Rucete ✏ AP Statistics In a Nutshell
1. Exploring One-Variable Data — Practice Questions
This chapter introduces how to display, describe, and summarize one-variable data using graphs, numerical measures, and clear statistical language.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following variables is categorical?
(A) Height of students in centimeters
(B) Number of siblings a student has
(C) Favorite school subject
(D) Time spent studying last night
(E) Weight of a backpack in kilograms
Answer
(C) — Favorite school subject is a label or category, not a numerical measurement.
2. Which of the following variables is quantitative?
(A) Type of pet owned
(B) Brand of laptop used
(C) Blood type
(D) Number of text messages sent today
(E) Eye color
Answer
(D) — A count is numerical, so it is a quantitative variable.
3. A relative frequency table differs from a frequency table because it shows
(A) categories in alphabetical order
(B) only the largest category
(C) proportions or percents rather than just counts
(D) the same information as a pie chart only
(E) only quantitative data
Answer
(C) — Relative frequency gives the proportion or percent in each category instead of just the count.
4. Which graph is most appropriate for displaying data on preferred ice cream flavor among students?
(A) Histogram
(B) Bar graph
(C) Boxplot
(D) Stemplot
(E) Scatterplot
Answer
(B) — Preferred flavor is categorical, and bar graphs are appropriate for categorical data.
5. A pie chart would be inappropriate when
(A) categories represent parts of a whole
(B) percentages add to 100%
(C) the variable is categorical
(D) the values do not represent parts of one whole
(E) there are only three categories
Answer
(D) — Pie charts require categories that partition a single whole.
6. Which statement about a bar graph for categorical data is true?
(A) The bars must touch each other
(B) The order of categories always determines the shape
(C) We usually describe the graph using skewness
(D) The categories do not have to follow a numerical order
(E) It is used only for continuous data
Answer
(D) — Categories in a bar graph usually do not need a numerical order unless there is a natural ordering.
7. Which of the following is a discrete quantitative variable?
(A) Weight of apples in a basket
(B) Time needed to finish a race
(C) Number of books on a shelf
(D) Temperature outside at noon
(E) Height of a tree
Answer
(C) — Number of books is counted in whole numbers, so it is discrete.
8. Which of the following is a continuous quantitative variable?
(A) Number of children in a family
(B) Number of goals scored in a game
(C) Number of AP classes taken
(D) Body temperature
(E) Number of cars in a parking lot
Answer
(D) — Temperature is measured and can take many values on a continuum.
9. A histogram is most appropriate for displaying
(A) favorite music genre
(B) political party preference
(C) lengths of fish in a lake
(D) blood type categories
(E) brand names of shoes
Answer
(C) — Length is quantitative, and histograms are used for quantitative data.
10. In a histogram, the area of a bar represents
(A) the variable name
(B) the exact data values
(C) the relative frequency for that interval
(D) the sample size only
(E) the median
Answer
(C) — Relative areas in a histogram correspond to relative frequencies.
11. Which display preserves individual data values most clearly?
(A) Histogram
(B) Stemplot
(C) Boxplot
(D) Pie chart
(E) Side-by-side bar graph
Answer
(B) — A stemplot keeps the individual values visible through stems and leaves.
12. Every stemplot should include
(A) a trend line
(B) a cumulative axis
(C) a key
(D) equal bar widths
(E) percentages only
Answer
(C) — A key explains how to read the stems and leaves.
13. If a stemplot skips a stem with no data values, the main problem is that
(A) the graph becomes too wide
(B) it may hide a gap in the data
(C) the median changes
(D) the mean becomes incorrect
(E) outliers disappear automatically
Answer
(B) — Missing stems can hide important gaps in the distribution.
14. A cumulative relative frequency plot is especially useful for estimating
(A) the mode only
(B) exact individual data values
(C) proportions below a given value
(D) the number of categories
(E) the color of bars
Answer
(C) — An ogive helps estimate the proportion at or below a value.
15. If a cumulative relative frequency plot shows 0.80 at x = 18, this means
(A) 80% of the data are above 18
(B) 20% of the data are at or below 18
(C) 80% of the data are at or below 18
(D) the mean is 18
(E) the standard deviation is 0.80
Answer
(C) — Cumulative relative frequency gives the proportion at or below the given value.
