Upper Level: Quantitative Reasoning Study Guide for the ISEE

Data Analysis and Probability

You will be dealing with quantitative data, either in the discrete (counted) or continuous (measured) form. Data are gathered through surveys, observations, or experiments, and the results are shown as statistics, graphs, and tables. It is essential that you know how to interpret these data representations and descriptions.

Different Ways of Representing Data

Data can be reported in tables with headings, columns, and rows. You’ll see many of these tables in data interpretation questions. They’re quite easy to understand—first read the title to have a general idea of the data being presented. Then read the headings and labels. If the table shows the sales of a car company for the 4 years, it may ask questions like, “Which car model had the highest average sales for at least two years?”

More often than not, data will be presented as a graph. Listed here are some of the common graphs you will encounter in this test. Aside from familiarizing yourself with these graphs, it is also important to know which graphs are appropriate for certain types of data and what inferences you can draw from them.

Bar graph— uses bars of different heights; it is ideal for showing discrete data for different categories, such as the number of advertisements for different categories (food, apparel, entertainment, gadgets, cars, educational resources, others)

Histogram— similar to bar graph because it uses bars; but it is ideal for representing continuous data along a range of numbers, such as the weight of male students across different age groups).

Pie graph— a circular graph that shows the relative sizes of data, such as the number of students engaged in different extra-curricular activities in a school.

Learn other common data graphs, such as dot plots, line graphs, pictographs, and stem-leaf plots. A scatterplot will also be interesting to learn, and it will be described further under bivariate data.

Measures of Central Value

While a set of data is represented visually through graphs, it may also be described by its measure of central value and spread of data. The central value of a data set is often described using the mean, median or mode.

The mean of a data set is computed using this formula:

Mean = sum of observed values ÷ number of values

The median is determined by sorting the data set in an ascending or descending order. The middle value is the median. If two values are in the middle, called the middle pair, obtain the average (or mean) of these values. The result is the median.

The mode is the value that occurs most frequently. It is also particularly useful in describing nominal data.

It is important to know which measure is useful for a given situation.

• A graph that shows values close together with an occasional outlier can use the mean to describe the system.
• A graph of data that shows scattered points but with visible grouping of points around a value or bracket may be more aptly described by a mode.
• If your purpose would be to see the range of values in which half of the set operates, then the median will be most useful.

Measures of Spread

The range and standard deviation are just two of the measures that describe the spread of data in a set.

Range is determined by subtracting the smallest value from the largest value. It is highly descriptive if the data set consists of tightly-grouped values, but can be misleading if the largest value or smallest value is an outlier.

Standard deviation is determined by a formula that measures how close the data set is to the mean.

Standard deviation, which uses the symbol sigma (\(\sigma\)), is the square root of the variance.

The variance, which uses the symbol \(\sigma^2\) is the average of the squared differences of the values from the mean.

Read further on interquartile range.

In ISEE Quantitative Reasoning, it is important to understand the concepts and the relationships between the measures of central value and spread. Some questions may even be answered without intensive calculations but with careful inspection, instead.

Types of Data and Variables

Data sets can have one type of data or variable, called univariate data. The pie graph mentioned above representing the number of students engaged in extra-curricular activities is such a data set. It represents only one variable—the number of students. Most graphs and data representations are of this type.

Bivariate data is a data set with two types of data or variables. It shows how one variable changes as another variable changes. This relationship between two variables is best captured by a scatterplot.

A scatterplot is a graph with \(xy\) coordinates and dots that represent the values. A scatterplot is useful in representing data such as, how literacy rate increases as employment rate decreases, or how final scores in a class improve with an increase in tutorial hours.

In the test, you may be shown a scatterplot and asked to predict a value based on graph. Learn about correlation, trends, line of best fit, interpolation, and extrapolation.

Probability

The concept of probability can be very challenging, but learning the very basic applications must be your focus.

Probability is the chance or likelihood that an event will happen. It is expressed as a number between \(\bf{0}\) and \(\bf{1}\). The closer the probability of an event is to 1, the higher is its likelihood of happening. A probability of \(\frac{1}{2}\) or \(0.5\) means that the chance of the event occurring equal to the chance of it not occurring. The basic formula for probability is:

Probability = number of ways an event can happen ÷ number of possible outcomes

Below are other relevant probability formulas:

The Probability Rule of Subtraction—This rule states that the probability of an event not occurring is equal to the probability of said event occurring subtracted from 1.

Probability that Event A will not occur = 1 – P(A)

The Probability Rule of Multiplication—The probability of two events occurring together is the product of their separate probabilities.

Probability that Event A occurs with Event B = P(A) \(\cdot\) P(B)

You may find additional resources on the Probability Rule of Addition.

Problem Solving

This involves using knowledge in the five previously listed concept areas to find solutions to problems. One problem may require you to utilize processes from more than one area, such as the need for both geometry and measurement skills. A solution may only be possible using skills from both areas. You may also be asked to make judgments and draw conclusions based on your knowledge of a variety of mathematical concepts.