The BriefA Blog about the LSAT, Law School and Beyond
Congratulations on your offers of admission! How do you decide where to go? Do you go with the best name brand school? The school that offered the largest scholarship? The one that sent you the nicest swag package?
If you are deciding between two or more schools, attending Admitted Students Day events is an excellent way to decide which program is best for you. Your visit could help you make a final decision about where you will spend the next three years of your life.
You will:
- Have an opportunity to speak with current students about their law school experiences (Do the students feel supported by the institution?)
- Speak with administrators from financial aid and career services and get a sense of the support and institutional commitment to the students
- Speak with faculty members to discuss your areas of legal interest and see their level of engagement and enthusiasm for teaching
- Likely get a chance to feel the 1L experience by participating in a mock classroom situation with actual professors
- Have an opportunity to explore the campus and surrounding neighborhood and decide for yourself if the environment is appealing to you (Does the school feel big enough for you? Is it cozy enough for you? Do you feel physically safe?)
- See what sort of housing opportunities are available to students and how much it will cost
- See what your transportation needs will be (Will you need a car? Does everyone Uber?)
- Interact with other admitted candidates (Is this a community that you want to join?)
You should also ask current students and the appropriate administrators about student resources like clerkship opportunities, public interest opportunities, clinical opportunities and the availability and competitiveness of other resume-building opportunities. Ask about the percentage of students who participate on Law Review and other journals and the process of joining. Ask what sort of networking opportunities and career services programming are provided and when.
Before you attend, email the admissions office and ask if the school will offer a travel stipend for you to attend the Admitted Students Day event. It never hurts to ask politely. Prepare your narrative and decide how you will introduce yourself to your future colleagues and faculty. Do some research into the city/town so you can make casual conversation.
On the day of the event, dress in business casual unless otherwise directed. You might feel awkward and shy, but EVERYONE is feeling the same way. Stand up straight, shake hands, talk less and smile more. You might feel apprehensive about appearing less than impressive to admissions or faculty. Remember that the event is a means for the school to impress YOU and get you to commit to their program.
The normal distribution is the following curve:
and it is important because there is a theorem in statistics (the "Central Limit Theorem") that tells us that if we repeat random experiments of a certain kind, the graph of the different outcomes will look more and more like a normal distribution the more we repeat the experiment. Thus, the outcomes of many random processes (e.g. individual heights), over time, come to be normally distributed.
Crucially, the distribution has the following three properties:
- It is symmetric about the mean. This means that half of the data is on one side of the mean, and the other half is on the other.
- The mean of the distribution is equal to the median
- There are certain facts about how the data is distributed. So:
- 68% of the data is within 1 standard deviation of the mean
- 95% of the data is within 2 standard deviations of the mean
- 99.7% of the data is within 3 standard deviations of the mean
Now, we will talk about how to find the standard deviation. There is a formula to do just that:
Suppose we have the following data: 
. Then, we can find the average:
![\[\overline{x} = \frac{\sum_{i=1}^n x_i}{n}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-8f4b8c59c646a631fc643697df6cfe1a_l3.png "Rendered by QuickLaTeX.com")
is the name commonly used for the average. Then, to find the standard deviation (often denoted 
), we simply calculate:
![\[\sigma = \sqrt{\frac{\sum_{i = 1}^n (x_i - \overline{x})^2}{n-1}}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-e923763f1f32af53e5d603d723683bad_l3.png "Rendered by QuickLaTeX.com")
Sometimes, problems will give you the data and ask you to find the standard deviation. Alternatively, problems may give you the normal distribution and ask you to find the probability of a certain values. Let's look at some examples of each.
Example 1
Over a period of ten days, the value of a bond at the close of trading has had values in accordance with the following table:
![\[\begin{center}\begin{tabular}{ |c|c| }\hlineDate & Lemonades Sold\\\hline 1/1/19 & 100 \\\hline 1/2/19 & 110 \\ \hline 1/3/19 & 120 \\ \hline 1/4/19 & 110 \\ \hline 1/5/19 & 130\\ \hline 1/6/19 & 130\\ \hline 1/7/19 & 110\\ \hline 1/8/19 & 70 \\ \hline 1/9/19 & 80\\ \hline 1/10/19 & 80\\ \hline\end{tabular}\end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-b909bfd0505e0a099510547e766eb629_l3.png "Rendered by QuickLaTeX.com")
Find the standard deviation of the bond over these past ten days.
We can calculate: 
Then, we simply apply the above formula to calculate that the standard deviation 
Example 2
Suppose we have a random variable Y which is normally distributed according to a distribution whose mean is 50 and whose standard deviation is 20. What is the probability of the event that Y has a value greater than 90?
Practice Problems
- Given the following data, find the mean and standard deviation: 12, 23, 75, 54, 34, 24, 13, 53, 24, 3, 33, 10.
