The BriefA Blog about the LSAT, Law School and Beyond
The essays below, which were all part of successful applications to Harvard Law, rely on humble reckonings followed by reflections. Some reckonings are political: an applicant grapples with the 2008 financial crisis; another grapples with her political party’s embrace of populism. Others are personal: a student struggles to sprint up a hill; another struggles to speak clearly. The writers have different ideologies, different ambitions, and different levels of engagement with the law. Yet all of them come across as thoughtful, open to change, and ready to serve.
Jump to a personal statement:
Essay 1: Sea Turtles
I stood over the dead loggerhead, blood crusting my surgical gloves and dark green streaks of bile from its punctured gallbladder drying on my khaki shorts. It was the fifth day of a five-week summer scholarship at the University of Chicago’s Marine Biological Laboratory (MBL), and as I shuffled downwind of the massive creature, the pungent scent of its decomposition wafted toward me in the hot summer breeze. Aggressive flies buzzed around my head, occasionally pausing to land on the wad of plastic we had extracted from the loggerhead’s stomach. The plastic had likely caused a blockage somewhere, and the sea turtle had died of malnutrition. When the necropsy was finished, we discarded the remains in a shallow hole under a thicket of trees, and with the last shovel of sand over its permanent resting place, its death became just another data point among myriad others. Would it make a difference in the long, arduous battle against environmental pollution? Probably not. But that dead loggerhead was something of a personal tipping point for me.
I have always loved the clean, carefully objective nature of scientific research, but when I returned to the US from my native XXXX to study biology, I began to understand that because of this objectivity, scientific data rarely produces an emotional effect. It is difficult to initiate change based on such a passive approach. My ecology professor used to lament that it was not science that would determine the fate of the environment, but politics. The deeper I delved into research, the more I agreed with her. Almost every day, I came across pieces of published research that were incorrectly cited as evidence for exaggerated conclusions and used, for example, as a rebuttal against climate change. Reality meant nothing when pitted against a provocative narrative. It was rather disillusioning at first, but I was never one to favor passivity. In an effort to better understand the issues, I began to look into the policy side of biological conservation. The opportunity at the MBL came at this juncture in my academic journey, and it was there that I received my final push to the path of law.
After weeks of sea turtle biology and policy debates at the MBL, we held a mock symposium on fishing and bycatch regulations. Participants were exclusively STEM majors, so before the debate even began, everyone in the room was already heavily in favor of reducing commercial fishing. I was assigned the role of the Chair of the New Bedford Division of Marine Fisheries, and my objective was clear: to represent the wishes of my constituents, and my constituents wanted more time out on the sea. However, that meant an increase in accidental bycatch, which could hurt endangered marine populations and fill up the bycatch quota for commercial fishermen before the season ended.
There were hundreds of pages of research data on novel technological innovations for bycatch reduction that I had to wade through, but with the help of my group, I was able to piece together a net replacement plan that just barely satisfied my constituents, the scientists, and the industry reps. Although the issue of widespread net replacement incentives for the commercial fishermen remained, there was no doubt that I enjoyed the mental stimulus of tackling this hypothetical challenge. I was able to use my science background to aid in brokering a compromise that would reduce the amount of damage done to the environment without endangering the livelihood of the people involved in the industry.
By the end of the symposium, I knew that I wanted to bridge the gap between presenting scientific data correctly and effecting change in the policy world. Although there are many ways for me to advocate for change, I believe that only legal and legislative enforcements will have a widespread and lasting effect on the heavy polluters of the world. I want to combine my legal education and a solid foundation in the biological sciences to tackle the ever-growing slew of environmental challenges facing us in the twenty-first century.
The night the symposium ended, we patrolled the beach for nesting females. As I walked beneath the stars, I thought of that sea turtle and of the repeating migration of my own life, from my birthplace in XXXX to my childhood in the US, back to XXXX and now the US again. With the guidance of the Earth’s magnetic fields, sea turtles are able to accurately return to their birthplace no matter how far they deviate, but I like to imagine that they, like me, do need to occasionally chart another course to get there. Standing on a beach in Woods Hole, thousands of miles from home, I knew that I was on the right path and ready to embark on a career in law.
