Straighterline Introduction to Statistics
SECTION 1: DESCRIPTIVE STATISTICS (Questions 1-8)
Q1: A teacher calculates the mean, median, and mode for a set of test scores. The mean is 84, the
median is 78, and the mode is 74. What can be concluded about the shape of the distribution?
A. Symmetric
B. Skewed left (negatively skewed)
C. Skewed right (positively skewed) [CORRECT]
D. Cannot be determined
Correct Answer: C
Rationale: [CORRECT] When mean > median > mode, the distribution has a right (positive) skew.
Outliers on the high side pull the mean upward. A is incorrect because symmetric distributions have
mean ≈ median ≈ mode. B is incorrect because left-skewed distributions have mean < median. D is
incorrect because the relationship between these three measures reliably indicates skewness direction.
Straighterline Tip: Remember "Mean follows the tail" — the mean is pulled toward the longer tail. Mean
> median always indicates right skew.
Q2: For the data set: 12, 15, 18, 22, 25, 30, 100, which measure of center best represents the "typical"
value?
A. Mean = 31.7
B. Median = 22 [CORRECT]
C. Mode = 18
D. Range = 88
Correct Answer: B
Rationale: [CORRECT] The value 100 is an extreme outlier that pulls the mean up to 31.7, which is higher
than 6 of the 7 data points. The median (22) is resistant to outliers and better represents the center of
the majority of data. A is incorrect because the mean is distorted by the outlier. C is incorrect because
there is no mode (all values appear once). D is incorrect because range is a measure of spread, not
center.
,Straighterline Tip: On the exam, always check for outliers first. If outliers exist, the median is the better
measure of center.
Q3: The interquartile range (IQR) of a data set is 12, Q1 = 20, and Q3 = 32. Which value would be
identified as an outlier using the 1.5 × IQR rule?
A. 38
B. 2 [CORRECT]
C. 28
D. 35
Correct Answer: B
Rationale: *CORRECT+ Lower fence = Q1 − 1.5×IQR = 20 − 18 = 2. Upper fence = Q3 + 1.5×IQR = 32 + 18 =
50. The value 2 is exactly at the lower fence boundary (typically considered an outlier or extreme value).
A (38) and D (35) are within the upper fence. C (28) is between Q1 and Q3.
Straighterline Tip: Be careful — values exactly on the fence are often treated as outliers. Always
calculate both fences.
Q4: A distribution of exam scores has a mean of 75 and a standard deviation of 5. According to the
Empirical Rule, approximately what percentage of scores fall between 65 and 85?
A. 68%
B. 95% [CORRECT]
C. 99.7%
D. 100%
Correct Answer: B
Rationale: *CORRECT+ 65 = 75 − 2(5) and 85 = 75 + 2(5), so this is μ ± 2σ. The Empirical Rule states
approximately 95% of data falls within 2 standard deviations of the mean for bell-shaped distributions. A
is for μ ± 1σ (70 to 80). C is for μ ± 3σ (60 to 90). D is incorrect because no rule guarantees 100%.
Straighterline Tip: The Empirical Rule ONLY applies to approximately normal (bell-shaped) distributions.
Check the shape first!
Q5: Which graphical display is most appropriate for comparing the distribution of a quantitative variable
across multiple groups?
, A. Pie chart
B. Bar chart
C. Side-by-side boxplots [CORRECT]
D. Stem-and-leaf plot
Correct Answer: C
Rationale: [CORRECT] Side-by-side boxplots display the five-number summary (min, Q1, median, Q3,
max) for multiple groups on the same scale, making comparisons of center, spread, and outliers easy. A
is for parts of a whole (categorical). B is for categorical data frequencies. D shows distribution shape for
one group but is cluttered for multiple groups.
Straighterline Tip: Boxplots are the go-to for comparing distributions. Look for keywords "compare
groups" or "identify outliers."
Q6: A data set has a variance of 25. What is the standard deviation?
A. 5 [CORRECT]
B. 625
C. 12.5
D. 50
Correct Answer: A
Rationale: *CORRECT+ Standard deviation is the square root of variance: √25 = 5. B is 25² (squaring
instead of rooting). C is 25/2. D is 25×2.
Straighterline Tip: Standard deviation is in original units; variance is in squared units. Always take the
square root, never square.
Q7: In a boxplot, the distance between Q1 and Q3 represents:
A. The range
B. The interquartile range (IQR) [CORRECT]
C. The standard deviation
D. The median
Correct Answer: B
Rationale: *CORRECT+ IQR = Q3 − Q1, representing the middle 50% of data. A is max − min. C is
calculated from all data points, not just quartiles. D is the line inside the box.
SECTION 1: DESCRIPTIVE STATISTICS (Questions 1-8)
Q1: A teacher calculates the mean, median, and mode for a set of test scores. The mean is 84, the
median is 78, and the mode is 74. What can be concluded about the shape of the distribution?
A. Symmetric
B. Skewed left (negatively skewed)
C. Skewed right (positively skewed) [CORRECT]
D. Cannot be determined
Correct Answer: C
Rationale: [CORRECT] When mean > median > mode, the distribution has a right (positive) skew.
Outliers on the high side pull the mean upward. A is incorrect because symmetric distributions have
mean ≈ median ≈ mode. B is incorrect because left-skewed distributions have mean < median. D is
incorrect because the relationship between these three measures reliably indicates skewness direction.
Straighterline Tip: Remember "Mean follows the tail" — the mean is pulled toward the longer tail. Mean
> median always indicates right skew.
Q2: For the data set: 12, 15, 18, 22, 25, 30, 100, which measure of center best represents the "typical"
value?
A. Mean = 31.7
B. Median = 22 [CORRECT]
C. Mode = 18
D. Range = 88
Correct Answer: B
Rationale: [CORRECT] The value 100 is an extreme outlier that pulls the mean up to 31.7, which is higher
than 6 of the 7 data points. The median (22) is resistant to outliers and better represents the center of
the majority of data. A is incorrect because the mean is distorted by the outlier. C is incorrect because
there is no mode (all values appear once). D is incorrect because range is a measure of spread, not
center.
,Straighterline Tip: On the exam, always check for outliers first. If outliers exist, the median is the better
measure of center.
Q3: The interquartile range (IQR) of a data set is 12, Q1 = 20, and Q3 = 32. Which value would be
identified as an outlier using the 1.5 × IQR rule?
A. 38
B. 2 [CORRECT]
C. 28
D. 35
Correct Answer: B
Rationale: *CORRECT+ Lower fence = Q1 − 1.5×IQR = 20 − 18 = 2. Upper fence = Q3 + 1.5×IQR = 32 + 18 =
50. The value 2 is exactly at the lower fence boundary (typically considered an outlier or extreme value).
A (38) and D (35) are within the upper fence. C (28) is between Q1 and Q3.
Straighterline Tip: Be careful — values exactly on the fence are often treated as outliers. Always
calculate both fences.
Q4: A distribution of exam scores has a mean of 75 and a standard deviation of 5. According to the
Empirical Rule, approximately what percentage of scores fall between 65 and 85?
A. 68%
B. 95% [CORRECT]
C. 99.7%
D. 100%
Correct Answer: B
Rationale: *CORRECT+ 65 = 75 − 2(5) and 85 = 75 + 2(5), so this is μ ± 2σ. The Empirical Rule states
approximately 95% of data falls within 2 standard deviations of the mean for bell-shaped distributions. A
is for μ ± 1σ (70 to 80). C is for μ ± 3σ (60 to 90). D is incorrect because no rule guarantees 100%.
Straighterline Tip: The Empirical Rule ONLY applies to approximately normal (bell-shaped) distributions.
Check the shape first!
Q5: Which graphical display is most appropriate for comparing the distribution of a quantitative variable
across multiple groups?
, A. Pie chart
B. Bar chart
C. Side-by-side boxplots [CORRECT]
D. Stem-and-leaf plot
Correct Answer: C
Rationale: [CORRECT] Side-by-side boxplots display the five-number summary (min, Q1, median, Q3,
max) for multiple groups on the same scale, making comparisons of center, spread, and outliers easy. A
is for parts of a whole (categorical). B is for categorical data frequencies. D shows distribution shape for
one group but is cluttered for multiple groups.
Straighterline Tip: Boxplots are the go-to for comparing distributions. Look for keywords "compare
groups" or "identify outliers."
Q6: A data set has a variance of 25. What is the standard deviation?
A. 5 [CORRECT]
B. 625
C. 12.5
D. 50
Correct Answer: A
Rationale: *CORRECT+ Standard deviation is the square root of variance: √25 = 5. B is 25² (squaring
instead of rooting). C is 25/2. D is 25×2.
Straighterline Tip: Standard deviation is in original units; variance is in squared units. Always take the
square root, never square.
Q7: In a boxplot, the distance between Q1 and Q3 represents:
A. The range
B. The interquartile range (IQR) [CORRECT]
C. The standard deviation
D. The median
Correct Answer: B
Rationale: *CORRECT+ IQR = Q3 − Q1, representing the middle 50% of data. A is max − min. C is
calculated from all data points, not just quartiles. D is the line inside the box.
