Upper Level: Mathematics Achievement Study Guide for the ISEE

Data Analysis and Probability

Drawing Inferences

Data from a random sample is useful for reasonably drawing inferences about a certain population without actually gathering data from the entire population. For instance, the owner of a bakeshop does not need to interview all \(2,200\) residents in the town to know the top \(3\) most popular bakery product favored by the residents. By conducting a survey on a reasonable sample of, say, \(100\) participants, this information can be inferred.

Good Study Design

A good study design is needed to make sure that conclusions and inferences drawn from the study are valid and reasonable. Since the inferred information depends only on data from the sampling, it is essential that the sample is representative of the population. It must be randomized to avoid having a skewed or biased sample. The method of selecting participants for the control and experimental groups must be randomized. Collecting data must be a systematic process, and analysis of results must be scientific.

Types of Data

Data can be numerical, categorical or ordinal.

Numerical data can either be measured (continuous data) or counted (discrete data). Height of students and weight of buyers are examples of continuous data. Number of pets owned by homeowners and number of children in the family are examples of discrete data.

Categorical data are non-numerical, such as gender, marital status, and home town.

Ordinal data are categorical data that are assigned with numbers, such as the data derived from a survey on customer satisfaction with ratings ranging from \(0\) to \(5\).

Data sets that have one variable are called univariate. Data sets containing two variables are called bivariate, such as improvement in scores of students and the number of tutorial hours spent.

Types of Data Representation

Data is represented in various ways. A good way to represent data visually is by graphing, and there are various types of graphs:

Pie graph— a circular representation that shows the sizes of data in relation to the whole data set, and this is usually indicated by percentages.

Line graph— a graph plotted on a vertical-and-horizontal axes system, showing points connected by lines. Line graphs usually show the variation of data over time.

Bar graph— a graph plotted using bars of different heights to compare data across several categories. This is particularly useful for representing discrete data.

Histogram— similar to a bar graph, a histogram uses bars of varying lengths to graph data. It is, however, more useful in representing continuous data.

Scatterplot— a graph that uses \(xy\)-coordinates and dots to represent bivariate data. Any point in the plot represents two values.

Parallel box plot— a graph that is based on the minimum value, the first to third quartiles, and the maximum value of the dependent variable. This is plotted parallel to the \(y\)-axis (value of the dependent variable) and uses lines connecting the minimum and maximum values to the rectangles or boxes (the quartiles).

Statistic Versus Parameter

Statistics and parameters are both calculated and used to describe a group, such as “\(35\%\) of the workers were satisfied with the working conditions in the factory”.

The difference between the two comes mainly from the source of the calculated description. A statistic describes a group based on the data gathered from a sample. A parameter describes a group based on the data gathered from an entire population.

Summarizing Data

A data set is summarized by describing its central value, such as the mean, median or mode. Know how to solve or determine these values from a given data set.

It may also be summarized by describing its spread of distribution. Range and standard deviations are measures of spread. They show the spread of the data in the set. The range, for instance, shows the difference between the smallest value and the largest value. The standard deviation describes how close the values are to the mean.

Predicting values based on the graphs is possible by making correlations, interpolations and extrapolations. A scatterplot, for example, can be inspected carefully and determined if a line of best fit can be drawn. Lines of best fit, also called trend line, are used for determining trends and predicting values (by interpolation or extrapolation).

Simulations

A study can be conducted by simulating or modeling real-world events. In predicting outcomes, such as the likelihood of an event occurring, simulation can be used. For a simulation to yield a valid result, the possible outcomes must be described and designated with random numbers, random number sources must be carefully chosen, and simulations must be repeated many times until outcomes show a consistent or stable pattern.

Probability

The probability of an event happening predicts the likelihood of such an event occurring. It is expressed as a fraction or a number between \(0\) and \(1\). An event with a probability of \(\frac{3}{4}\) has a bigger chance of occurring compared to an event with a probability of \(\frac{1}{4}\). Probability can be computed using this formula:

Probability that an event will happen \(=\) number of ways it can happen \(\div\) number of possible outcomes

However, it’s not always possible to answer all probability questions using that basic formula alone. Here are other relevant rules that will help in analyzing questions:

The probability that an event will not occur and the probability that it will occur will always be equal to \(1\).
The probability that one event (A) will occur together with another event (B) is equal to the product of their individual probabilities.

Problem-Solving

Much of what you’ll encounter on the ISEE math tests involves solving problems. The essence of problem-solving can be summarized by three main questions:

What do you have?
What do you want?
How do you get it?

Following is a list of strategies you can use to answer these questions. You may not need to do all of them for each problem. Just use as many as you need to find the answer. You may need just a few steps for simple questions or all of them for more difficult problems. Think of them as resources in your problem-solving journey.

Tip 1: Read the Problem With Pen/Pencil in Hand.

The obvious first step is to read the problem, very likely multiple times, if it is at all complex. This should get you started thinking about the first two questions.
Mark up the question. Underline or circle the important facts, especially the exact question being asked. To help you make sense of the information, write down the relevant numbers with labels and units.
If there are more than two or three numbers involved, think about putting them into a nicely organized table.

