Directions:	 Using the scores below compute for the percentile, decile and quartile using ungrouped data.

10         11         12         14          16          18         19          20        21         22          24          25

1.	P90

2.	P40

3.	D9

4.	Q2

5.	D8

To solve for the percentiles (P), deciles (D), and quartiles (Q) from ungrouped data, we need to follow these steps:

1. **Organize the data in ascending order:**

The data is already in ascending order:  
\[10, 11, 12, 14, 16, 18, 19, 20, 21, 22, 24, 25\]

2. **Determine the position of the value:**

For percentiles, deciles, and quartiles, the formula to calculate the position is:

\[
\text{Position} = \left( \frac{k}{100} \right) \times (n + 1)
\]

where:
- \( k \) is the required percentile, decile, or quartile (depending on the problem),
- \( n \) is the number of data points (here \( n = 12 \)).

### 1. **P90 (90th percentile)**

\[
\text{Position for P90} = \left( \frac{90}{100} \right) \times (12 + 1) = 0.9 \times 13 = 11.7
\]

This means that the 90th percentile lies between the 11th and 12th values. To find the exact value, we interpolate:

\[
P90 = 24 + 0.7 \times (25 - 24) = 24 + 0.7 \times 1 = 24.7
\]

### 2. **P40 (40th percentile)**

\[
\text{Position for P40} = \left( \frac{40}{100} \right) \times (12 + 1) = 0.4 \times 13 = 5.2
\]

This means that the 40th percentile lies between the 5th and 6th values. We interpolate:

\[
P40 = 16 + 0.2 \times (18 - 16) = 16 + 0.2 \times 2 = 16.4
\]

### 3. **D9 (9th decile or 90th percentile)**

Since \( D9 \) corresponds to \( P90 \), the value is the same:

\[
D9 = 24.7
\]

### 4. **Q2 (2nd quartile or median, equivalent to the 50th percentile)**

\[
\text{Position for Q2} = \left( \frac{50}{100} \right) \times (12 + 1) = 0.5 \times 13 = 6.5
\]

The median lies between the 6th and 7th values:

\[
Q2 = 18 + 0.5 \times (19 - 18) = 18 + 0.5 \times 1 = 18.5
\]

### 5. **D8 (8th decile or 80th percentile)**

\[
\text{Position for D8} = \left( \frac{80}{100} \right) \times (12 + 1) = 0.8 \times 13 = 10.4
\]

The 80th percentile lies between the 10th and 11th values. We interpolate:

\[
D8 = 22 + 0.4 \times (24 - 22) = 22 + 0.4 \times 2 = 22.8
\]

### Summary of Results:
1. \( P90 = 24.7 \)
2. \( P40 = 16.4 \)
3. \( D9 = 24.7 \) (same as \( P90 \))
4. \( Q2 = 18.5 \) (median)
5. \( D8 = 22.8 \)

Would you like a further breakdown or any clarifications?

#### Related Questions:
1. How do you compute percentiles from a larger dataset?
2. What is the difference between percentiles, deciles, and quartiles?
3. How does interpolation work when calculating percentiles?
4. Can deciles or percentiles be calculated differently for grouped data?
5. What does the value of a certain percentile indicate in a data distribution?

#### Tip:
Always remember to interpolate between values when the position calculated is not a whole number. This gives a more accurate result, especially with small datasets.

Directions: Using the scores below compute for the percentile, decile and quartile using ungrouped data.
10 11 12 14 16 18 19 20 21 22 24 25

1. P90
2. P40
3. D9
4. Q2
5. D8

Learn how to compute percentiles, deciles, and quartiles using ungrouped data with scores 10, 11, 12, 14, 16, 18, 19, 20, 21, 22, 24, 25. Detailed steps include percentile (P90, P40), decile (D9, D8), and quartile (Q2) calculations, with interpolation methods. Suitable for high school students in Grades 9-12.

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