Q: A test score is considered a potential outlier if it is more than $1.5 \times IQR$ above $Q_3$ or below $Q_1$. If $Q_1 = 60$, $Q_3 = 80$, and $IQR = 20$, what is the upper limit for potential outliers?

$Q_3 + 1.5 \times IQR = 80 + 1.5 \times 20 = 110$

Q: Suppose a data set has $Q_1 = 15$, $Q_3 = 35$, and $IQR = 20$. Which of the following values would be considered a potential outlier?

$Q_1 - 1.5 \times IQR = 15 - 30 = -15$. Any value less than $-15$ is a potential outlier. So, $-20$ is a potential outlier.

Question 1

Which of the following best describes the 80th percentile of a data set?

Accepted Answer

It is the value below which 80% of the observations fall.

Question 2

Given the data set: 10, 15, 20, 25, 30, 35, 40, what is the median?

Accepted Answer

$25$

Question 3

A data set has $Q_1 = 20$ and $Q_3 = 50$. What is the interquartile range (IQR)?

Accepted Answer

$IQR = Q_3 - Q_1 = 50 - 20 = 30$

Question 4

Which of the following is NOT part of the five-number summary used in a box plot?

Accepted Answer

The mean

Question 5

A test score is considered a potential outlier if it is more than $1.5 	imes IQR$ above $Q_3$ or below $Q_1$. If $Q_1 = 60$, $Q_3 = 80$, and $IQR = 20$, what is the upper limit for potential outliers?

Accepted Answer

$Q_3 + 1.5 	imes IQR = 80 + 1.5 	imes 20 = 110$

Question 6

Which step is performed first when calculating the $k$th percentile of a data set?

Accepted Answer

Arrange all data values in ascending order.

Question 7

A student scored 90 on a test where the mean was 80 and the standard deviation was 5. What is the student's $z$-score?

Accepted Answer

$z = \frac{90 - 80}{5} = 2$

Question 8

Suppose a data set has $Q_1 = 15$, $Q_3 = 35$, and $IQR = 20$. Which of the following values would be considered a potential outlier?

Accepted Answer

$Q_1 - 1.5 	imes IQR = 15 - 30 = -15$. Any value less than $-15$ is a potential outlier. So, $-20$ is a potential outlier.

Question 9

Which of the following statements about box plots is TRUE?

Accepted Answer

Box plots are useful for identifying potential outliers in a data set.

Question 10

A researcher wants to compare the exam performance of two students from different classes. Student A scored 75 on a test with mean 70 and standard deviation 2. Student B scored 88 on a test with mean 85 and standard deviation 4. Who performed better relative to their class?

Accepted Answer

Student A: $z = \frac{75-70}{2} = 2.5$; Student B: $z = \frac{88-85}{4} = 0.75$. Student A performed better relative to their class.

Question 11

If a data set has an odd number of observations, how is the first quartile ($Q_1$) determined?

Accepted Answer

It is the median of the lower half of the data, excluding the overall median.

Question 12

A data set has the following five-number summary: Min = 10, $Q_1 = 20$, Median = 30, $Q_3 = 40$, Max = 100. Which value is most likely a potential outlier?

Accepted Answer

$100$

Question 13

Which of the following best explains why $z$-scores are useful in comparing scores from different distributions?

Accepted Answer

They standardize scores by accounting for differences in means and standard deviations.

Question 14

If a value in a bell-shaped distribution has a $z$-score of $-3.5$, what does this indicate?

Accepted Answer

The value is a potential outlier, being more than 3 standard deviations below the mean.

Question 15

Given the data set: 5, 7, 12, 14, 15, 22, 25, 30, 36, 42, 53, what is the median?

Accepted Answer

$22$

Question 16

Which of the following is the correct formula for calculating a $z$-score?

Accepted Answer

$z = \frac{	ext{Observation} - 	ext{Mean}}{	ext{Standard deviation}}$

Question 17

A data set has 20 values. To find the 75th percentile, what is the first step after arranging the data in order?

Accepted Answer

Multiply $0.75 	imes 20 = 15$ to find the index.

Question 18

If the IQR of a data set is $12$, $Q_1 = 18$, and $Q_3 = 30$, what is the lower limit for potential outliers?

Accepted Answer

$Q_1 - 1.5 	imes IQR = 18 - 1.5 	imes 12 = 18 - 18 = 0$

Question 19

Which of the following is a limitation of box plots?

Accepted Answer

They do not clearly show the shape of the distribution, such as mounds or gaps.

(Lecture 4) Measures of Position: Percentiles, Quartiles, Box Plots, and Outlier Detection

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