- Given the following data, find the mean and standard deviation: 73, 60, 1021, 584, 12, 807, 1700, 6, 321, 49.
- The random variable X is distributed normally, with a mean at 43. The probability that X is greater than 50 is .3. What is the probability that X is weakly greater than 36?
-
The random variable Y is normally distributed according to a distribution whose mean is 36 and whose standard deviation is 2. Find the probability that Y is greater than 36.
-
The random variable Y is normally distributed according to a distribution whose mean is 20 and whose standard deviation is 3. Find the probability that Y is less than than 11.
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:
![\[\begin{center} \begin{tabular}{ |c|c| } \hline Coffees & Words Written\\ \hline 0 & 290 \\ \hline 1 & 360 \\ \hline 1.8 & 520 \\ \hline 2.6 & 560\\ \hline 4.4 & 470 \\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-5f146cd824a92b6a9b98f41e8ec8b033_l3.png "Rendered by QuickLaTeX.com")
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:
![\[\begin{center} \begin{tabular}{ |c| } \hline \hline Score \\ \hline 92 \\ \hline 94 \\ \hline 93 \\ \hline 100\\ \hline 82\\ \hline 83\\ \hline 85 \\ \hline 72\\ \hline 63\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-8ef2edb0cb3b14f492dfb7f6ae5c683b_l3.png "Rendered by QuickLaTeX.com")
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:
![\[\begin{center} \begin{tabular}{ |c|c| } \hline Coffees & Words Written\\ \hline 0 & 290 \\ \hline 1 & 360 \\ \hline 1.8 & 520 \\ \hline 2.6 & 560\\ \hline 4.4 & 470 \\ \hline 4.2 & 492 \\ \hline 5.3 & 455 \\ \hline 4.9 & 462 \\ \hline .8 & 310 \\ \hline .1 & 270 \\ \hline 1.3 & 280 \\ \hline .5 & 255 \\ \hline 2.1 & 540 \\ \hline 2.4 & 510 \\ \hline 3.2 & 572 \\ \hline 2.9 & 580 \\ \hline 4.7 & 400 \\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-a741031763726c92c653352caff96238_l3.png "Rendered by QuickLaTeX.com")
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.
Practice Problems
1. In the below histogram, what if anything can we conclude about the median of the data?
2. In the following chart, what can we conclude about the range of the data?
. Subtracting the largest possible minimum from the smallest possible maximum, we get 
Those are our maximum and minimum possible ranges, respectively. The true range must lie between them.3. A new kid, Richard, joins the class. Richard is 4'8. Given that the graph below accurately depicts the heights of Richard's classmates, what is the maximum possible number of classmates that are taller than Richard? What is the minimum possible number?
4. In the below chart, what if anything can you conclude about the median height of the class?
5. The following chart represents the books read by the 500 fifth-graders 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).
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?
be the total amount of sales made in January 2010. We get that ![\[.23 x = 23000\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-f761f87b252eef83a869b30d1f0b9f3c_l3.png "Rendered by QuickLaTeX.com")
Then, as canned foods make up 7% of all sales that month, we conclude that the store sold $7000 worth of canned food.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.
Dividing, we conclude that 
And since in 2011, the proportion of dairy products sold was 21%, we conclude that 
dollars worth of dairy products were sold in 2011. 
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?
![\[.25 x = .1x + 10000\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-8cc7cbba7a4766606e7b5d8f5c4f6f15_l3.png "Rendered by QuickLaTeX.com")
![\[.15x = 10000\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-cf93dc12e61020990ef55a72809f137c_l3.png "Rendered by QuickLaTeX.com")
.
be the total sales in January 2011. The question gives us: ![\[.15x = 2(.11)y\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-e9e37d6ec22e7bbf5789733e84c2ab67_l3.png "Rendered by QuickLaTeX.com")
.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 fifth-graders 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 fifth-graders 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 fifth-graders 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 fifth-graders 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:
![\[100, 98, 68, 88, 79, 92, 90, 85, 86\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-bc7db9de0ef7b14bb4d7cccf1e9908cf_l3.png "Rendered by QuickLaTeX.com")
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?
. It's a little hard to see what the exact value are just based on the graph, so anything from 126 and 136 would be reasonable.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?
