Mathematics Study Guide for the ParaPro Assessment

Data Analysis

As humans, we are naturally curious about the world we live in. We developed numbers and shapes to describe things we see and feel. We started to notice patterns based on those numbers and shapes. From those patterns (formulas, equations, etc) we can draw conclusions and make predictions. This is the essence of data analysis: to take information and draw meaning from it.

Reading Graphs and Tables

An easy way to collect data is using a table with columns and rows. But patterns and relationships really appear when you graph the data. There are two main types of graphs: a line graph and a bar graph.

Here’s a table showing how many points 4 different teams scored in 2 periods of a basketball game.

Below, the same data is represented in a line graph. Line graphs are usually good at showing trends. But there is no reason to look for a trend for this particular data. For instance, one might think: as the team number gets higher (Team 2, Team 3, Team 4) then the scoring in the second period tends to increase. That doesn’t really make sense as the teams were probably randomly numbered.

A better way to represent the data would be by using the bar graph below. Bar graphs are great at showing differences. You can easily scan and see that Team 1 scored significantly more points in period 2 than it did in period 1, for instance.

For both graphs, it’s important to have 2 major things: labels, and a key. For both graphs, the label on the vertical axis is “Points Scored” and the team numbers are the labels on the bottom axis. Numbers and values should be evenly spaced out. The key is the part in the upper right corner telling which color is used for each period.

Reading Pie Charts

Pie charts are helpful for noticing parts of a whole or percentages. Here’s a table showing how many students in an English class have certain hair colors.

We can represent this type of data best using the pie chart below. The entire circle represents the total number of students. Each pie-shaped section of the circle represents the group of students with a particular color of hair and takes up a proportional amount of space in the circle according to the percentage.

Interpreting Trends

As mentioned above, line graphs are best at interpreting trends. To interpret a trend in the data, look for an upward or downward slope to the right. Let’s look at the following example about the distance between a toy car and the wall (position 0).

You can clearly see that the graph “goes uphill” to the right. This indicates a positive correlation (upward trend). If it trends downhill, we say the data has a negative correlation. If it jumps around everywhere or stays flat, we say the data has, no correlation.

Because there is a correlation, we can make predictions. Where will the car be at 6 seconds from the start? To make predictions, you should look for the rate of change: how much distance changes per second of change. In this case, for each second that goes up, the distance goes up anywhere from 11 to 15 meters.

Let’s take the middle and say the rate of change is 13 meters per second. To make the prediction you can start at the right edge of the graph (position = 60 meters) and add 13 meters every second you move to the right. Since we have to move 2 more seconds, the position will be 60 + 13 +13 = 86 meters (or somewhere around there).

Creating Charts, Graphs, and Tables

To create line or bar graphs from tables, the first thing you need to do is decide which variable should go on which axis. Usually, the first column should go on the \(x\)-axis. The next step should be to look at the range of the numbers.

Look at the table from the previous example: The range of time is from \(0\) to \(4\) seconds. The range of position is from \(10\) to \(60\) meters. On the \(x\)-axis, a mark should be made for each second, starting at \(0\) from the left side. On the \(y\)-axis, you should also start at \(0\) (even though it’s not part of the range, this is common practice). It would be hard to make \(60\) dashes for each position, so you should make a choice. Maybe you make a dash every \(10\), or every \(15\), or even every \(20\).

Now, you plot the data points (as ordered pairs). The first point is \((0, 10)\). So start at the origin, move right \(0\), up \(10\), and place a point. For \((1, 21)\), start at the origin again, move right one, up \(21\) (just over \(20\)), and place a point. Continue. Connect the dots for a line graph or use the dots as the tops for columns in a bar graph.

For a pie chart, you’ll need to find percentages. First, find the total number of students (for our example): \(110\) students.
Now, find the percentage by dividing the number of students of a hair color by the total number of students and multiply by \(100\).
Blonde: \(24 \div 110 \cdot 100 = 21.8\%\)
Brown: \(36 \div 110 \cdot 100 = 23.7\%\) etc.

For a hand-drawn pie chart, draw a circle and lightly split it into \(4\) equal parts, each \(25\%\) of the circle. The blonde section should take up a little less than one of those parts, so draw a slightly smaller pie. Let this be the blonde section.

The brown hair section should take about \(\frac{1}{3}\) of the circle (same with the black hair). So split the circle into \(3\) parts and let two of the parts be brown and black hair. The remaining section will be red hair.

Mean, Median, and Mode

When analyzing data, it’s often helpful to use one number to represent the whole set. Usually, we choose a centrally-located number. There are three main ways to choose this number: mean, median, and mode. Let’s look at the following set of numbers and find each of the three values: \(\{2, 3, 4, 6, 7, 8, 8, 9, 10\}\).

To find the mean (or average), you should add the values (\(57\)) and divide by how many values there are (\(9\)):

\(57 \div 9 = 6.33\) or \(6\)

To find the median (or middle), line the numbers up from least to greatest. Starting from the outside, take one step toward the center. Continue until there is one number left: \(7\).
Note: If there are \(2\) numbers in the middle, take the average of the two. Example: If \(6\) and \(5\) are in the middle of a set, the median is \(\frac{(6+5)}{2} = 5.5\)

To find the mode, just find the value that occurs most often. In this case, \(8\) appears twice and is, therefore, the mode. Note, sometimes you’ll have more than one mode, and sometimes you’ll have no mode.