Percentiles and Quartiles

In a corporate evaluation of $200$ employees, performance scores range from $0$ to $100$. An employee with a score of $84$ scored strictly higher than $150$ employees and strictly lower than $42$ employees, while the remaining employees received a score of exactly $84$. If the percentile rank of a score is defined as the percentage of all scores strictly below it plus half the percentage of all scores equal to it, what is the percentile rank of a score of $84$?

A data set consists of $60$ distinct test scores arranged in ascending order. The $70\text{th}$ percentile of this data set is equal to the $42\text{nd}$ score. If $15$ new test scores, all strictly lower than the lowest score in the original data set, are added to form a new data set of $75$ scores, what is the percentile rank of the score that was the $70\text{th}$ percentile of the original data set?

A quality control analyst measured the shelf life, in days, of a sample of $80$ manufactured batteries, where each battery had a distinct shelf life. Battery $X$ had a shelf life at the $65\text{th}$ percentile of the sample. If $20$ additional batteries are subsequently tested and every one of these $20$ batteries has a shelf life strictly greater than Battery $X$, what is the percentile rank of Battery $X$'s shelf life in the combined sample of $100$ batteries?

In a dataset of $N$ distinct test scores, score $S_1$ is at the 75th percentile and score $S_2$ is at the 40th percentile. A group of 60 new distinct scores is added to the dataset: 12 are strictly less than $S_2$, 18 are strictly between $S_2$ and $S_1$, and 30 are strictly greater than $S_1$. If score $S_1$ is at the 60th percentile of the combined dataset, what is the percentile rank of score $S_2$ in the combined dataset?

In a class of $40$ students, each student received a distinct score on a final examination. If a particular student's score is at the $85\text{th}$ percentile of the class scores, how many students scored lower than this student?

A dataset consists of 20 distinct numerical values arranged in increasing order. The lower quartile ($Q_1$) of the dataset is equal to the average of the 5th and 6th values, and the upper quartile ($Q_3$) is equal to the average of the 15th and 16th values. Given that the lower quartile $Q_1 = 42$, the interquartile range ($\text{IQR} = Q_3 - Q_1$) is 38, and the 15th value in the dataset is 74, what is the value of the 16th term?

In a corporate firm of $N$ employees, each employee earns a distinct annual salary. An employee earning a salary of $\$78,000$ is at the $60\text{th}$ percentile of all salaries in the firm. The firm subsequently hires $10$ new employees, each of whom earns an annual salary strictly less than $\$78,000$. If $\$78,000$ is now at the $70\text{th}$ percentile of all salaries in the expanded firm, how many employees were originally in the firm?

A dataset contains $20$ distinct test scores arranged in ascending order:

$10, 12, 15, 17, 20, 22, 25, 28, 30, 32, 35, 38, 40, 43, 45, 48, 50, 52, 55, 60$

If the $p\text{th}$ percentile of a dataset of $N$ values is defined as the value at position $k = \frac{p}{100} \times N$ when ordered from least to greatest, what is the $75\text{th}$ percentile of these test scores?

The table below shows the distribution of scores achieved by $50$ candidates on a professional certification assessment:

What is the interquartile range (IQR) of the assessment scores?

In a dataset of $N$ distinct values arranged in ascending order, the $p\text{th}$ percentile is defined as the value at position $\left\lceil \frac{p}{100} \times N \right\rceil$.

Initially, a dataset contains $100$ distinct test scores. The $40\text{th}$ percentile score is $S_1$ and the $70\text{th}$ percentile score is $S_2$. A researcher adds $x$ new scores that are strictly less than $S_1$, and $y$ new scores that are strictly between $S_1$ and $S_2$, where $x$ and $y$ are positive integers.

If the $40\text{th}$ percentile score of the expanded dataset remains $S_1$ and the $70\text{th}$ percentile score of the expanded dataset remains $S_2$, what is the maximum possible value of $x + y$?