Essay 2: Joining the Arsonists To Become a Fireman
On the morning of the 2004 presidential election, my sixth-grade teacher told me to watch out for John Kerry voters in the hallways because our school was a polling station. I nodded and went to the water fountain, thinking to myself that my parents were voting for John Kerry, and that as far as I could tell, they posed no risk to students. It was a familiar juxtaposition—the ideas at my dinner table in conflict with the dogmas I encountered elsewhere in my conservative Missourian community. This dissonance fostered my curiosity about issues of policy and politics. I wanted to figure out why the adults in my life couldn’t seem to agree.
Earlier in 2004, Barack Obama’s now famous DNC keynote had inspired me to turn my interests into actions. Even at age twelve, I was moved by his ideas and motivated to work in public service. When Obama ran for president four years later, I heeded his call to get involved. I gave money I had made mowing lawns to my parents to donate to his campaign and taped Obama-Biden yard signs to my old Corolla, which earned it an egging and a run-in with silly string in my high school parking lot.
While I knew in high school that I wanted to involve myself in public service, I wasn’t sure what shape that involvement would take until signs of the financial crisis—deserted strip malls and foreclosed homes—cropped up in my hometown. I was amazed by the disaster and shaken by the toll it took on my community. As I saw it, the crisis wasn’t about Wall Street, but about people losing their jobs, homes, and savings. I didn’t understand what Lehman Brothers had to do with the fact that my neighbor’s appliance store had to lay off most of its employees.
Intent on understanding what had happened, I started reading up, inhaling books about financial crises and articles on mortgage-backed securities and rating agencies. Along the way, I also developed an affinity for the policymakers fighting the crisis. I admired how time and again these unknown bureaucrats struggled to choose the best among bad options, served as Congressional piñatas on Capitol Hill, and went back across the street to face the next disaster. I decided that I too wanted to work in financial regulation. I thought then and believe today that if I can help protect consumers and mitigate the downturns that force people from their jobs and homes, I will have done something worthwhile.
Strange though it may seem, this decision led me to join Barclays as an investment banking analyst after college. While in a sense I was “joining the arsonists to become a fireman,” as one skeptical friend put it, banking gave me immediate experience working with the firms and people who had played key roles in the response to the financial crisis years before. I was initially worried that I would discover financial rules and regulations to be impotent platitudes, without the power to change the financial system, but my experience taught me the opposite. New regulations catalyzed many of the transactions on which I worked, from bank capital raises to divestitures aimed at de-risking. Ironically, becoming a banker made me even more of an idealist about the power of policy.
I envisioned spending years in the industry before moving to a government role, and I left banking for private equity investing with that track in mind. When I began making get-out-the-vote calls on behalf of the Clinton presidential campaign, however, I realized that I needed to change my plans. I cared more about contacting voters, about the result of the election, and about its policy implications than anything I did at work. Although I’m grateful for what I’ve learned in the private sector, I don’t want to spend more time on the sidelines of the policy debates and decisions that matter to me.
That’s why I am pursuing a J.D. I want to help shape the policies that will make the financial system more resilient and equitable, and to do so effectively, I need to understand the foundation upon which the financial system is built: the law. The post-crisis regulatory landscape is already in need of recalibration; large banks still pose systemic risks, and regulation lags even further behind in the non-bank world. Advances in financial technology, from online lending platforms to blockchain technology, are raising new questions about everything from capital and liquidity to smart contracts and financial privacy. Policymakers need to confront these issues proactively and pursue legal and regulatory frameworks that foster public trust while encouraging innovation. A J.D. will give me the training I need to be involved in this process. I don’t claim to have a revolutionary theory of financial crisis, but I do hope to be a part of preventing the next one.
Essay 3: Populism
Growing up, I felt that I existed in two different worlds. At home, I was influenced by my large, conservative Arizonan family, who shaped my values and understanding of the world. During middle school, my family moved, and I enrolled in a small, left-leaning school with an intense focus on globalism and diversity. I enjoyed being surrounded by people who challenged my beliefs, and I prided myself on my ability to dwell comfortably in both spaces.
