# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/statistics-foundations/percentile-data-group Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/statistics-foundations/percentile-data-group/en.mdx Calculate percentiles in grouped data using interpolation formulas. Learn to find data positions and interpret percentile rankings with examples. --- ## What Are Percentiles? You're already familiar with [quartiles](/en/subjects/mathematics/statistics-foundations/quartile-data-group), which divide data into $$4$$ equal parts, right? Well, **percentiles** are like quartiles' sibling, but they're even more detailed! Visible text: You're already familiar with [quartiles](/en/subjects/mathematics/statistics-foundations/quartile-data-group), which divide data into 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. Visible text: If quartiles divide data into chunks, percentiles divide ordered data into equal chunks. That's a lot, huh? Like dividing a chocolate bar into 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 $$P_{99}$$. Visible text: Each chunk is separated by a percentile value. There are percentile values, starting from , , , ..., up to . - $$P_{10}$$ ($$10$$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 $$15\%$$. Visible text: - (th Percentile) means this value separates the smallest of the data from the remaining . - (th Percentile) is exactly the same as the **Median** or the **Second Quartile ( )**, because it divides the data right in the middle ( below, above). - (th Percentile) means this value separates the smallest of the data from the largest . 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. Visible text: Just like finding quartiles for grouped data, we also use **interpolation** to find the value of a percentile () 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: ```math \text{Position of } P_i = \text{the } \frac{i}{100} \times n \text{-th data point} ``` - $$i$$ = Which percentile are we looking for? (e.g., $$10, 50, 85$$) - $$n$$ = Total number of data points Visible text: - = Which percentile are we looking for? (e.g., ) - = 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. Visible text: Once we have the position, we look at the cumulative frequency table () 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: ```math P_i = T_b + \left( \frac{\frac{i}{100}n - F_{kum}}{f_i} \right) p ``` 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 Visible text: - = Value of the i-th Percentile (what we're looking for) - = Lower boundary of the i-th percentile class - = Which percentile (e.g., ) - = Total frequency - = Cumulative frequency **BEFORE** the i-th percentile class - = Frequency of the i-th percentile class - = Class width Notice, the formula is very similar to the quartile formula, the only difference is the $$\frac{i}{100}n$$ part (quartiles use $$\frac{i}{4}n$$). Visible text: Notice, the formula is very similar to the quartile formula, the only difference is the part (quartiles use ). ## Finding Math Test Scores For example, let's say we have the math test scores of $$40 \text{ students}$$: Visible text: For example, let's say we have the math test scores of : | Test Score | Frequency ($$f$$) | Cumulative Frequency ($$F_k$$) | Lower Boundary ($$T_b$$) | Class Width ($$p$$) | | :--------: | :---------------------------------: | :----------------------------------------------: | :----------------------------------------: | :-----------------------------------: | | $$61\text{-}70$$ | $$4$$ | $$4$$ | $$60.5$$ | $$10$$ | | $$71\text{-}80$$ | $$10$$ | $$14$$ | $$70.5$$ | $$10$$ | | $$81\text{-}90$$ | $$16$$ | $$30$$ | $$80.5$$ | $$10$$ | | $$91\text{-}100$$ | $$10$$ | $$40$$ | $$90.5$$ | $$10$$ | | **Total** | $$40$$ | | | | Visible text: | Test Score | Frequency () | Cumulative Frequency () | Lower Boundary () | Class Width () | | :--------: | :---------------------------------: | :----------------------------------------------: | :----------------------------------------: | :-----------------------------------: | | | | | | | | | | | | | | | | | | | | | | | | | | **Total** | | | | | We want to find the value of the $$85$$th Percentile ($$P_{85}$$). Visible text: We want to find the value of the th Percentile (). 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 $$P_{85}$$:** ```math P_{85} = T_b + \left( \frac{\frac{85}{100}n - F_{kum}}{f_{85}} \right) p ``` ```math P_{85} = 90.5 + \left( \frac{34 - 30}{10} \right) 10 ``` ```math P_{85} = 90.5 + \left( \frac{4}{10} \right) 10 ``` ```math P_{85} = 90.5 + 4 ``` ```math P_{85} = 94.5 ``` Visible text: 1. **Find the Position of :** The position of is the -th data point. 2. **Determine the Class of :** Look at the column. Where is the th data point? The class has (not enough). The class has (data points through are here). So, the class is . 3. **Gather Ingredients for the Formula:** - (Lower boundary of class ) is - - - (Cumulative frequency before class ) = - (Frequency of class ) is - (Class width) is 4. **Calculate :** So, the $$85$$th Percentile value is $$94.5$$. This means $$85\%$$ of the students scored $$94.5$$ or less, and $$15\%$$ scored above $$94.5$$. Visible text: So, the th Percentile value is . This means of the students scored or less, and scored above . ## Exercise Try calculating the value of the $$20$$th Percentile ($$P_{20}$$) from the math test score data above! Visible text: Try calculating the value of the th Percentile () 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 $$P_{20}$$:** ```math P_{20} = T_b + \left( \frac{\frac{20}{100}n - F_{kum}}{f_{20}} \right) p ``` ```math P_{20} = 70.5 + \left( \frac{8 - 4}{10} \right) 10 ``` ```math P_{20} = 70.5 + \left( \frac{4}{10} \right) 10 ``` ```math P_{20} = 70.5 + 4 ``` ```math P_{20} = 74.5 ``` Visible text: 1. **Position of :** The position of is the -th data point. 2. **Class of :** Look at . The th data point is in the class (because the previous class's is , and this class's is ). 3. **Formula Ingredients:** - - - - - - 4. **Calculate :** The $$20$$th Percentile value is $$74.5$$. Visible text: The th Percentile value is .