SAT one- and two-variable data: review and practice

• Find the mean, median or range from a list, a frequency table or a dot plot.
• Compare the spread of data sets and predict what happens when a value is added, removed or changed.
• Read a scatterplot and use a line of best fit to make a prediction.
• Explain what the slope or starting value of a model means in context.
• Tell whether data grow by a fixed amount (linear) or by a fixed percent (exponential).

• Treating each row of a frequency table as one value. If the score 7 has a frequency of 3, the value 7 appears three times in the data and counts three times in the mean and median.
• Judging standard deviation by the size of the numbers. {100, 101, 102} has a smaller standard deviation than {1, 5, 9}. Spread is about how far values sit from the mean.
• Finding the median before sorting. The median of 9, 2, 7, 4, 5 is 5, not 7. Put the list in order first.
• Calling percent growth linear. Interest of 3% of the current balance each year is exponential, because the amount added keeps growing. A fixed $5 a year is linear.
• Swapping the variables. If a line of best fit predicts height (y) from age (x), plug the age in as x.
• Reading cause into a scatterplot. A strong trend shows two variables move together. It doesn't show one causes the other.

1. Question 1 · Easy

   A data set consists of the values 11, 14, 22, 25, 25, and 29. What is the mean of the data set shown?

   • A21
   • B23.5
   • C25
   • D25.2
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   A. The sum is 11 + 14 + 22 + 25 + 25 + 29 = 126, so the mean is 126/6 = 21. The value 23.5 is the median, 25 is the mode, and 25.2 results from dividing 126 by 5 instead of 6.

2. Question 2 · Easy

   Which of the following data sets has the smallest standard deviation?

   • A30, 30, 32, 34, 34
   • B31, 31, 32, 33, 33
   • C29, 31, 32, 33, 35
   • D28, 31, 32, 33, 36
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   B. All four means are 32, and the sums of squared deviations for choices D, B, A, and C are 34, 4, 16, and 20, respectively. Dividing each by five and taking the square root preserves this order, so choice B has the smallest standard deviation. Choice D has the widest spread, choice A repeats values farther from the mean, and choice C has more distant endpoints than choice B.

3. Question 3 · Easy

   A savings plan begins with $240. At the end of each year, interest equal to 1.8% of the balance at the start of that year is added to the plan, and no other transactions occur. Which of the following models best describes how the balance changes over time?

   • AAn increasing linear function
   • BA decreasing linear function
   • CA decreasing exponential function
   • DAn increasing exponential function
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   D. Each new balance is the preceding balance plus 0.018 times that balance. The yearly multiplier is 1.018, which is greater than 1, so the model is increasing and exponential. Choice A uses a fixed added amount instead of a repeated multiplier; Choice B uses fixed decreases instead of repeated growth; Choice C reverses the growth.

4. Question 4 · Medium

   Packages delivered = 2, Frequency (one-hour periods) = 2. Packages delivered = 4, Frequency (one-hour periods) = 5. Packages delivered = 6, Frequency (one-hour periods) = 4. Packages delivered = 8, Frequency (one-hour periods) = 3. Packages delivered = 10, Frequency (one-hour periods) = 2. Packages delivered = 26, Frequency (one-hour periods) = 1. The table shows the number of packages delivered during each of 17 one-hour periods. The entry of 26 packages was recorded in error and is removed. Which statement correctly compares the median of the remaining 16 entries with the median of the original 17 entries?

   • AThe median of the remaining entries is equal to the original median.
   • BThe median of the remaining entries is less than the original median.
   • CThe median of the remaining entries is greater than the original median.
   • DThere is not enough information to compare the medians.
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   A. The frequencies total 17, and the value 6 occupies positions 8 through 11, so the original ninth entry and the remaining eighth and ninth entries are all 6. Both medians are therefore 6; removing the maximum does not force a decrease, the relevant positions do not shift to higher values, and the frequency table provides all needed information.

5. Question 5 · Hard

   The number r is a real constant. Which of the following data sets has the largest standard deviation?

   • Ar, r, r + 1, r + 7, r + 8
   • Br, r + 2, r + 3, r + 3, r + 8
   • Cr, r + 1, r + 4, r + 7, r + 8
   • Dr, r, r + 6, r + 8, r + 8
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   D. Subtracting r gives means 16/5, 4, 22/5, and 16/5 for choices A, C, D, and B. Using the mean of the squares minus the square of the mean gives variances 314/25, 250/25, 336/25, and 174/25, so choice D has the largest standard deviation. Choice A has a smaller weighted spread despite its repeated minimum, choice C confuses a central mean with greater spread, and choice B shares the same endpoints but clusters three values near its mean.

6. Question 6 · Hard

   A data set contains eight positive integers, each less than 41. Seven of the integers are 19, 23, 24, 27, 28, 29, and 32, and their mean is 26. The mean of all eight integers is an integer greater than 26. What is the value of the largest integer in the data set?

   • A27
   • B32
   • C34
   • D40
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   C. The known sum is 7(26) = 182; for an integer mean m ≥ 27, the missing value is x = 8m − 182, so the candidates begin 34, 42, and 50 and increase by 8. Only 34 is less than 41, and it exceeds the listed maximum 32, so the largest value is 34, attained by adjoining 34 to the seven listed integers. The value 27 is the new mean, 32 ignores the missing value, and 40 gives a noninteger mean (182 + 40)/8 = 27.75.

For a short list, type mean([3, 8, 8, 12]) or median([3, 8, 8, 12]). For a frequency table, put the values in one list and the frequencies in another, like v = [1, 2, 3] and f = [4, 6, 2], then type total(v f)/total(f) for the mean. For paired data, enter the points in a table and type y₁ ~ mx₁ + b to get the line of best fit, then plug in the x-value the question gives.

Ashore's SAT course starts with a 15-minute check that finds the question types costing you the most points, then gives you lessons and practice for each.

• Linear Equations and Inequalities
• Linear Functions and Systems
• Ratios, Rates, Percentages, and Units
• Probability and Statistical Inference
• Equivalent Expressions and Quadratics
• Nonlinear Functions and Exponentials
• Lines, Angles, and Triangles
• Right Triangles and Trigonometry
• Circles, Area, and Volume
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