In 2015, American political reality disrupted the happy balance between my two worlds. The Republican presidential primary, in a gust of populism, was proposing ideas that I didn’t recognize and wouldn’t condone, like a hardline immigration stance, opposition to free trade, and a tolerance for harassment. I resented this populist wave for hijacking the party, and the voters who created it. I didn’t understand them, and I didn’t think I could.
Despite my skepticism, I decided to make an attempt. As the founder of the Bowdoin College Political Union, a program that promotes substantive, inclusive conversations about policy and politics among students, I brought speakers with diverse ideologies to campus and hosted small group discussions with members of the College Democrats, the College Republicans, and students somewhere in between. In the winter of my senior year, I helped organize a summit that brought together students with a broad spectrum of views from dozens of universities throughout the eastern United States.
As a resident assistant during the 2016 presidential election, I held open-door discussions for individuals from across the political spectrum and around the globe. Facilitating these discussions felt like a natural extension of my role on campus, and I learned not only that having space for open dialogue can ease tensions, but also that the absence of that space does not erase political difference. Instead, it creates feelings of isolation and fosters ignorance.
But it was the death of a family member in early 2016 that helped me understand another perspective, namely the populist views beginning to overwhelm the Republican Party. After the death of my mother’s cousin from cancer, I called my second cousins, all three of whom are around my age, to offer my condolences. I was surprised to learn that none of them had finished high school. Instead, they had worked to help pay for their mother’s treatment. While I had been worrying about which summer internships to apply for, they were worried about maintaining their family home. In the past, I’d thought that their views on economic policy and immigration came from a place of ignorance or spite. I realized over the course of our conversation that I had no idea what it was like to not have a high school degree and compete for employment in a rural area where wages are low. For the first time, I was engaging with people in the demographic that was generating the populist wave that was sweeping the country. This conversation led me to expand my studies in politics and to think beyond the left-right spectrum to consider class and urban-rural divides within my own party. Ultimately, reconnecting with my extended family informed my decision to write my senior thesis on populist movements and why economics drives them. It also changed the way I thought about politics and its effect on people like my second cousins.
After my college graduation, I took a job with a political and opposition research firm called XYZ in Washington, because I felt that my understanding of 2016’s populism was still lacking. XYZ gave me the opportunity to work with people from different parts of the Republican Party: both establishment operatives and grassroots operations. This enabled me to work within the framework of Republican politics that resembles my own, while being exposed to the perspectives of people working to represent people like my second cousins. My time at XYZ helped me see the power of the populist movement, but also understand the limitations of its proposed solutions, like a resurgence of manufacturing. Now that I have interacted with populist groups, I see that ultimately, the valid frustrations of many working-class Americans need to be addressed by empathetic leadership and challenging but necessary evaluations of policy in the areas of economics, education, and culture.
I want to apply my passion for political discourse in law school and in my career as a lawyer. My passion for engaging with others will serve me well in the classroom and in a career at the intersection of law and politics. I hope to continue to make connections between people of diverse backgrounds and viewpoints and to engage in meaningful, bipartisan discourse.
Essay 4: Pop Warner
One summer, when I was eight years old, I signed up to play Pop Warner Football for my hometown. After the calisthenics, scrimmages, and the rest of practice concluded in the midst of the sweltering early August sun, I would sprint thirty yards up a hill steep enough to go sledding down. I had to lose nine pounds in order to make weight for my junior pee-wee football team. I wanted nothing more than to be on the team, so it didn’t faze me that I was the only one running up and down the hill. A dirt path marked the grassy knoll from my countless trips up and down. I usually managed to hold back the tears just long enough until I got home. As an eight-year-old, this was the most difficult challenge I had ever been tasked with. But the next day, I would get down in a three-point stance and sprint up the hill under the red sky of the setting sun.
When I finally made the team, I was elated; I had achieved a goal I often felt impossible in those moments of sweat and tears. The excitement was, nonetheless, short-lived. The other kids still called me “Corey the Cupcake,” a nickname I thought I’d left behind with the extra pounds. In every game of the season, my first playing football, I received my eight minimum plays and rode the bench the rest of the game. It was an unusually wet September, and I caught a cold a few times from standing there for two and a half hours in the nippy morning rain. I hated it, but I kept playing.
