MOD 3.3 | VID 4 | Comprehensive Guide to Percentiles and Percentile Rank Calculations

Overview of Percentiles and Quartiles

• Quartiles and percentiles describe the shape of a data distribution:

  • Quartiles divide a data set into four equal parts (fourths).

  • Percentiles divide a data set into one hundred equal parts (hundredths), providing a more detailed description of the distribution.

• Definition of the $p\text{th}$ percentile:

  • For a number $p$ between $1$ and $99$, the $p\text{th}$ percentile separates the lowest $p\%$ of the data from the highest (100 - p)\%$.\n * The 1st percentile separates the lowest 1\%$of the data from the highest$99\%$.

  • The 98th percentile separates the lowest $98\%$ of the data from the highest 2\%$.\n* Computing percentiles can be done using several methods, all of which yield similar results.\n\n# Method 1: Computing a Percentile Value from a Given Percentile Rank\n\n* Procedure for finding the p\text{th}$percentile given a percentile rank$p:\n * Step 1: Arrange the data set values in increasing order.\n * Step 2: Let n$represent the total number of values in the data set. Compute the location index$l:\n\nl = \frac{p}{100} \times n\n\n * Step 3: Evaluate the value of l:\n * If l$is a whole number, the$p\text{th}$percentile is the average of the value in position$l$and the value in position$l + 1$.

  • If $l$ is not a whole number, round $l$ up to the next higher whole number. The percentile is the value located in this rounded-up position.

Example: Calculating the 60th Percentile of Los Angeles Rainfall

• Context and Data Parameters:

  • Dataset: Annual rainfall in inches in Los Angeles during the month of February over several years.

  • Total number of data values ($n$): $45$ values (pre-arranged in increasing order).

  • Target percentile ($p$): $60$

• Step-by-Step Calculation:

  • Compute position index $l$:

$l = \frac{60}{100} \times 45 = 27$

• Since $l = 27$ is a whole number, the 60th percentile is the average of the numbers in the 27th and 28th positions.

• Identify position values:

  • Number in the 27th position = $3.58$

  • Number in the 28th position = $3.71$

• Calculate average:

$\text{60th Percentile} = \frac{3.58 + 3.71}{2} = 3.645$

• Interpretation:

  • $60\%$ of the annual February rainfall data values fall below $3.645\,\text{inches}$.

  • $40\%$ of the annual February rainfall data values fall above $3.645\,\text{inches}$.

Method 2: Computing the Percentile Rank Corresponding to a Given Data Value

• Procedure for determining the percentile rank of a specific data value $x$:

  • Step 1: Arrange the data set in increasing order.

  • Step 2: Identify $x$ as the specific data value whose percentile rank is to be computed, and let $n$ be the total number of values in the set.

  • Step 3: Apply the percentile rank formula:

$\text{Percentile} = 100 \times \frac{\text{number of values less than } x + 0.5}{n}$

• Step 4: Round the resulting percentile value to the nearest whole number.

Example: Calculating the Percentile Rank for a Los Angeles Rainfall Value

• Context and Data Parameters:

  • Dataset: Annual February rainfall in Los Angeles ($n = 45$ values, pre-arranged in increasing order).

  • Target data value ($x$): $1.90\,\text{inches}$ (recorded in February 1989).

• Step-by-Step Calculation:

  • Count the number of values strictly less than $1.90$: $17$ values.

  • Apply the percentile rank formula:

$\text{Percentile} = 100 \times \frac{17 + 0.5}{45}$

• Compute intermediate result:

$\text{Percentile} = 100 \times \frac{17.5}{45} = 38.9$

• Round $38.9$ to the nearest whole number: $39$

  • Conclusion:

• The rainfall value of $1.90\,\text{inches}$ corresponds to the 39th percentile.