On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I look at two sets of numbers together, I ask: when one goes up, what does the other do? I think about a phone battery. The longer I use the phone, the lower the battery goes. If I wrote down the hour and the battery percent in a table, the numbers would not be perfect, because a game uses more power than a message. But I could still draw a line through the middle of them and say, about 10% is lost every hour. With that line I can guess what happens at hour 5, even if I never measured it. This is what the SAT asks in Two-Variable Data. You read a pattern between two things, use a line or a curve as a model, and use it carefully to predict or explain.
These questions can look scary because of the graphs and tables, but most of them only ask you to read one number and use one rule. You can do this.
What is inside this part
Describing the direction and strength of the pattern between two variables.
positive or negative association · outliers · strong or weak pattern
Using or interpreting a line of best fit, including its slope, intercept and predictions.
meaning of slope · predicting y · finding the equation from points
Comparing actual data with the values predicted by a model.
residual = actual − predicted · largest residual · above or below the line
Using an exponential or quadratic model to describe data that does not follow a straight line.
growth or decay factor · equal ratios · curve with a highest point
How to know a question is from this part
The question gives pairs of numbers, a scatterplot, or a model equation with two variables and asks you to describe, predict or interpret.
- The words scatterplot, line of best fit, model, trend or predicted.
- A table of x and y values, often followed by 'which equation best models the data'.
- 'Which of the following is the best interpretation of the slope / y-intercept?'
- The word residual, or 'difference between the actual and predicted value'.
- A model with a power such as 24000(0.85)ᵗ or 500(1.2)ᵗ.
The lesson
1. What two-variable data is
Two-variable data means each item has two numbers. For example, for each student: hours studied and test score. We call one number x (the input, or explanatory variable) and the other y (the output, or response variable).
We often show the pairs in a table or as dots on a scatterplot. Each dot is one pair (x, y). The x-value says how far to the right, and the y-value says how far up.
2. Reading a scatterplot: association
Look at the general pattern of the dots. If the dots go up from left to right, there is a positive association: when x is bigger, y tends to be bigger. If the dots go down, it is a negative association. If there is no clear pattern, there is little or no association.
The closer the dots are to a straight line, the stronger the association. Points far from the pattern are called outliers.
Association is not the same as cause. Two things can move together without one causing the other.
3. The line of best fit
A line of best fit is a straight line that goes through the middle of the dots, with about as many dots above as below. It is a model: a simple rule that describes the pattern. Its equation is y = mx + b.
The slope m tells you how much y is predicted to change when x increases by 1. The y-intercept b is the predicted y when x = 0. Always say these in the words of the story.
A line of best fit for plant height is y = 4x + 12, where y is height in cm and x is weeks. What does 4 mean?
Show the steps
- The slope is 4.
- So each extra week, the predicted height goes up by 4 cm.
- The 12 is the predicted height at week 0, the start.
The predicted height increases by 4 cm each week.
4. Using a model to predict
To predict, put your x-value in the equation and compute y. If the x-value is inside the range of the data, this is called interpolation, and it is usually reliable. If the x-value is far outside the data, it is extrapolation, and the model may be wrong.
If a question gives you two points on the line instead of the equation, first find the slope: m = (y2 − y1) ÷ (x2 − x1). Then use y = mx + b, or use the point-slope idea: y − y1 = m(x − x1).
A model for test score is y = 3x + 40, where x is hours studied. What is the predicted score for 5 hours?
Show the steps
- Put x = 5: y = 3(5) + 40.
- y = 15 + 40 = 55.
55
5. Residuals: how far the model is wrong
A residual is actual value minus predicted value. A positive residual means the dot is above the line. A negative residual means the dot is below the line.
To find it, put the dot's x into the model to get the predicted y, then subtract: residual = actual y − predicted y. The dot with the largest distance from the line has the largest residual (ignoring the sign).
The model is y = 2x + 10. A real data point is (6, 25). What is the residual?
Show the steps
- Predicted y = 2(6) + 10 = 22.
- Actual y = 25.
- Residual = 25 − 22 = 3.
3 (the point is above the line)
6. Choosing the right kind of model
Linear model: y = mx + b. The y-value goes up or down by the SAME AMOUNT for each step in x. Look for equal differences.
Exponential model: y = a·bˣ. The y-value is multiplied by the SAME FACTOR for each step in x. Look for equal ratios. If b > 1 it is growth, and if 0 < b < 1 it is decay. A decay factor 0.85 means a 15% decrease each step.
Quadratic model: y = ax² + bx + c. The graph is a curve with a highest or lowest point (a parabola). Use it when the data rises then falls, or falls then rises.
A table gives f(0) = 200 and f(2) = 288. Assume an exponential model. What is f(3)?
Show the steps
- Over 2 steps the value is multiplied by 288 ÷ 200 = 1.44.
- So each step multiplies by √1.44 = 1.2.
- f(3) = 200 × 1.2³ = 200 × 1.728 = 345.6.
345.6
7. Finding the best equation from a table
If the SAT gives a few points and asks for the best model, test the signs and the size first. Does y go down as x goes up? Then the slope must be negative. Is y about 10 when x is 0 or 1? Then the intercept must be near 10.
You can also test each option by putting two x-values in the equation and checking that the answers are close to the table. Wrong options usually fail on the sign or on the size.
8. Units and meaning in context
Read the units. If x is days and you want the change per week, multiply the slope by 7.
The SAT likes sentences such as 'the best interpretation of the slope' or 'the predicted value of y when x = 30'. Always answer in the words of the story, with the unit, and say 'predicted' or 'estimated', never 'exactly'.
9. Common traps
Trap 1: mixing up slope and intercept; the intercept is where x = 0.
Trap 2: treating an exponential table as linear (equal differences are linear; equal ratios are exponential).
Trap 3: using the wrong sign for a residual; it is actual minus predicted.
Trap 4: forgetting that x and y may have different units from what the question asks.
Trap 5: trusting a prediction far outside the data.
See it
- 1Read what x and y mean, with units.
- 2Identify the model: line, exponential or curve.
- 3Find the rule: slope from two points, or a growth factor from a ratio.
- 4Put the x-value in and compute y.
- 5Say the answer with its unit and the word 'predicted'.
- Equal DIFFERENCES: 5, 8, 11, 14
- y = mx + b
- Slope m is the change per step
- Graph is a straight line
- Equal RATIOS: 5, 10, 20, 40
- y = a·bˣ
- b is the factor per step
- Graph curves up (b > 1) or down (0 < b < 1)
The dots go up from left to right (positive association). The line goes through the middle of them.
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