I continued to play every fall through high school. My freshman year, during a varsity practice, I broke both the radius and ulna bones in my left arm and simultaneously dislocated my wrist, which required a plate and four screws to repair. To this day, I can’t help but flash back to that frigid November afternoon when I look at the five-inch scar on my left arm or when the breaking point is hit precisely. Sophomore year, I was introduced to a coach who frequently criticized me for “not being black enough,” or sometimes, contradictorily, for acting “too black.” I was even benched for my entire junior year for being unable to attend football camp over the summer.
Why did I play football for eleven years? It might have been for the Friday nights in front of the school, as there was nothing more thrilling than making a crucial catch and hearing the whole town cheer. It might have been because I wanted to fit in with my athletic classmates. It might have been because I felt that I was improving after each catch, each hit, and each drill. But I believe, above all else, it was because I just don’t like to give up.
My first job as a project assistant at a large law firm was somewhat similar to my experiences as a young football player; both required grit and determination to push through difficult circumstances. Late one evening, two days before Thanksgiving, my supervisor asked me to complete and organize the service of eighteen subpoenas for the following day. The partners and associates were so busy with internal politics—one of the head partners was leaving the firm—that no one was available to walk me through the process. I felt ridiculous when I Googled “How to fill out and serve a subpoena,” but it was important to me that I complete the project properly.
I am appreciative of the challenges that I faced as a project assistant. If it weren’t for those experiences, it is unlikely that I would have been fortunate enough to be hired by the Delaware Office of the Attorney General, where I work today. My job here has confirmed that law is exactly what I want to do. I realized this through several opportunities to draft written discovery. I loved fashioning objections to each individual request in a given set. Developing legitimate grounds for disputing discovery on its merits and intent was inspiring to me. I can’t wait to do this more and on a larger scale as an attorney.
The steadfastness that I obtained as a young athlete defines who I am. I couldn’t see it at the time, but every day on which I gave something my best effort, whether it was on the practice field or in my tiny office on the twenty-seventh floor, I became a little bit stronger, a little bit wiser. I am confident that my perseverance and dedication will facilitate my future success, both in law school and afterwards.
Essay 5: Speech Therapy
When I was very young, I was diagnosed with a severe phonological disorder that hindered my ability to verbalize the most basic sounds that make up words. It didn’t take my parents long to notice that as other children my age began speaking and communicating with each other, I remained quiet. When I did speak, my words were mostly incomprehensible and seemed to lack any repetition. I was taken to numerous speech therapists, many of whom believed that I would never be able to communicate effectively with others.
From the age of three until I was in seventh grade, I went to speech therapy twice a week. I also regularly practiced my speech outside of therapy, eventually improving to such an extent that I thought I was done with therapy forever. This, however, was short-lived. By tenth grade, I realized my impediment was back and was once again severely limiting my ability to articulate words. That was also the year my family moved from Vancouver, Canada to Little Rock, Arkansas, which complicated matters for me.
I knew that my speech was preventing me from making new friends and participating in classroom discussions, but I resisted going back into therapy. I thought that a renewal of speech therapy would be like accepting defeat. It was a part of my life that had long passed. With college approaching, though, I was desperate not to continue stuttering words and slurring sentences. I knew that I would have to become more confident about my speech to make friends and to be the student I wanted to be. During the summer before my freshman year, I reluctantly decided to reenter speech therapy.
I see now that this decision was anything but an acceptance of defeat. In fact, refusing to reenter therapy would have been a defeat. With my new therapist, I made significant strides and the quality of my speech improved greatly. Using the confidence that I built in therapy that summer, I pushed myself to meet new people and join extracurricular organizations when I entered college. In particular, I applied to and was accepted into a competitive freshman service leadership organization called Forward.
The other members of Forward were incredibly outgoing, and many of them had been highly involved in their high school communities—two things I was not. I made a concerted effort to learn from those who were different from me. I was an active participant in discussions during meetings, utilizing my unique background to provide a different perspective. My peers not only understood me, but also cared about what I had to say. I even began taking on leadership roles in the program, such as directing a community service project to help the elderly. My time in Forward made it clear to me that my speech disorder wouldn’t be what held me back in college; as long as I made the effort, I could succeed. The confidence I gained led me to continue to push past the boundaries I had set for myself in high school, and has guided the bold approach I have taken to new challenges in college.
