of our numbers.
In these cases, we say that the median is the average of the middle two numbers. Thus, the median in the above case is 
So in summary, the median is either the middle number or, if you have two middle numbers, the average of those two middle numbers. Now, we try to find the median for our lemonade example:
| Date | Lemonades Sold |
| 7/2/19 | 1 |
| 7/3/19 | 2 |
| 7/4/19 | 5 |
| 7/5/19 | 2 |
| 7/6/19 | 3 |
![\[1, 2, 2, 3, 5\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-f3a215bb6600f100ab0808519b7a906e_l3.png "Rendered by QuickLaTeX.com")
Finally, the mode is the most straightforward of the three: it is the value that occurs most often. So, if you have the data:
![\[1, 2, 2, 3, 4, 5\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-2375db16c8c4129404ba61c15be74dbf_l3.png "Rendered by QuickLaTeX.com")
Then the mode is 
.
Now, a set of numbers can have more than one mode, as in this example:
![\[1, 1, 1, 2, 3, 4, 4, 4\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-720a7386585a173b4421724abc37214b_l3.png "Rendered by QuickLaTeX.com")
Here, since both 
and occur three times, they are both modes.
Often, to find the mode it can help to put the numbers in ascending order (so you can see how often certain values are repeated). To return to our lemonade example:
| Date | Lemonades Sold |
| 7/2/19 | 1 |
| 7/3/19 | 2 |
| 7/4/19 | 5 |
| 7/5/19 | 2 |
| 7/6/19 | 3 |
Finally, one property of the median is that it is "more resilient to outliers." What does this mean? First, let's be clear on what an outlier is: an outlier is a data value that is very far from many of the other observations. So suppose you have some data on how many days in a month it rains:
![\[12, 13, 14, 14, 15, 16, 17, 19\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-f15bbb7ac8b4933db8f0f5e02306ff03_l3.png "Rendered by QuickLaTeX.com")
But one month, you have constant downpours and so you get 
as a new observation. Your new data is:
![\[12, 13, 14, 14, 15, 16, 17, 19, 31\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-131afa44abbc5428c2de1866446957c8_l3.png "Rendered by QuickLaTeX.com")
But the is quite far from all of the other observations. If we were to graph this data on a boxplot:
We would see that sticks out like a sore thumb. That's how you know it is an outlier. (There are more precise, formal definitions of an outlier, e.g. more than 1.5 times the interquartile range outside the 1st or 3rd quartiles, but you won't need to know those kinds of definitions for the GRE).
Now, when we say that the median is "more resilient" to outliers, we mean that if we add an outlier to our data, the median is less affected than the mean. So the change to the median will be less than the change to the mean. Let's confirm that this is the case in our above example. To do so, we will compute the old mean/median, the new mean/median and compare them.
The old mean is: 
The old median is: 
since both 14 and 15 are in the middle of 
The new mean is: 
.
The new median is: 
.
So we can see that the new mean is 
higher than the old mean, whereas the median only increased by 
and so the median changed less than the mean did, as expected.
Now, you might wonder why the median is more resilient than the mean to outliers. You won't need to know this for the GRE, but it might help in remembering which one is more resilient to outliers. The reason is that when you are adding a new value to the data set, the median doesn't really care how large/small the new number is. All that matters is whether it is larger than the previous median or smaller. If it is larger than the previous median, then the new median moves a number to the right. If it is smaller, the new median moves a number to the left. But the mean cares about how large/small the new number is. So if the new number is an outlier, then it is, by definition, really large or really small compared to the other numbers. So the mean responds a lot to this new number whereas the median just moves a number to the right/left. That's why outliers affect the mean more than they do the median.
Practice Problems
In this post, we'll talk about ranges, quartiles, and percentiles. These are all ways of getting a sense of what the overall distribution looks like and what the possible outcomes look like.
Let's start with the range. The range is the difference between the largest and the smallest value in your data. So suppose you have the following data:
| Date | Sold |
| 7/2/19 | 1 |
| 7/3/19 | 2 |
| 7/4/19 | 5 |
| 7/5/19 | 2 |
| 7/6/19 | 3 |
| 7/7/19 | 2 |
| 7/8/19 | 5 |
| 7/9/19 | 4 |
| 7/10/19 | 3 |
| 7/11/19 | 3 |
Then, the range is $5 - 1 = 4$. Thus, the range tells you the interval over which your data is distributed.
Quartiles are more complicated. As the name suggests, quartiles are a way of dividing up the data in to four parts. So, if we have the following data:
The quartiles are the points that divide up the data in four segments, each with the same number of observations. For the above data, we would get:
You can see how in each section, we have exactly three points which is one fourth of our total data (made up of 12 points).
More formally, the first quartile is the value that separates the bottom 25% of the data from the top 75%; the second quartile is the median and splits the data in half; the third quartile separates the bottom 75% of the data from the top 25%.
Now how can we find the quartiles?
Well, we already know how to find the second quartile since, recall, that's just the median! And if we know what the median is, we can separate the data into two halves: the lower half is made up of all the values less than our median, whereas the higher half is all the values greater than our median.
Then, we find the median of the lower half and that will be the first quartile value, and the median of the upper half will be the second quartile value! This makes some intuitive sense since 25% is exactly at the middle between 0 and 50%, which is the data that makes up our lower half, whereas 75% is the middle of 50% and 100%, which is the data that makes up our upper half.
Example 1
Find the first, second, and third quartiles of:
| 21 |
| 13 |
| 37 |
| 45 |
| 5 |
| 1 |
| 9 |
| 17 |
| 33 |
| 41 |
| 25 |
| 29 |
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