16. A distribution that stretches far to the right is described as
(A) symmetric
(B) skewed left
(C) uniform
(D) skewed right
(E) bimodal
Answer
(D) — A long tail on the high-value side indicates right skew.
17. A distribution with two clear peaks is called
(A) uniform
(B) bell-shaped
(C) bimodal
(D) symmetric
(E) left-skewed
Answer
(C) — Two distinct peaks indicate a bimodal distribution.
18. When describing a distribution, SOCS stands for
(A) shape, outliers, center, spread
(B) sample, outlier, category, standard deviation
(C) skewness, ogive, center, sample
(D) shape, order, cluster, symmetry
(E) spread, outliers, center, skewness
Answer
(A) — SOCS is a standard checklist for describing quantitative distributions.
19. Which of the following is an unusual feature in a distribution?
(A) Units
(B) Sample size
(C) Gap
(D) Mean only
(E) Axis label
Answer
(C) — Gaps are unusual features because they show intervals with no observations.
20. The median of the ordered data set 3, 5, 8, 12, 15 is
(A) 5
(B) 8
(C) 9
(D) 12
(E) 15
Answer
(B) — With five values, the middle value is 8.
21. The mean of 4, 6, and 8 is
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9
Answer
(B) — The mean is (4 + 6 + 8) / 3 = 6.
22. Which measure of center is more resistant to extreme values?
(A) Mean
(B) Median
(C) Range
(D) Standard deviation
(E) Variance
Answer
(B) — The median is resistant because it depends on order, not the size of extreme values.
23. The range of the data set 2, 5, 7, 11, 14 is
(A) 12
(B) 11
(C) 9
(D) 7
(E) 5
Answer
(A) — Range = maximum − minimum = 14 − 2 = 12.
24. The interquartile range is defined as
(A) maximum − minimum
(B) mean − median
(C) Q1 − Q3
(D) Q3 − Q1
(E) standard deviation squared
Answer
(D) — The IQR measures the spread of the middle 50% of the data.
25. According to the 1.5(IQR) rule, a high outlier is any value greater than
(A) Q3 + 1.5(IQR)
(B) Q1 + 1.5(IQR)
(C) mean + 1.5(SD)
(D) median + IQR
(E) maximum + IQR
Answer
(A) — This is the standard upper fence used to identify outliers.
26. A histogram shows scores from 50 to 100 and has bars of equal width. The tallest bar is from 70 to 80. Which statement is best supported?
(A) The exact median is 75
(B) The interval 70 to 80 has the greatest frequency
(C) The mean must be 75
(D) All scores are above 70
(E) The distribution is uniform
Answer
(B) — The tallest bar indicates the highest frequency for that interval.
27. A cumulative relative frequency plot rises slowly at first and then rises steeply for larger values. The underlying distribution is most likely
(A) skewed right
(B) skewed left
(C) uniform
(D) perfectly symmetric
(E) bimodal with equal peaks
Answer
(B) — Slow early rise and steeper later rise is typical of a left-skewed distribution.
28. Two classes have the same mean test score. Class X has a much larger standard deviation than Class Y. This means
(A) Class X must have a higher median
(B) Class Y must have more students
(C) scores in Class X are more spread out from the mean
(D) scores in Class X are more symmetric
(E) Class Y must have no outliers
Answer
(C) — A larger standard deviation means greater variability around the mean.
29. If 7 is added to every value in a data set, which quantity stays the same?
(A) Mean
(B) Median
(C) Range
(D) Maximum
(E) Minimum
Answer
(C) — Adding a constant shifts all values equally, so the spread measured by range stays unchanged.
30. If every value in a data set is multiplied by 3, the standard deviation becomes
(A) one-third as large
(B) unchanged
(C) 3 times as large
(D) 9 times as large
(E) increased by 3 units only
Answer
(C) — Multiplying all data values by a constant multiplies the standard deviation by that constant.
31. A data set has mean 40 and standard deviation 5. The z-score of 50 is
(A) -2
(B) -1
(C) 1
(D) 2
(E) 10
Answer
(D) — z = (50 − 40) / 5 = 2.
32. A value with z-score -1.5 is
(A) 1.5 standard deviations above the mean
(B) 1.5 units below the mean
(C) 1.5 standard deviations below the mean
(D) at the mean
(E) an automatic outlier
Answer
(C) — A negative z-score indicates the value is below the mean by that many standard deviations.
33. Which statement about percentile rank is correct?
(A) It tells the exact distance from the mean
(B) It gives the percent of data at or below a value
(C) It is always equal to the z-score
(D) It can be used only for symmetric distributions
(E) It is the same as the standard deviation
Answer
(B) — Percentile rank describes the percentage of observations at or below a particular value.
34. Which situation best calls for comparing relative frequencies rather than raw frequencies?
(A) Comparing two classes of the same size
(B) Comparing one class to itself
(C) Comparing two schools with different enrollments
(D) Finding the exact median in one data set
(E) Finding the exact maximum in one data set
Answer
(C) — Relative frequencies make fair comparisons when group sizes differ.
35. Which statement is most accurate when describing a real-world distribution?
(A) It is always perfectly symmetric or perfectly skewed
(B) It should be described only by the mean
(C) It is often approximately rather than perfectly shaped in a common way
(D) It cannot have both clusters and outliers
(E) It should never mention context
Answer
(C) — Real data are usually described as roughly or approximately symmetric, bell-shaped, or uniform.
36. A data set in order is 4, 7, 9, 10, 12, 15, 18. Find the median.
Answer
10 — With seven ordered values, the fourth value is the median.
37. Find the mean of the data set 6, 8, 10, 12, 14.
Answer
10 — The sum is 50, and 50 ÷ 5 = 10.
38. Find the range of the data set 13, 17, 20, 20, 24, 31.
Answer
18 — Range = 31 − 13 = 18.
39. For the ordered data set 2, 4, 6, 8, 10, 12, 14, 16, find Q1.
Answer
5 — The lower half is 2, 4, 6, 8, so Q1 is the median of that half: (4 + 6) / 2 = 5.
40. For the ordered data set 2, 4, 6, 8, 10, 12, 14, 16, find Q3.
Answer
13 — The upper half is 10, 12, 14, 16, so Q3 = (12 + 14) / 2 = 13.
41. Using your answers from Questions 39 and 40, find the interquartile range.
Answer
8 — IQR = Q3 − Q1 = 13 − 5 = 8.
42. A data set has Q1 = 12 and Q3 = 20. Find the IQR.
Answer
8 — The interquartile range is 20 − 12 = 8.
43. A data set has Q1 = 12 and Q3 = 20. Find the upper fence using the 1.5(IQR) rule.
Answer
32 — IQR = 8, so upper fence = 20 + 1.5(8) = 20 + 12 = 32.
44. A data set has Q1 = 12 and Q3 = 20. Find the lower fence using the 1.5(IQR) rule.
Answer
0 — IQR = 8, so lower fence = 12 − 1.5(8) = 12 − 12 = 0.
45. The mean is 70 and the standard deviation is 8. Find the z-score of 86.
Answer
2 — z = (86 − 70) / 8 = 16 / 8 = 2.
46. The mean is 70 and the standard deviation is 8. Find the raw score with z = -1.
Answer
62 — x = mean + z(SD) = 70 + (-1)(8) = 62.
47. A cumulative relative frequency graph shows a value of 0.25 at x = 14. What percent of the observations are at or below 14?
Answer
25% — A cumulative relative frequency of 0.25 means 25% are at or below that value.
48. In a class of 40 students, 30 students scored at or below 82 on a test. What is the percentile rank of a score of 82?
Answer
75th percentile — 30 ÷ 40 = 0.75, so 82 is at the 75th percentile.
49. A student's score is 1.8 standard deviations above the mean. State the student's z-score.
Answer
1.8 — A z-score is the number of standard deviations from the mean, so the z-score is 1.8.
50. The values in a data set are all increased by 5. State what happens to the mean.
Answer
The mean increases by 5. — Adding the same constant to every value shifts the mean by that same amount.