Example problem with a useful table:

Jordan is \(4\) times older than Claire and Claire is \(2\) years older than Ryan. If the sum of their ages is \(70\), how old will Claire be in \(5\) years?

This table would be a good way to organize the information. The given information has been filled in.

Name	Expression	Ages
Jordan	\(4(x+2)\)
Claire	\(x+2\)
Ryan	\(x\)
Sum		\(70\)

After working out that \(x=10\) using \(6x+10=70\), the rest of the table can be completed.

Name	Expression	Ages	Age in 5 yrs
Jordan	\(4(x+2)\)	\(48\)
Claire	\(x+2\)	\(12\)	\(12+5=17\)
Ryan	\(x\)	\(10\)
Sum	\(6x+10\)	\(70\)

Tip 2: Recognize Clue Words

Look for clue phrases to help you figure out what you will need to do. Some are pretty obvious:

Sum of or total of a number and \(11\) or increased by \(11\): \(x+11\)
Difference between a number and \(6\) or decreased by \(6\): \(x-6\)
\(5\) less than a number: \(x-5\)
A number exceeds \(7\) by \(3\): \(x=7+3\)
Product of a number and \(5\): \(x\times5\) or \(5x\)
Per means divide, as in miles per hour: \(\dfrac{200 \text{ miles}}{3 \text{ hours}}\)

Tip 3: Draw the Problem

Very often a labeled drawing will help you understand a problem. The act of drawing helps to anchor the information in your mind and it gives you a visual representation that your brain loves.

Example problem:

A wagon left a farm at \(11\text{:}04\) and traveled at a constant speed until it reached the town at \(12\text{:}24\). If it went a total of \(12\) miles, how far had it traveled by \(11\text{:}34\)?

You can see that it took a time of \(1\text{:}20\), which would be \(\dfrac{4}{3} \text{ hr.}\)

The rate would be \(\dfrac{12 \text{ mi.}}{\frac{4}{3}\text{ hr.}} = 9 \dfrac{\text{mi.}}{\text{hr.}}\)

At \(11\text{:}34\), the cart will have traveled for \(30\) minutes, which is \(0.5\) hour. The distance traveled by then is:

\[9 \dfrac{\text{mi.}}{\text{hr.}} \times 0.5 \text{ hr.} = 4.5 \text{ mi.}\]

Tip 4: Work Backward

Another technique that can help you figure things out is to do what might be called working the problem backward. Take the previous problem as an example.

A. The question asks for the distance traveled between \(11\text{:}04\) and \(11\text{:}34\). How do we calculate distance? One way is to use \(d=rt\). If we know \(r\) and \(t\) for the first part of the trip, we can calculate \(d\) and be done.

B. Now, how can we get \(t\), the time, for the first part of the trip?
It’s easy: \(11\text{:}34-11\text{:}04= 30 \text{ min} = 0.5\text{ hr.}\)

C. We still need \(r\), the rate of speed. We can find it by considering the entire trip and using

\[r=\dfrac{d}{t} \text{, where}\] \[d=12\text{ miles} \text{ and}\] \[t=12\text{:}24-11\text{:}04=1\text{:}20=\dfrac{4}{3}\text{ hr.}\] \[r=\dfrac{12\text{ mi.}}{\frac{4}{3}\text{ hr.}}\] \[r=9 \dfrac{\text{ mi.}}{\text{hr.}}\]

D. Now we have \(r\) and \(t\) for the first part of the trip, so we can go back to where we started (I) and calculate \(d\) for the first part of the trip.

\[d=rt\] \[d=(9 \dfrac{\text{ mi.}}{\text{hr.}})(0.5 \text{ hr.})\] \[d=4.5 \text{ mi.}\]

(Be careful. We’ve used two different values of \(d\) in this problem. Don’t mix them up.)

Tip 5: When Nothing Seems to Work

Sometimes, you may be at a loss as to what to do. A useful strategy can be this: If you can’t see how to get the final answer, try to figure out anything you can. You may come up with an intermediate answer that you can then use to find the final answer.

Other Tips and Tricks

Consider First, Then Choose the Answer

Be careful to work the problem out carefully and check your work before choosing an answer. Some of the incorrect answer choices may represent the answer you would get if you miscalculated or used the wrong procedure.

Remember that All Questions Have Equal Value

No question is worth any more than another, so go through and answer the ones you know quickly. You can then come back to items requiring more effort. Also note that you can mark in your test booklet, so you could mark questions to which you want to return.

Use All Given Tools

Be sure to study both the question and any graph or image given carefully before attempting to answer the question. If a question seems impossible, there’s a good chance you have overlooked a piece of information.

Remain Calm!

Remember that your score on all sections is only compared with students at your grade level. You may see some questions that deal with mathematics from upper-level courses that you have not yet taken. Just don’t worry, make some sort of mark beside them, and move on. If you have time at the end of the testing session, you can always go back and see if you can figure them out.

A final piece of advice, if you have time to do it, is to persevere. Wrestle with the problem enough and you may suddenly get an “Aha!—flash of insight.” Good luck.