When I first finished therapy in seventh grade, I pretended that I had never had a speech disorder in the first place. Having recently finished therapy again, I can accept that my speech disorder has shaped the person I am today. In many ways, it has had a positive effect on me. My struggle to communicate, for example, has made me a better listener. My inability to ask questions has forced me to engage with problems on a deeper level, which has led me to develop a methodical approach to reasoning. I believe these skills will help me succeed in law school, and they are part of what motivates me to apply in the first place. Having struggled for so long to speak up for myself, I look forward to the day when I can speak up for others.
Essay 6: Ting Hua
“Ting hua!” I heard it when I scalded my fingers reaching above the kitchen counter to grab at a steaming slice of pork belly before it was served; I heard it when I hid little Twix bars underneath the bags of Chinese broccoli in the grocery store shopping cart; I heard it when I brought sticks back home to swing perilously close to the ceiling fan. Literally translated, “ting hua” means “hear my words.” Its true meaning, though, is closer to “listen to what I mean.” Although the phrase was nearly ubiquitous in my childhood, that distinction—between hearing and listening—did not become clear for me until much later in life.
That childhood began in Shanghai, where I was born, and continued in Southern California, where we moved shortly after I turned four. Some things stayed the same in the US. We still ate my mom’s chive dumplings at the dinner table. On New Year’s, I could still look forward to a red envelope with a few dollars’ worth of pocket money. But other things changed. I stopped learning Chinese, and my parents never became proficient in English. Slowly, so slowly I almost didn’t realize, it became harder and harder for me to communicate with them.
Because I didn’t feel like I could talk to them, I could never resist opening my mouth with others. I talked to good friends about Yu-Gi-Oh, to not-so-good friends about Pokemon, and to absolute strangers about PB&J, the Simpsons, and why golden retriever puppies were the best dogs ever. Even alone, I talked to my pet turtle Snorkel and tried out different war cries—you know, in case I woke up one morning as a mouse in Brian Jacques’s Redwall.
The way I communicated with my parents didn’t change until I came back for Thanksgiving my freshman year of college. I was writing for the school newspaper—a weekly column on politics. I had written an article in support of gay marriage. My parents had asked me about it, and in the way I was wont to do, I answered briefly before moving on to talk about my friends and my floor and my classes.
While I was brushing my teeth that night, my dad came into the restroom. He stood in the doorway and said, “Hey. I read the article you wrote about gay marriage… you should be careful saying things like that.”
His words—you should be careful saying things like that—sounded to me like homophobia. I knew that in China, same-sex relationships were illegal, stigmatized, banned, so I thought I understood where my dad was coming from, even though I also thought it was bigotry. I was about to brush him off, to accept that we had different views, but when I looked up, I didn’t see the judgment I was expecting. In the way he stood slightly hunched in the doorway, in the way he touched his chin, in the way his eyebrows drew together, I saw love. So I swallowed down “don’t worry about it” and asked what he meant. He told me about a cousin of his, someone I would have called Uncle, who was expelled from his school and sent to the countryside for his political comments. In that moment, I realized that my dad wasn’t concerned about my politics—he was concerned about me. Had I not stopped to listen, rather than just to hear, I would not have understood that. I would not have known why he told me to be careful.
Although I still enjoy talking to other people about PB&J sandwiches, I have learned to listen, to actively engage with my parents when we communicate. More importantly, whether I’m interviewing witnesses on the stand in mock trial, resolving disagreements between friends, or sitting in a chair while teachers and professors give me advice, I’ve made an effort to remember those words my mom has spoken since I was a toddler: “ting hua.”
📌 Check out our full, free admissions course.
Who We Are
7Sage was founded in 2012 to provide affordable, on-demand LSAT test prep to students in underserved markets. We’ve forgone the traditional model of expensive three-month classes in favor of an online curriculum that students can access anywhere, any time, for as long as they need. Our courses include over 4,000 video lessons and problem sets, explanations of every LSAT question ever made, analytic tools, and more.
