Q: Which formula is used to calculate the percentile of a data value $$x in a $$dataset?

$$\text{Percentile of value } x = \frac{\text{number of values less than } x}{\text{total number of values}} \times 100$$

Q: In a dataset of 45 walking times, what is the locator $$l$$ for the 90th percentile?

$$l = \frac{90}{100} \times 45 = 40.5$$

Q: Using the IQR method, what is the upper fence for outliers in the electricity price data (Q3 = 14.3, IQR = 3.1)?

$$\text{Upper fence} = Q3 + 1.5 \times IQR = 14.3 + 1.5 \times 3.1 = 18.95$$

Q: A value of 30.3 is observed in the electricity price data. Is this value an extreme outlier, and why?

Yes, because it is greater than $$Q3 + 3 \times IQR = 14.3 + 9.3 = 23.6$$

Question 1

A sample of 45 students has a mean walking time of 14.69 minutes and a standard deviation of 1.31 minutes. What is the z-score for a student who walked a mile in 13 minutes?

Accepted Answer

$$z = \frac{13 - 14.69}{1.31} = -1.29$$

Question 2

Which of the following statements about z-scores is TRUE?

Accepted Answer

A z-score indicates how many standard deviations a value is from the mean.

Question 3

Given a normal distribution with mean $$\mu = 98.2$$ and standard deviation $$\sigma = 0.62$$, is a body temperature of $$100^\circF $$unusual?

Accepted Answer

Yes, because $$z = \frac{100 - 98.2}{0.62} = 2.90$$, which is greater than 2.

Question 4

Which formula is used to calculate the percentile of a data value $$x in a $$dataset?

Accepted Answer

$$	ext{Percentile of value } x = \frac{	ext{number of values less than } x}{	ext{total number of values}} 	imes 100$$

Question 5

In a dataset of 45 walking times, what is the locator $$l$$ for the 90th percentile?

Accepted Answer

$$l = \frac{90}{100} 	imes 45 = 40.5$$

Question 6

If the locator $$l$$ for a percentile is a whole number, what should you do to find the corresponding data value?

Accepted Answer

Take the average of the value at position $$l$$ and the next value.

Question 7

Which of the following is NOT a property of z-scores?

Accepted Answer

Z-scores are always positive.

Question 8

Given the five-number summary: Min = 9.7, Q1 = 11.2, Median = 12.35, Q3 = 14.3, Max = 30.3, what is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q3 - Q1 = 14.3 - 11.2 = 3.1$$

Question 9

Using the IQR method, what is the upper fence for outliers in the electricity price data (Q3 = 14.3, IQR = 3.1)?

Accepted Answer

$$	ext{Upper fence} = Q3 + 1.5 	imes IQR = 14.3 + 1.5 	imes 3.1 = 18.95$$

Question 10

A value of 30.3 is observed in the electricity price data. Is this value an extreme outlier, and why?

Accepted Answer

Yes, because it is greater than $$Q3 + 3 	imes IQR = 14.3 + 9.3 = 23.6$$

Question 11

Which of the following best describes the five-number summary?

Accepted Answer

Minimum, Q1, Median, Q3, Maximum

Question 12

If a data value has a z-score of -2.5, what can you conclude about this value?

Accepted Answer

It is significantly low and considered unusual.

Question 13

A dataset has 12, 13, 14, 15, 16, 17, and 18 as values. What is the percentile of the value 15?

Accepted Answer

$$	ext{Percentile} = \frac{3}{7} 	imes 100 \approx 43\%$$

Question 14

Which of the following is a reason why different calculators or software may give different results for percentiles and quartiles?

Accepted Answer

There is no universal agreement on the procedure for calculating percentiles and quartiles.

Question 15

Which measure of center and variation is considered the most stable?

Accepted Answer

Median and IQR

Question 16

In a modified boxplot, how are outliers represented?

Accepted Answer

By special symbols such as asterisks or points.

Question 17

If the 25th percentile (Q1) of a dataset is 11.2 and the 75th percentile (Q3) is 14.3, what is the range of the middle 50% of the data?

Accepted Answer

$$Q3 - Q1 = 14.3 - 11.2 = 3.1$$

Question 18

A student scored at the 90th percentile on a test. What does this mean?

Accepted Answer

The student scored higher than 90% of the other students.

Measures of Position: z-Scores, Percentiles, Quartiles, and Boxplots

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