Unit E: Statistics and Probability

Chapter 1: Statistics

Calculating Percentile Rank

The formula to calculate percentile rank is

\begin{array}{rcl} percentile rank & = & \frac{number of values below the chosen value}{total number of values} \times 100 % \\ = & \frac{b}{n} \times 100 % \end{array}

where
b is the number of values below the chosen value
n is the number of values in the data set

The first step to finding the percentile is to order the values in ascending order. Once the values are ordered, find b by counting the number of values that are lower than the chosen value.

The steps to find percentile rank are as follows:

Step 1: Arrange the values in ascending order.
Step 2: Count the number of values less than the chosen value to find b.
Step 3: Count the number of values in the set of data to find n.
Step 4: Substitute these values into the equation.

percentile rank = \frac{b}{n} \times 100 %

Example 3

Natasha has a set of Russian dolls.

What is the percentile rank for the height of the second tallest Russian doll?
What does the percentile rank indicate?

Solutions

Step 1: The dolls are already arranged in ascending order.

Step 2: To find b, count the number of dolls shorter than the chosen value. The value of b is 6.

Step 3: The value for n is 8 since there are 8 dolls in the set of data.

Therefore, b = 6 and n = 8.

Step 4: Substitute b and n into the percentile rank formula.

$\begin{array}{rcl} percentile rank & = & \frac{b}{n} \times 100 % \\ = & \frac{6}{8} \times 100 % \\ = & 0.75 \times 100 % \\ = & 75 % \end{array}$

The second tallest doll is at the 75th percentile.
The second tallest doll (the chosen value) is taller than 75% of the dolls in the set.

Example 4

Carlee has a free-throw average of 67%. Calculate Carlee's percentile rank. Her basketball team has the following statistics:

Free-Throw Average

Team Member	Free-Throw Average (%)
Danica	54
Genie	59
Trista	61
Leslie	63
Flo	63
Nancy	65
Penny	66
Olive	67
Carlee	67
Kristen	70
Janny	71
Sadie	71

Solution

Step 1: The free-throw averages are already arranged in ascending order.

Step 2: To find b, count the number of free-throw averages smaller than the chosen value of 67%.

Free-Throw Average

Team Member	Free-Throw Average (%)
Danica	54 $1$
Genie	59 $2$
Trista	61 $3$
Leslie	63 $4$
Flo	63 $5$
Nancy	65 $6$
Penny	66 $7$
Olive	67
$Carlee$	$67$ $chosen value$
Kristen	70
Janny	71
Sadie	71

Note: Olive would not be counted when finding b since she has the same free-throw average as Carlee. Olive's score is not below Carlee's score.

The number of values below Carlee's score is 7. Therefore, b = 7.

Step 3: The total number of values in the set of data is 12. That means, n = 12.

Step 4: Substitute b and n into the formula for percentile rank.

\begin{array}{rcl} percentile rank & = & \frac{b}{n} \times 100 % \\ = & \frac{7}{12} \times 100 % \\ = & 0.583 \times 100 % \\ = & 58 % \end{array}

Carlee's rank is at the 58th percentile.

Below is a video solution of Example 4.

Recall that the terms percentile and percent have different meanings. For example, if a student received 21 out of 25 on a test, the percent earned is

\frac{21}{25} \times 100 = 84 %

. A percentile rank of 84% on a test indicates that the chosen value is higher than 84% of the other values found in the set of data.