From the very beginning, we’ve partnered with PreProBono, a non-profit that helps economically disadvantaged minorities, apply to law school.
3. Syllabus and Video Explanations
4. Digital Tester
6. Raw to Scaled Score Conversion
7. Question Bank
👉 Video overview of the admissions process: https://7sage.com/law-school-admissions-primer/
👉 Law school rankings, medians, and acceptance rates: https://7sage.com/top-law-school-admissions/
👉 Compilation of application requirements: https://7sage.com/admissions/lesson/application-requirements-for-top-schools/
👉 Free admissions course: https://7sage.com/admissions/progress/
Five star rating on AppStore (700+ reviews)
Five star rating on Facebook (100+ reviews)
Five star rating on Yelp (20+ reviews)
It's frustrating and disorienting not to know how the Flex test will be scored. The LSAT is stressful enough without worrying about a new format.
But, the truth is, you've already been given the best converter in existence from the LSAC itself: the regular 4 section PrepTest. Take 4 section PTs. That will be the best predictor of how you will do on a 3 section Flex test. On test day, frame the loss of 1 LR section to yourself as a treat: 1 fewer stress inducing nerve-racking task to do.
We've debated creating a "Flex score converter" and a "Flex PT" and we've been hesitant to do so because of how speculative it would inherently be.
The truth is that only LSAC can create a "Flex score converter" or a "Flex PT." LSAC has not given any significant details on how they will score LSAT Flex. Anything we try to do on that front will necessarily be guesswork and misleading.
Having said that, we made this "Flex Score Estimator" based on requests by students to see what their score would be if LR, RC, and LG were weighted the same. Feel free to play around with it and don’t take it seriously!
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.
- 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:
where is the name commonly used for the average. Then, to find the standard deviation (often denoted ), we simply calculate:
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.
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:
Find the standard deviation of the bond over these past ten days.
Let's first find the value of the average, We can calculate: Then, we simply apply the above formula to calculate that the standard deviation
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?
Recall the third feature of normal distributions that we discussed above:
3. 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 know that 90 is exactly 2 standard deviations away from the mean. Thus, there is a 5% chance of the event that Y will be 2 standard deviations away from the mean. But we have to remember that it is also possible that Y will be 2 standard deviations below the mean (in this case, below 10). And since, by fact 1, the normal distribution is symmetric about the mean, we know that P(Y > 90) = P(Y < 10). And since P(Y > 90) + P(Y < 10) = .05, we conclude that P(Y > 90) = .025. So there is a 2.5% chance that Y will be greater than 90.
- Given the following data, find the mean and standard deviation: 12, 23, 75, 54, 34, 24, 13, 53, 24, 3, 33, 10.
By applying the above formula, we get that the mean is approximately 29.83 and the standard deviation is approximately 21.36.
- Given the following data, find the mean and standard deviation: 73, 60, 1021, 584, 12, 807, 1700, 6, 321, 49.
By applying the above formula, we get that the mean is 463.3 and the standard deviation is approximately 566.9.
- 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?
Recall that the normal distribution is symmetric about the mean. And since 36 and 50 are both 7 numbers apart from 43, we know that P(X is less than 36) = P(X is greater than 50). Thus, P(X is less than 36) = .3. But if X is not less than 36, then X must be weakly greater than 36 (i.e. greater than or equal to 36). Thus, P(X is weakly greater than 36) = 1 - .3 = .7.
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.
Note that 36 is the mean of the distribution. Thus, half of the values are above 36. Thus, the probability is 1/2.
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.
Note that 11 is three standard deviations below the mean. We know that there is only a .003 chance that the variable will be three standard deviations from the mean, and it is equally likely to be above the mean as it is to be below the mean. Thus, we divide by two to get that the probability is .0015.
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?
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?
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:
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:
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?
To answer this question, we simply count up the number of dots that have an x-value of 1 or smaller. Thus, we find 5 such people.
Of the people who drank two or more cups of coffee, how many wrote more than 500 words?
Now, we want to look for the people who have an x-value of 2 or greater and a y-value that exceeds 500 words. Again, we get 5.
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.
According to the trend line in our graph, approximately how many words will someone who drank 2 cups of coffee write?
