# The BriefA Blog about the LSAT, Law School and Beyond

A bar graph tells you how many objects in a given category you have. For example, in the following bar graph:

Each bar tells you how many children fall into that height range. So, for example, there are 2 children between 0 and 4 feet; there are 7 between 4 feet 1 inch and 4 feet 6 inches, and so on.

__Example 1__

How many children are between 4 feet 7 inches and 6 feet? How many children are taller than 6 feet?

We can also have segmented bar graphs like the following:

These bar graphs allow us to compare different groups, in this case the group of 1 year olds against the group of 2 year olds, 3 year olds, and so on. We can thus see how their preferences shift over time.

To read the graph, the horizontal signs tell us which categories we have (here, the different ages 1 to 5) and then the vertical component tell us how many members of that category (e.g. how many one year-olds) in accordance with the key to the right. Thus, the left-most blue bar tells us how many 1 year-olds have "Mighty Man" as their favorite superhero. The right-most yellow bar tells us how many 5 year-olds have "Valiant Vanessa" as their favorite hero and so on.

__Example 2__

Which superhero gains the most fans as the children mature from being 1 years-old to 5 years-old?

**Example 3**

Which superhero has the most supporters overall (i.e. across all the age groups)?

**Example 4**

Which of the following tables goes with the given bar graph:

A.

B.

C.

D.

**Practice Problems**

- Which of the following tables is compatible with the given graph?

A.B.

C.

D.

- 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:

**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:

We order it from least to greatest to get:

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:

We order it from least to greatest:

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:

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:

**Finding the Mode**

Finally, the **mode** is the value that occurs most often. So, if you have the data:

Then the mode is .

Now, a set of numbers can have more than one mode, as in this example:

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:

But one month, you have constant downpours and so you get as a new observation. Your new data is:

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?

In this post, we'll talk about ranges, quartiles, and percentiles. These are all ways of getting a sense of what the overall distribution looks like and what the possible outcomes look like.

Let's start with the range. The **range** is the difference between the largest and the smallest value in your data. So suppose you have the following data:

Then, the range is . Thus, the range tells you the interval over which your data is distributed.

Quartiles are more complicated. As the name suggests, quartiles are a way of dividing up the data in to four parts. So, if we have the following data:

The **quartiles** are the points that divide up the data in four segments, each with the same number of observations. For the above data, we would get:

You can see how in each section, we have exactly three points which is one fourth of our total data (made up of 12 points).

More formally, the **second quartile** is the median. Then, the **first quartile** is the median of all the values that are less than the median (or second quartile). The **third quartile** is the median of all the values that are greater than the median.

Now how can we find the quartiles?

Well, we already know how to find the second quartile since that's just the median. After finding the median, separate the data into two halves: the lower half is made up of all the values smaller than our median, whereas the higher half is all the values greater than our median.

Then, find the median of both halves. The median of the lower half will be the first quartile, and the median of the upper half will be the second quartile value.

This makes some intuitive sense since 25% is exactly at the middle between 0 and 50%, which is the data that makes up our lower half, whereas 75% is the middle of 50% and 100%, which is the data that makes up our upper half. So if we were looking for a process to split the data into four parts, each containing 25% of our observations, this method (to find the quartiles) would be an excellent candidate for that task.

Finally, the **interquartile range** is just the value of the third quartile minus the value of the first quartile.

__Example 1__

Find the first, second, and third quartiles of: 21, 13, 37, 45, 5, 1, 9, 17, 33, 41, 25, 29. Then find the interquartile range.

**Practice Problems**

- Find the first, second, and third quartiles of: 30, 22, 90, 43, 28, 65, 2, 12. Then, find the interquartile range. Also, find the range.

- Find the first, second, and third quartiles for: 111, 210, 291, 240, 287, 534, 323, 222, 401. Then, find the interquartile range. Also, find the range.

- For the following numbers, is the interquartile range larger than the range? 22, 12, 32, 53, 46, 23, 21, 19, 35, 42, 31, 21, 24

In the next few posts, we will introduce different kinds of graphs. These graphs allow you to visually represent data in ways that emphasize different aspects of the data. In preparation, let's look at some data represented with different graphs:

Tables of Lemonades Sold

$$\begin{center}

\begin{tabular}{ |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 7/7/19 & 2\\

\hline 7/8/19 & 5\\

\hline 7/9/19 & 4 \\

\hline 7/10/19 & 3\\

\hline 7/11/19 & 3\\

\hline

\end{tabular}

\end{center}

$$

Here are some of the difference kinds of graphs you may encounter:

Bar graph:

Circle graph:

Box plot:

Scatterplot:

In subsequent posts, we will talk more about how to read each of these graphs.

