Percentiles & Quartiles: Videos & Practice Problems

The goals scored by the $50$ winning teams in a football tournament final are represented in the ogive below. What score represents the $60$^th percentile? What interpretation could be drawn from this?

A scientist measures the temperature of a chemical reaction and finds it is at the third quartile for all recorded reactions. What does this indicate about the temperature compared to the other reactions?

Determine whether the statement is true or false.

The first quartile represents the value below which $50\%$ of the data fall.

Consider the following set of test scores: $68$, $72$, $65$, $70$, $69$, $74$, $70$, $71$, $67$, $73$, $80$, $72$, $70$, $75$, $77$. Find the first quartile $Q_1$, the median $Q_2$, and the third quartile $Q_3$.

Consider the following set of test scores: $68,72,65,70,69,74,70,71,67,73,80,72,70,75,77$. Identify the outliers, if any.

The following are the ages of $24$ participants in a workshop: $18$, $19$, $20$, $21$, $22$, $23$, $24$, $25$, $26$, $27$, $28$, $29$, $30$, $31$, $32$, $33$, $34$, $35$, $36$, $37$, $38$, $39$, $40$, $41$. Which ages are above the $75$^th percentile?

The following data set lists the number of hours spent studying by a group of college students for a final exam: $12$, $8$, $15$, $10$, $14$, $9$, $11$, $13$, $7$, $10$, $12$, $8$, $14$, $7$, $13$, $10$, $12$, $9$, $11$, $13$. What is the $50$th percentile value for hours studied? How would you interpret this result?

Given the data set $12$, $15$, $10$, $18$, $14$, $20$, $13$, $17$, $11$, and $16$, find the midquartile.

A chemistry exam score of $60$ is at the $30$^th percentile. What percentage of students scored higher than $60$?

The ogive shown below displays the percentiles of math scores for a sample of students. Which score marks the $50$th percentile according to the ogive? What does this percentile indicate?

The number of defective products recorded daily over $16$ days is as follows: $\text{5, 7, 6, 8, 10, 6, 9, 50, 7, 8, 9, 7, 6, 8, 7, 7}$. Identify any outliers among the data.

The measure defined by $Q_3-Q_1$ is best described as which of the following?

A company tracks the number of daily page views (in thousands) for a product over $12$ days and flags unusual spikes using the upper fence.

Data: $60,62,65,68,70,72,73,75,76,80,82,140$

What is the upper fence (cutoff) above which a day should be flagged?

An online retailer tracks how long customers spend in live chat (in minutes) to resolve an issue. For a recent month, the summary for a sample of chats is: $Q_1=18$, $Q_2=24$, $Q_3=35$. Determine the interquartile range ($IQR$).

A clinic records the time (in minutes) patients waited. The quartiles of the data are $Q_1=7$, $Q_2=12$, and $Q_3=20$. What is the interquartile range ($IQR$)?

On a professional licensing exam, the scores are approximately normally distributed with a mean of $\mu =630$ and a standard deviation of $\sigma =85$. What raw score is required to be in the top $5\%$ of test takers?

Which of the following best describes the three quartiles of an ordered data set?

A growth chart shows that a $10$-year-old boy weighing $65$ pounds is at the $70$^th percentile for weight. Which interpretation is correct?

A pediatric growth chart reports that an $8$-year-old girl's weight of $56.2$ pounds is at the ${40}^{}$^th percentile. Which statement best interprets this percentile?

For a symmetric bell-shaped distribution, which pair correctly names the measure of center and the measure of dispersion used together to describe the distribution?

True or False: Chebyshev's theorem gives a guaranteed lower bound on the proportion of observations within $k$ standard deviations for any distribution with finite variance, whereas the Empirical Rule's specific percentages $(68\%,95\%,99.7\%)$ hold only for distributions that are approximately normal.

The ordered dataset is $4$, $7$, $9$, $12$, $12$, $15$, $18$, $21$, $25$, $30$. Using the exclusive (strictly-below) convention, find the percentile rank of the value $18$, where the percentile rank is defined as $P=\frac{L}{n}\times 100$ and $L$ is the number of observations strictly less than $18$.

An ordered dataset has 16 values: $52$, $55$, $59$, $61$, $64$, $67$, $70$, $72$, $75$, $78$, $81$, $85$, $88$, $91$, $94$, $99$. Using the averaging-adjacent-values convention described in class, find $p_{25}$.

For the ordered dataset $8$, $10$, $13$, $14$, $17$, $19$, $22$, $24$, $29$, $31$, $36$, $40$, compute the interquartile range using $q_1=p_{25}$ and $q_3=p_{75}$ with the averaging convention.

A dataset is $41$, $33$, $28$, $35$, $47$, $39$, $31$, $43$, $35$, $50$, $29$, $45$. Find the percentile rank of the value $39$ using the convention:

$\text{percentile rank}=\left(\frac{\text{number of observations less than or equal to the value}}{\text{total number of observations}}\right)\times 100$

For the ordered dataset $102$, $108$, $111$, $115$, $119$, $123$, $127$, $131$, $138$, $145$, $152$, $160$, the 75th percentile can be reported either as the dataset value with 9 observations below it or as the average of the 9th and 10th values. Find the numerical difference between these two reported values.

The ordered completion times are $12$, $14$, $15$, $18$, $20$, $21$, $24$, $27$, $28$, $32$, $35$, $41$,$45$, $50$, $53$, $60$, $66$, $72$, $80$, $95$. Using the averaging-adjacent-values convention, find $p_{60}$.

For the ordered dataset $5,7,9,11,14,18,20,23,27,31,34,38$, first compute $q_1$ and $q_3$ using the percentile method with averaging. Then find the percentile of the midpoint $\frac{q_1+q_3}{2}$.

A class has 25 exam scores. A particular score is at the 68th percentile. Using the definition $\text{percentile rank}=\left(\frac{\text{number of scores strictly below the value}}{n}\right)\times 100$, how many scores are below that value?