All of Problem Solving and Data Analysis on the SAT

Summary: Problem Solving and Data Analysis on the SAT

The video reviews the main skills in the SAT’s Problem Solving and Data Analysis domain. Its overall advice is to recognize what a question is testing, use graphs and units intelligently, and avoid unnecessary calculations by hand. The instructor also demonstrates using Desmos for lists, statistics, equations, and calculations.

1. One-Variable Data

One-variable data questions concern a single set of values. They may ask about its center, spread, or individual extremes, or present the data in a plot or table.

Measures of Center and Spread

• Mean: Add all the values and divide by the number of values. The instructor also describes the mean as the data’s balance point—a useful way to estimate it from a graph.
• Median: Put the values in numerical order and find the middle one.
  • With an odd number of values, the median is the single middle value.
  • With an even number, it is the mean of the two middle values.
• Minimum and maximum: The smallest and largest values.
• Range: Maximum minus minimum.
• Standard deviation: A measure of how spread out the data is, or how much values vary from the mean. More spread means a higher standard deviation; tighter clustering means a lower one.
  • SAT questions often ask which data set has the greater standard deviation rather than requiring an exact calculation.
• Mode: The most frequent value. A data set may have more than one mode, or no uniquely most-frequent value.

Outliers and Comparing Mean with Median

• The mean is sensitive to outliers: an unusually large or small value can pull it toward itself.
• The median is more resistant: its position depends on the ordered values, not on how far away an outlier is.
• In a skewed distribution, the mean is generally pulled toward the tail or unusual values, while the median stays closer to the middle of the main cluster.
• The video encourages estimating the mean and spread visually instead of entering every data point when only a comparison or rough estimate is needed.

How Adding or Multiplying Data Affects Statistics

If the same number is added to or subtracted from every data point:

• Measures of center—mean, median, and mode when applicable—shift by that amount.
• Range and standard deviation do not change, because the data’s relative spread stays the same.

If every data point is multiplied or divided by the same factor:

• Measures of center change accordingly.
• Measures of spread change as well. For a positive factor, range and standard deviation are scaled by that factor.

These transformations apply when the same operation is applied to all values, not just one.

Common Data Displays

• Dot plot: Each dot represents one observation. Repeated values appear as stacked dots.
• Bar graph: Bars show counts or amounts for separate categories.
• Frequency table: Lists each value or category and its frequency (count). Do not confuse the frequency with the value itself. The instructor recommends sketching the corresponding dot plot when needed to compare distributions.
• Histogram: Groups numerical data into intervals or bins. A bar tells you how many values fall in a range, not the exact values.
  • To find the greatest possible mean, assume every observation in each bin is at the highest value allowed by that interval.
  • To find the least possible mean, use the lowest allowed value in each bin.
  • Check how the question defines the interval boundaries.
• Box plot (box-and-whisker plot): The line inside the box marks the median, not the mean. In the examples discussed, the endpoints of the whiskers indicate the minimum and maximum.

2. Two-Variable Data

Two-variable data questions involve paired values, commonly shown as points on a scatterplot with an (x)-axis and a (y)-axis.

• Dots are observed data points—the actual values in the sample.
• A line of best fit models the overall trend and can be used to estimate or predict a value.
• When a question asks for an actual value, read the relevant dot. When it asks for a predicted value, use the line.
• A question may ask for the difference between an observed value and the value predicted by the line.

For graphs showing change over time:

• A variable is increasing when its plotted value rises and decreasing when it falls.
• Rate of change refers to slope. A constant rate means the slope is constant over the stated interval.
• When comparing “average rates,” compare slopes over the relevant intervals. The magnitude of a rate is its size, regardless of whether the slope is positive or negative.
• Read answer choices carefully: if a statement says two conditions must both hold, verify both rather than stopping after confirming one.

3. Unit Conversions

The central method is to set up conversion factors so unwanted units cancel.

1. Identify the units given and the unit requested.
2. Use the conversion relationships provided in the question or reference sheet.
3. Write each relationship as a fraction in the orientation that cancels the unit you have.
4. Multiply the conversion factors, checking that only the desired unit remains.
5. Calculate and round only when the question asks you to.