Talk of "data" is ubiquitous -- what does that word mean in the context of the GRE? We will think about **data** as observations of given variables. Now what does that mean?

Well suppose we are trying to help our child increase her earnings from a lemonade stand. Some days, she sells a lot and some days she sells almost nothing. To figure out why, we might start keeping track of how much she sells on a given day. So we would start making a table that looks like:

Now, each of the rows in the table is an observation. And we have our variables at the top of the columns: the date and the number of lemonades sold. And more generally: **variables** are just the characteristics that we keep track of, while an **observation** is some particular occasion when we record the values of our variables.

Now, a very simple way to keep track of data is via a **frequency distribution**. This is a table that records, on the left-hand side, possible values of a given variable, and on the right-hand side, it records how often those values appeared. Applied to the above data, we would get:

This table answers questions like "How often did my child sell three lemonades?" To find the answer, we go to the "Lemonades Sold" column and look for the row with three lemonades sold. In that row, the right hand column (corresponding to the number of days) says one. So there was one day where the child sold three lemonades.

And, in addition to a frequency distribution, we can also create a **relative frequency distribution** which records, on the left-hand side, possible values of a given variable, and on the right hand-side, the percentage of all observations where that value occurred. So, in the above example, we would get:

since the total number of days is and 20% and 40% and so on through the table.

Now, to really get a handle on what is driving her lemonade sales, we should probably add some more variables (e.g. daily temperature, day of week) and collect some more observations:

Here are some practice problems on the above concepts:

**Practice Problems:**

- Using the above data, construct a frequency table that tells you how often a certain number of lemonades was sold:

- Using the above data, how many lemonades were sold on Tuesdays?

- Using the above data, how many days had more than 3 lemonades sales?

- On days where the temperature was at least 73 degrees, how many lemonades did she sell on average?

- On days where the temperature was less than 73 degrees, how many lemonades did she sell on average?

In the next few posts, we’ll unpack some of the statistical terms that the GRE throws around. These terms can be seen as ways to answer two central questions:

- What’s Typical?
- What’s Possible?

In response to the question of "What's Typical?" we can find the median, mean, or mode of the data. These give you a sense of what the average value is, or what the most common value is; in short, they describe what a typical value might be.

And in answering "What's Possible?" we can find the range, quartiles, or percentiles of the data. These give you a sense of the possibilities and how the possibilities are distributed.

In later posts, we will give actual definitions of these concepts. But first, some overarching advice about learning these concepts.

There are four main ways you’ll get tested on these concepts.

- Given some data, find the value of one of these concepts (e.g. the mean, median, range, etc.).
- Given some data, alter the data in some way (e.g. by adding 10 to every value), and then find the value of one of these concepts.
- Given some graph/chart, estimate the value of one of these concepts.
- Answer some question about some concept's general properties (e.g. that the interquartile range is always less than or equal to the range).

Thus, you want to understand the formal definition well enough to do questions of the first and second types. But you also want to have some intuitive sense for the concepts, so that you can do questions that fall in the third and fourth categories.

Around 20% of GRE questions involve "data analysis." These questions typically involve looking at data/graphs/tables in order to find certain values, like the median or percent change. Other questions ask about important statistical concepts, like the normal distribution.

All the posts in this series can be found here:

- What is Data? Variables and frequencies
- Statistical Concepts
- Reading Graphs: An Introduction
- Normal distributions

Content for other parts of the math GRE can be found here:

For reasons known only to ETS, number line problems are categorized with geometry. Such problems are basically algebra problems: they present you with some diagram or some facts about certain variables and then ask whether some equations or inequalities could be true, given certain relationships among the variables. Here is a straightforward example:

__Example 1
__

__x__

__y__

A. is greater than

B. is greater than

C. The two quantities are equal

D. It cannot be determined which, if any, is greater.

Now, a number line simply gives all the numbers from least to greatest. So the numbers to the right of the number line are greater than those to the left. And if there are markings on the number line, then you may (provided the question says nothing to the contrary) assume that the markings are evenly spaced. What does this mean? Well in the following diagram:

you know that the distance between and is only half of the distance between and . Using algebra, we could express this as:

Now, how do we solve number line problems? It will depend on what the question asks for. Some questions ask for which of the following *could* be true, whereas others ask about which of the following *must* be true.

If the questions asks which equations could be true, then we are looking for either a set of numbers that makes the equation true and is compatible with the diagram/given information, or we want to show that the equation is somehow inconsistent with our diagram/given information.

If the questions asks which equations must be true, then we are looking for either a set of numbers that is compatible with the diagram/given information but makes our equation false, or we want to show that the equation somehow follows from the diagram/given information.

Whether you should look for a specific example or some proof of the consistency/inconsistency of the given equation is a judgment call that you need to make in the moment. It's hard to give general principles about when to do which, but after doing some practice problems and just thinking about the equation at hand, you should develop some sense of how to decide if an equation is consistent or not. For example:

__Example 2__

Let and . Must the following be true?

After doing some practice problems, you get a sense of which numbers to consider when you come to one of these problems. And by running through those numbers quickly, you can often show that some equation either can be satisfied (which answers whether the equation could be true) or can be violated (which answers whether the equation must be true).

The only real piece of advice I have here is two-fold: First, always try easy, concrete numbers. Don't just think about "one negative and one positive value." It's a lot more general to think that way, but also a lot harder to evaluate. Try, instead, or and . Nice, easy numbers to calculate. And if you want a small number, try or Second, it is easy to think about what happens if both are normal, natural numbers (e.g. 1 and 2). But also think about what happens if they are small, positive numbers (e.g. and ) or only one is positive (e.g. 1 and -1) or both are negative (e.g. -1 and -2). If the equation works (or is always violated) in all of those cases, then probably it's always true (or always false).

When it comes to proving that some equation is either always true/false given the diagram/information in the question, look for easy inferences. And if you can't see a way to prove it quickly, consider just flagging it and coming back. If you've gone through a few diverse examples and they all come out the same way, probably the equation actually is always true (or always false) and the time spent confirming that might be better spent on other problems.

__Practice Problems:__

1. The markings on the below number line are evenly spaced:

Which of the following must be true (select all that apply):

A.

B.

C.

2. Suppose we know and Which of the following must be true (select all that apply):

A.

B.

C.

3. Suppose we know Which of the following must be true (select all that apply):

A.

B.

C.

D.

E.

4. The markings in the below number line are equally spaced. Which of the following could be true:

A.

B.

C.

5. The markings in the below number line are equally spaced. Which of the following must be true:

A.

B.

C.

Geometry questions sometimes involve various graphs. For example:

__Example 1__

Find the area of the following square:

Now, these problems do not differ substantially from what we have before. Essentially, graphing problems give you the same information in a slightly different way. So instead of telling you how long the sides of a particular shape are, they might give you the coordinates of the endpoints. But of course, we know how to infer the length of that side, for we know the Pythagorean Theorem:

__Example 2__

A rectangle has a side whose vertices are at (1, 3) and (2, 2). Find the length of that side.

In this vein, a more realistic problem might look like:

__Example 3__

A triangle has vertices at (1, 4), (2,3), and (0,0). Find its perimeter.

Similarly, graphing problems can involve angles:

__Example 4__

A triangle has vertices at and . Find the degree of angle .

In short, if you have a geometry problem that involves certain points or a diagram, you can generally use those points or that diagram to infer aspects like the lengths of certain sides or the degrees of various angles, and then use that information as you normally would.

**Practice Problems:**

1. A rectangle (shown below) has vertices at (0,1), (2, 0), and (2,5). Where is its fourth vertex?

2. A circle has its center at point and its circumference includes the point . Find its radius.

3. A triangle has vertices at (1, 4), (2,3), and (0,0). Find its area.

4. The square in the following diagram has points at (2, 1) and (1, 4) as labelled. What is the area of the square?

There are also two relationships between angles and circles which are crucial. First, let's define the term "central angle":

** Central Angle:** A central angle is an angle located at the center of a circle with endpoints on the circumference of that circle

Now, the crucial fact to learn is:

** Central Angles and Arcs:** In the following diagram:

The degree of angle AOB =

Thus, we can determine the degree of any central angle by looking at the length of its arc, and vice versa.

**Example 1**

The circle below has a radius of 2 and has a length of . What is the degree of angle ?

__Example 2__

The circle below has a radius of 6 and the degree of is 47 degrees. What is the length of ?

A related fact to the above is:

** Circumference Angle Theorem:** In the following circle with radius :

The degree of angle ACB = .

This theorem is also called the Inscribed Angle Theorem.

In essence, if you have an angle on the circumference of a circle, its degree will be one-half the degree of a central angle with the same endpoints. In pictures:

Now, what may be surprising about this is that it does matter where on the circumference the angle begins from. In other words:

the angles and above have the same degree.

Finally, you must be careful to make sure that you are comparing the angle on the circumference to the correct central angle. For example:

and do not have the same degree. That is because they give rise to different central angles:

Thus, it is not enough to look at the endpoints of your angle. You also need to look at where on the circumference the angle is, so you can be sure to compare it to the relevant central angle.

__Practice Problems__

1. The below circle has a center at . Find :

2. Find the value of

3. In the following diagram, Find

4. The radius of the following circle is Find

5. The radius of the following circle is 1. Find

6. The radius of the below circle is 1. Find

7. In the below circle, which is larger?

__ __

A.

B.

C. They are equal.

D. It cannot be determined.