This is just a matter of knowing how to read a line on a graph. We look at our trend line and see where it has an x-value of 2. At that point, we are at around 500 words.
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.
By (approximately) how much has the S&P 500 increased from 1/1/1999 to 1/1/12?
In looking at the graph, we see that the S&P 500 was at about 1250 in 1/1/1999 and about 1250 in 1/1/12. Thus, the approximate increase is $0.
By (approximately) what percentage has the S&P 500 increased from 1/1/95 to 1/1/18?
We see that on 1/1/95, the S&P 500 was at about 500. On 1/1/18, the S&P 500 was about 2700. Thus, the percentage increase is:
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:
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:
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:
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
Notice that none of the data points are below 200. So their average cannot be below 200. So the average is . 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:
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
If we divide each of our observations by 2, then our mean will drop by half. So, since the average of the original, uncorrected data must be 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:
Find the median number of words written in the above scatterplot.
We can mark off pairs of the highest and lowest points in our graph. This is just like, when we find the median of some data, we order the values least to greatest and strike off pairs of the highest and lowest values. Doing so here, we get that some value around 475 will be our median.
And again, we can modify the given data and find the new median:
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.
We simply multiply our answer in the previous question by 2 to get 950.
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.).
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.
1. In the below histogram, what if anything can we conclude about the median of the data?
We cannot conclude the exact median, but we know that it will be between 80 and 89. This is because there are four students above that range and there are two students below that range. So the median must be the average of two values between 80 and 89.
2. In the following chart, what can we conclude about the range of the data?
We don't know the exact range but we can give some values that the range must lie in between. We know that the minimum value is somewhere between 141 and 143.9. We know that the maximum value is somewhere between 178.7 and 181.6. Subtracting the smallest possible minimum value from the largest possible maximum, we get . 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?
We know that every kid that is 5'1 or taller will be taller than Richard. Thus, at a minimum, 8 people have to be taller than Richard. But it is not clear how many of the children between 4'7 and 5' are taller than Richard. If none of them are, then 8 people will be taller than Richard. If all of them are, then 17 people will be taller than Richard. Thus, the maximum possible number is 17 and the minimum possible number is 8.
4. In the below chart, what if anything can you conclude about the median height of the class?
While you cannot conclude the exact value of the median, you know that it must occur somewhere in the range of 4'7 to 5'. That is because you know that there are 9 students shorter than 4'7 and removed students at either extreme, you would have to end up with someone in the 4'7 to 5' range.
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:
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 know from the above diagram that 23% of all sales in January of 2010 were of frozen foods. Thus, let be the total amount of sales made in January 2010. We get that
We know from the above diagram that 23% of all sales in January of 2010 were of frozen foods. Thus, let be the total amount of sales made in January 2010. We get that
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.
From January 2010 to January 2011, which category grew the most as a proportion of total sales?
We can go through the different categories to determine how much each grew/shrank. Pharmaceuticals: 1 Produce: -1 Dry Foods: 1 Canned Foods: -5 Dairy Products: 1 Frozen Food: +3 Thus, frozen foods grew the most.
We can go through the different categories to determine how much each grew/shrank.
Dry Foods: 1
Canned Foods: -5
Dairy Products: 1
Frozen Food: +3
Thus, frozen foods grew the most.
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:
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.
Now, in January 2011, the proportion of canned foods sold was 2%. Let be the total amount of sales in 2011. Thus, 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. Thus, the absolute difference is
We know that in January 2010, the percentage of goods sold that were dairy products was 20%. Thus, the total amount of dairy products sold was
Now, in January 2011, the proportion of canned foods sold was 2%. Let be the total amount of sales in 2011. Thus, 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.
Thus, the absolute difference is
- 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?
Let be the total amount of sales for all goods. The prompt gives us the following equation:
Then we get
- Overall sales in January 2010 were $100,000. How much more revenue was generated by dry foods as compared to canned foods?
We can calculate that sales of dry foods were $15,000 in January 2010, while sales of canned foods in January 2010 were $7,000. Thus, there was $8,000 more in revenue from dry foods than from 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?
Let be the total sales in January 2010 and be the total sales in January 2011. The question gives us: