PERCENTILES FOR GROUPED DATA AND PERCENTILE RANK.pptx - Free Printable

To solve this problem, we need to find the percentile ranks of two students who scored 42 and 35 on a mathematics test, based on grouped data. The goal is to determine whether these scores are above the 75th percentile, which is the passing score for the scholarship.

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📌 Given:

- Grouped frequency distribution of test scores.
- Total number of students: N = 50
- We want to find the percentile rank of:
- Student A: Score = 42
- Student B: Score = 35
- Determine if their scores are at or above the 75th percentile (i.e., qualify for scholarship).

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🔍 Step 1: Understand Percentile Rank Formula

For grouped data, the percentile rank (PR) of a score is calculated using:

$$
\text{Percentile Rank} = \left( \frac{L + \frac{(X - L)}{w} \times f}{N} \right) \times 100
$$

But more commonly used formula is:

$$
PR = \left( \frac{\text{Cumulative Frequency below } X + \frac{(X - L)}{w} \times f}{N} \right) \times 100
$$

Where:
- $ X $ = the score
- $ L $ = lower limit of the class interval containing $ X $
- $ w $ = width of the class interval
- $ f $ = frequency of that class
- Cumulative frequency below $ X $ = sum of frequencies of all classes below the one containing $ X $

Alternatively, use the interpolation method:

$$
PR = \left( \frac{CF_{\text{below}} + \frac{(X - L)}{w} \times f}{N} \right) \times 100
$$

We'll compute cumulative frequencies first.

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📊 Step 2: Build Cumulative Frequency Table

| Class Interval | Frequency (f) | Cumulative Frequency (CF) |
|----------------|---------------|----------------------------|
| 46–50 | 4 | 4 |
| 41–45 | 8 | 4 + 8 = 12 |
| 36–40 | 11 | 12 + 11 = 23 |
| 31–35 | 9 | 23 + 9 = 32 |
| 26–30 | 12 | 32 + 12 = 44 |
| 21–25 | 6 | 44 + 6 = 50 |

Total: N = 50

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✔ Student A: Score = 42

- Score 42 falls in the class interval 41–45
- This interval has:
- Lower limit $ L = 41 $
- Width $ w = 5 $ (since 45 - 41 = 4, but intervals are inclusive — usually assume 41–45 means 41 ≤ x < 46, so width = 5)
- Frequency $ f = 8 $
- Cumulative frequency below this class: $ CF_{\text{below}} = 4 $ (from 46–50 class? Wait — no!)

Wait! Let's double-check the order of the table.

The table is listed from highest to lowest score. But cumulative frequency should be built from lowest to highest.

So let’s reorder the table from lowest to highest for clarity:

| Class Interval | Frequency (f) | Cumulative Frequency (CF) |
|----------------|---------------|----------------------------|
| 21–25 | 6 | 6 |
| 26–30 | 12 | 6 + 12 = 18 |
| 31–35 | 9 | 18 + 9 = 27 |
| 36–40 | 11 | 27 + 11 = 38 |
| 41–45 | 8 | 38 + 8 = 46 |
| 46–50 | 4 | 46 + 4 = 50 |

Now correct.

So now:
- For score 42, it lies in 41–45
- $ L = 41 $
- $ w = 5 $
- $ f = 8 $
- $ CF_{\text{below}} = 38 $ (sum of all below 41–45)

Now apply formula:

$$
PR = \left( \frac{CF_{\text{below}} + \frac{(X - L)}{w} \times f}{N} \right) \times 100
$$

$$
PR_A = \left( \frac{38 + \frac{(42 - 41)}{5} \times 8}{50} \right) \times 100
= \left( \frac{38 + \frac{1}{5} \times 8}{50} \right) \times 100
= \left( \frac{38 + 1.6}{50} \right) \times 100
= \left( \frac{39.6}{50} \right) \times 100
= 0.792 \times 100 = 79.2%
$$

✔ So Student A's percentile rank = 79.2

This is above 75%, so qualifies.

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✔ Student B: Score = 35

- Score 35 falls in 31–35
- $ L = 31 $
- $ w = 5 $
- $ f = 9 $
- $ CF_{\text{below}} = 18 $ (from 21–25 and 26–30)

Now:

$$
PR_B = \left( \frac{18 + \frac{(35 - 31)}{5} \times 9}{50} \right) \times 100
= \left( \frac{18 + \frac{4}{5} \times 9}{50} \right) \times 100
= \left( \frac{18 + 0.8 \times 9}{50} \right) \times 100
= \left( \frac{18 + 7.2}{50} \right) \times 100
= \left( \frac{25.2}{50} \right) \times 100
= 0.504 \times 100 = 50.4%
$$

So Student B's percentile rank = 50.4%

This is below 75%, so does not qualify.

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✔ Final Answer:

- Student A (score = 42): Percentile Rank = 79.2% → Qualifies
- Student B (score = 35): Percentile Rank = 50.4% → Does NOT qualify

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🟩 Conclusion:

Only Student A qualifies for the scholarship, since their score is above the 75th percentile.

> Answer:
> - Student A: 79.2th percentile → ✔ Qualifies
> - Student B: 50.4th percentile → ✘ Does not qualify