Percentiles for Grouped Data

What Are Percentiles?

You're already familiar with quartiles, which divide data into $4$ equal parts, right? Well, percentiles are like quartiles' sibling, but they're even more detailed!

If quartiles divide data into $4$ chunks, percentiles divide ordered data into $100$ equal chunks. That's a lot, huh? Like dividing a chocolate bar into $100$ tiny squares.

Each chunk is separated by a percentile value. There are $99$ percentile values, starting from $P_1$, $P_2$, $P_3$, ..., up to $P_{99}$.

• $P_{10}$ ($10$th Percentile) means this value separates the smallest $10\%$ of the data from the remaining $90\%$.
• $P_{50}$ ($50$th Percentile) is exactly the same as the Median or the Second Quartile ($Q_2$ ), because it divides the data right in the middle ($50\%$ below, $50\%$ above).
• $P_{85}$ ($85$th Percentile) means this value separates the smallest $85\%$ of the data from the largest $15\%$.

Percentiles are very useful for seeing the position of a specific value relative to the entire dataset, like test-score rankings in one group or a child's growth compared to peers of the same age.

How to Find Percentile Values for Grouped Data

Just like finding quartiles for grouped data, we also use interpolation to find the value of a percentile ($P_i$) when the data is grouped.

The steps are very similar:

Find the Percentile Class Position

First, we determine which data point corresponds to the i-th percentile. The formula is:

• $i$ = Which percentile are we looking for? (e.g., $10, 50, 85$)
• $n$ = Total number of data points

Once we have the position, we look at the cumulative frequency table ($F_k$) to find out which class interval this percentile falls into.

Calculate the Percentile Value using the Interpolation Formula

Once we know the class, we use this magic interpolation formula:

Where:

• $P_i$ = Value of the i-th Percentile (what we're looking for)
• $T_b$ = Lower boundary of the i-th percentile class
• $i$ = Which percentile (e.g., $10, 85$)
• $n$ = Total frequency
• $F_{kum}$ = Cumulative frequency BEFORE the i-th percentile class
• $f_i$ = Frequency of the i-th percentile class
• $p$ = Class width

Notice, the formula is very similar to the quartile formula, the only difference is the $\frac{i}{100}n$ part (quartiles use $\frac{i}{4}n$).

Finding Math Test Scores

For example, let's say we have the math test scores of $40 \text{ students}$:

We want to find the value of the $85$th Percentile ($P_{85}$).

1. Find the Position of $P_{85}$:

   The position of $P_{85}$ is the $\frac{85}{100} \times 40 = \frac{3400}{100} = 34$-th data point.

2. Determine the Class of $P_{85}$:

   Look at the $F_k$ column. Where is the $34$th data point? The $81\text{-}90$ class has $F_k = 30$ (not enough). The $91\text{-}100$ class has $F_k = 40$ (data points $31$ through $40$ are here). So, the $P_{85}$ class is $91\text{-}100$.

3. Gather Ingredients for the Formula:

   • $T_b$ (Lower boundary of class $91\text{-}100$) is $90.5$
   • $i = 85$
   • $n = 40$
   • $F_{kum}$ (Cumulative frequency before class $91\text{-}100$) = $30$
   • $f_{85}$ (Frequency of class $91\text{-}100$) is $10$
   • $p$ (Class width) is $10$

4. Calculate $P_{85}$:

   $$P_{85} = T_b + \left( \frac{\frac{85}{100}n - F_{kum}}{f_{85}} \right) p$$

   $$P_{85} = 90.5 + \left( \frac{34 - 30}{10} \right) 10$$

   $$P_{85} = 90.5 + \left( \frac{4}{10} \right) 10$$

   $$P_{85} = 90.5 + 4$$

   $$P_{85} = 94.5$$

So, the $85$th Percentile value is $94.5$. This means $85\%$ of the students scored $94.5$ or less, and $15\%$ scored above $94.5$.

Exercise

Try calculating the value of the $20$th Percentile ($P_{20}$) from the math test score data above!

Answer Key

1. Position of $P_{20}$:

   The position of $P_{20}$ is the $\frac{20}{100} \times 40 = \frac{800}{100} = 8$-th data point.

2. Class of $P_{20}$:

   Look at $F_k$. The $8$th data point is in the $71\text{-}80$ class (because the previous class's $F_k$ is $4$, and this class's $F_k$ is $14$).

3. Formula Ingredients:

   • $T_b = 70.5$
   • $i = 20$
   • $n = 40$
   • $F_{kum} = 4$
   • $f_{20} = 10$
   • $p = 10$

4. Calculate $P_{20}$:

   $$P_{20} = T_b + \left( \frac{\frac{20}{100}n - F_{kum}}{f_{20}} \right) p$$

   $$P_{20} = 70.5 + \left( \frac{8 - 4}{10} \right) 10$$

   $$P_{20} = 70.5 + \left( \frac{4}{10} \right) 10$$

   $$P_{20} = 70.5 + 4$$

   $$P_{20} = 74.5$$

The $20$th Percentile value is $74.5$.