Important cases:

• Squared or cubed units: If a length conversion is (1) inch (= 2.54) centimeters, a volume conversion requires cubing the conversion factor: (1\text{ in}^3 = (2.54)^3\text{ cm}^3). Similarly, square the factor for area. The exponent applies to the conversion factor as well as the unit.
• Density and volume: Track the units in the given relationship. If the calculation produces cubic meters but the question asks for meters, convert the volume first and then take a cube root.
• Radians and degrees: The SAT reference sheet gives (360^\circ = 2\pi) radians. Use this as a conversion factor rather than relying on memorization.
• Unit-checking equations: In word problems, comparing units on both sides of an equation can help rule out answer choices that combine incompatible quantities.

4. Percentages

The instructor emphasizes distinguishing a direct percentage from a percent increase or decrease.

• A direct question such as “What is 20% of (x)?” asks for (0.20x).

• An increase of (p\%) gives a final value of: [ \text{original}\times\left(1+\frac{p}{100}\right). ]

• A decrease of (p\%) gives a final value of: [ \text{original}\times\left(1-\frac{p}{100}\right). ]

• In reverse problems, keep the percentage attached to the value it is applied to. For example, a sales tax is applied to the pre-tax price; a later tax after a discount is applied to the discounted price.

• For multiple successive percentage changes, model each change in sequence, with each percentage applied to the correct current or stated amount.
• If the problem gives only percentage relationships and no starting quantity, the instructor demonstrates choosing a convenient starting value, often 100, to compare the resulting amounts.
• For problems mixing two solutions, the instructor presents a weighted-average setup: concentration times amount for each solution, added together, equals the final concentration times the total amount.

Desmos can calculate expressions involving percentages and solve equations, but the setup still needs to reflect what each percentage applies to.

5. Probability

The video’s main framework is:

[ \text{Probability}=\frac{\text{target outcomes}}{\text{total possible outcomes}}. ]

A reliable procedure is:

1. Identify the total set of outcomes being considered.
2. Identify the target outcomes within that total.
3. Divide target by total.
4. Simplify if needed.

For conditional questions containing phrases such as “given that,” the condition limits the total. For example, if the question asks for the probability of selecting a particular type given that the item is hardcover, the denominator includes only hardcover items. The target must also come from that restricted group.

For a table, use only the cells included in the specified total and target; do not count entries outside the group the question says to select from.

6. Sampling and Generalization

A sample is used to estimate a larger population when surveying or measuring everyone would be impractical.

• To avoid biased results, a sample should be randomly selected from the population of interest.
• A sample taken from a particular subgroup may not represent the broader population. For example, surveying families at a playground may overrepresent households with children.
• Results can be generalized only to the population from which the sample was drawn, subject to the sample being appropriately selected. If a random sample includes only fourth-grade students at one school, the results do not automatically apply to all students at that school, all fourth graders in a county, or students elsewhere.
• In numerical estimation questions, the sample proportion or rate is used to estimate the corresponding count in the larger population.

7. Margin of Error

• Margin of error describes uncertainty around a sample-based estimate.
• If an estimate is (23\%) with a margin of error of (4) percentage points, the plausible interval described is (19\%) to (27\%).
• A larger sample generally produces a smaller margin of error, because more data tends to make an estimate more reliable.
• A smaller sample generally produces a larger margin of error.

8. Cause and Effect

The video distinguishes evidence of an association from evidence of a causal relationship.

• To establish cause and effect in an experiment, participants or subjects should be randomly assigned to the treatment and control groups.
• Random assignment helps reduce systematic differences between groups that could otherwise explain the results.

9. Study Advice

The instructor recommends reviewing any section that did not fully make sense, then practicing Problem Solving and Data Analysis questions—especially difficult ones—so that identifying the right method becomes reliable under test conditions. The recurring practical themes are to read carefully, track units and conditions, use visual estimates when appropriate, and verify answer choices before selecting one.

Speakers and Sources Featured

• James Lu — the sole speaking instructor, presenting SAT lessons and worked examples.
• Desmos — shown as a calculator and graphing tool; it is a tool featured in the demonstrations, not a speaker.
• SAT materials — SAT-style questions, graphs, and the SAT reference sheet are discussed or shown as instructional sources.