Interpreting Percentiles, Quartiles, and Median

Learning Outcomes

Interpret percentiles, medians and quartiles in context

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Data value percentages are less than or equal to the $p$ th percentile. For example, $15$ % of data values are less than or equal to the $15$ th percentile.

Low percentiles always correspond to lower data values
High percentiles always correspond to higher data values

A percentile may or may not correspond to a value judgment about whether it’s “good” or “bad.” The interpretation of whether a certain percentile is “good” or “bad” depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies.

Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.

Guidelines

When writing the interpretation of a percentile in the context of the given data, the sentence should contain the following information.

information about the context of the situation being considered
the data value (value of the variable) that represents the percentile
the percent of individuals or items with data values below the percentile
the percent of individuals or items with data values above the percentile

Example

On a timed math test, the first quartile for time it took to finish the exam was $35$ minutes. Interpret the first quartile in the context of this situation.

Show Solution

Twenty-five percent of students finished the exam in $35$ minutes or less
Seventy-five percent of students finished the exam in $35$ minutes or more
A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If you take too long, you might not be able to finish.)

Try It

For the $100$ -meter dash, the third quartile for times for finishing the race was $11.5$ seconds. Interpret the third quartile in the context of the situation.

Show Solution

Twenty-five percent of runners finished the race in

$11.5$ seconds or more. Seventy-five percent of runners finished the race in

$11.5$ seconds or less. A lower percentile is good because finishing a race more quickly is desirable.

Example

On a $20$ question math test, the $70$ th percentile for number of correct answers was $16$ . Interpret the $70$ th percentile in the context of this situation.

Show Solution

Seventy percent of students answered $16$ or fewer questions correctly
Thirty percent of students answered $16$ or more questions correctly
A higher percentile could be considered good, as answering more questions correctly is desirable

Try It

On a $60$ point written assignment, the $80$ th percentile for the number of points earned was $49$ . Interpret the $80$ th percentile in the context of this situation.

Show Solution

Eighty percent of students earned

$49$ points or fewer. Twenty percent of students earned 49 or more points. A higher percentile is good because getting more points on an assignment is desirable.

Example

At a community college, it was found that the $30$ th percentile of credit units that students are enrolled for is seven units. Interpret the $30$ th percentile in the context of this situation.

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Try It

During a season, the $40$ th percentile for points scored per player in a game is eight. Interpret the $40$ th percentile in the context of this situation.

Show Solution

Example

Sharpe Middle School is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed $15$ anonymous students to determine how many minutes a day the students spend exercising. The results from the $15$ anonymous students are shown.

$0$ minutes; $40$ minutes; $60$ minutes; $30$ minutes; $60$ minutes

$10$ minutes; $45$ minutes; $30$ minutes; $300$ minutes; $90$ minutes;

$30$ minutes; $120$ minutes; $60$ minutes; $0$ minutes; $20$ minutes

Determine the following five values.

Min = $0$
$Q_1$ = $20$
Med = $40$
$Q_3$ = $60$
Max = $300$

If you were the principal, would you be justified in purchasing new fitness equipment?

Show Solution

Since $75$ % of the students exercise for $60$ minutes or less daily, and since the $IQR$ is $40$ minutes $(60 – 20 = 40)$ , we know that half of the students surveyed exercise between $20$ minutes and $60$ minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.

However, the principal needs to be careful. The value $300$ appears to be a potential outlier.

$Q_3$ + $1.5$ ( $IQR$ ) = $60 + (1.5)(40) = 120$ .

The value $300$ is greater than $120$ so it is a potential outlier. If we delete it and calculate the five values, we get the following values:

Min = $0$
$Q_1$ = $20$
$Q_3$ = $60$
Max = $120$

We still have $75$ % of the students exercising for $60$ minutes or less daily and half of the students exercising between $20$ and $60$ minutes a day. However, $15$ students is a small sample and the principal should survey more students to be sure of his survey results.

Module 2: Descriptive Statistics