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?
children. 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
Which of the following tables goes with the given bar graph:
A.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 4'6 & 4 \\ \hline 4'7 - 5' & 9\\ \hline 5'1 - 5'6 & 8 \\ \hline 5'7 - 6' & 3\\ \hline 6' - 6'6 & 0\\ \hline 6'7 - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-279753a7e8393b8377241cb57aea6efe_l3.png "Rendered by QuickLaTeX.com")
B.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 4'6 & 7 \\ \hline 4'7 - 5' & 9\\ \hline 5'1 - 5'6 & 5 \\ \hline 5'7 - 6' & 3\\ \hline 6' - 6'6 & 0\\ \hline 6'7 - 8' & 1\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-3c0c0621418dcd28ba222ceaf4059a2a_l3.png "Rendered by QuickLaTeX.com")
C.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 4'6 & 7 \\ \hline 4'7 - 5' & 9\\ \hline 5'1 - 5'6 & 5 \\ \hline 5'7 - 6' & 3\\ \hline 6' - 6'6 & 0\\ \hline 6'7 - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-b31ceb72e545ad4f247966c3a86e4f53_l3.png "Rendered by QuickLaTeX.com")
D.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 4'6 & 7 \\ \hline 4'7 - 5' & 9\\ \hline 5'1 - 5'6 & 4 \\ \hline 5'7 - 6' & 3\\ \hline 6' - 6'6 & 0\\ \hline 6'7 - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-9f7915ed66d64261261e7a9cd27acf88_l3.png "Rendered by QuickLaTeX.com")
Practice Problems
- Which of the following tables is compatible with the given graph?
A.![\[\begin{center} \begin{tabular}{ |c|c| } \hline Height & Number of Children\\ \hline 0 - 4'00 & 1 \\ \hline 4'1 - 5' & 13 \\ \hline 5'1 - 6' & 8 \\ \hline 6' - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-ece2fecd4caf75713514dc55b0020620_l3.png "Rendered by QuickLaTeX.com")
B.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 5' & 16 \\ \hline 5'1 - 6' & 8 \\ \hline 6' - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-d2a4a8102714a3e75f58340f8e24b3a7_l3.png "Rendered by QuickLaTeX.com")
C.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 5' & 16 \\ \hline 5'1 - 6' & 8 \\ \hline 6' - 8' & 2\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-5a4c5e7555325e1260ad4f0c6d1780a4_l3.png "Rendered by QuickLaTeX.com")
D.
![\[\begin{center} \begin{tabular}{ |c|c| } \hline Height & Number of Children\\ \hline 0 - 4'00 & 2 \\ \hline 4'1 - 5' & 16 \\ \hline 5'1 - 6' & 7 \\ \hline 6' - 8' & 0\\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-41fefb7568e18a60b1af3a079dcc0b32_l3.png "Rendered by QuickLaTeX.com")
- A new kid, Richard, joins the class. Richard is 4'8. Given that the graph below accurately depicts the heights of Richard's classmates, what is the maximum possible number of classmates that are taller than Richard? What is the minimum possible number?
- In the below chart, what percentage of three-year olds take Tornado Terry to be their favorite hero?
- In the below chart, which superhero loses the most fans as children age from 1 to 5?
- In the below chart, what if anything can you conclude about the median height of the class?
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.
Finding the Mean
The mean (or average) of a variable is found by adding up all of the values of that variable and then dividing by the number of observations. So to go back to our lemonade example:
![\[\begin{center} \begin{tabular}{ |c|c|c| } \hline Date & Lemonades Sold \\ \hline 7/2/19 & 1 \\ \hline 7/3/19 & 2 \\ \hline 7/4/19 & 5 \\ \hline 7/5/19 & 2 \\ \hline 7/6/19 & 3 \\ \hline \end{tabular} \end{center}\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-aa47268fbcd9dbdd3c14ee9f9eea4816_l3.png "Rendered by QuickLaTeX.com")
Finding the Median
Now, finding the median is a little trickier. Imagine lining up all of the values for your variable from least to greatest. Then, the median is 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")
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, both the left-most and right-most observation, to get:
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
And we repeat:
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
Until we get:
1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6, 7, 8, 9
And thus the median is 3.
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:
![\[1, 2, 2, 3, 5\]](https://7sage.com/wp-content/ql-cache/quicklatex.com-f3a215bb6600f100ab0808519b7a906e_l3.png "Rendered by QuickLaTeX.com")
Finding the Mode
Finally, the mode 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 (and no other number occurs three or more 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:
Median is More Resilient to Outliers
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 definitions of an outlier, but you won't need them for the GRE).
Now, when we say that the median is "more resilient" to outliers than the mean, we are saying that if we add an outlier to our data, the median is affected less than the mean. In other words, 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 
. Thus, the median changed less than the mean did, exactly as predicted.
Why is the Median More Resilient?
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.
Here's why. When you add 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 does care 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
- Suppose we have the following heights of a kindergarten class (in inches): 29, 31, 33, 33, 37, 28, 29, 30. Find the mean, median, and mode of this data.
- Suppose the ruler we used to measure everyone's height was mislabeled: in fact, everyone is five inches taller than the ruler suggests. Take the data in question 1, adjust for this fact, and find the new mean, median, and mode.
- A new kid who is remarkably tall (60 inches) joins the class. Which statistic is more affected by this change, the median or the mean?
(You could either add up all of our new values and divide by 8, or simply add 5 to the mean we got in question 1).