Read scatterplots and lines of best fit, and interpret what a model predicts compared with what was observed.
You read data plotted as points and a model drawn through them. Questions ask you to estimate a predicted value from a line or curve of best fit, to compare a predicted value with an actual data point, to interpret the slope or intercept of the model in context, or to say which kind of model, linear or exponential, fits the shape of the data. Some questions give the equation of the best-fit line and never show the plot at all.
Scatterplot and two-variable data questions usually appear about 1 to 3 times among the 44 math questions.
Written for this skill, with the answer explained. 116 more are in the app.
| Age group | Running | Swimming | Cycling |
|---|---|---|---|
| 18 to 29 | 42 | 35 | 23 |
| 30 to 44 | 38 | 41 | 29 |
| 45 to 59 | 25 | 33 | 34 |
The table shows the preferred exercise of the people surveyed in three age groups. How many of the people surveyed chose swimming?
A data set: 5, 5, 5, 5. What is the standard deviation?
Why: Standard deviation measures how far the values sit from their mean. The mean of 5, 5, 5, 5 is 5, and every value is exactly 5, so every deviation is 0. That makes the standard deviation 0. 5 is the mean and the common value, not a measure of spread. 25 is 5 squared, which no step here calls for. 1 would require the values to differ from each other, and they do not.
The scatterplot shows the average daily screen time and the average nightly sleep, both in hours, for 12 teenagers, along with a line of best fit. Which of the following is the best interpretation of the slope of the line of best fit in this context?
Why: The slope is how much the predicted value changes for each 1-hour increase in screen time. Read two points on the line, (0, 9.5) and (8, 5.5): sleep falls 4 hours while screen time rises 8, so the slope is -4/8 = -0.5. So each extra hour of screen time lowers predicted sleep by 0.5 hours. Saying "For each additional hour of daily screen time, the predicted nightly sleep decreases by 9.5 hours." uses 9.5, which is where the line meets x = 0, the intercept rather than the slope. Saying "For each additional hour of daily screen time, the predicted nightly sleep increases by 0.5 hours." has the right size and the wrong direction, and the line falls from left to right. Saying "For each additional hour of daily screen time, the predicted nightly sleep decreases by 2 hours." uses 2, which is the drop across a 4-hour increase in screen time, not across one hour.
A study of 10 green roofs recorded the planting depth, in centimeters, and the stormwater retained, in liters per square meter. The scatterplot shows the data along with a line of best fit. According to the line of best fit, by how many liters per square meter does the predicted retention increase for every 10-centimeter increase in planting depth?
Why: The line passes through (0, 15) and (30, 105), so its slope is (105 - 15) / 30 = 3 liters per square meter for each centimeter of depth. Over a 10-centimeter increase that is 3 x 10 = 30 liters per square meter. The slope itself, 3, is the rise for a single centimeter, and 6 is the rise for two.
It is the straight line that comes closest to the overall pattern of the plotted points. You use it to predict a y-value for a given x-value. Individual data points can sit above or below it, and the distance between a point and the line is the error for that observation.
The slope is the predicted change in the vertical quantity for each increase of one unit in the horizontal quantity. Always attach both units. If the horizontal axis is years and the vertical axis is dollars, the slope is dollars per year.
No. A scatterplot can show a strong association without establishing cause. Causation can only be claimed from a well-designed experiment with random assignment, not from observed data alone.
Questions on two-variable data and scatterplots alone, with explanations on every one, and the app tracks which ones you keep missing.
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