Five restaurants, coded R1, R2, R3, R4 and R5 gave integer ratings to five gig workers – Ullas, Vasu, Waman, Xavier and Yusuf, on a scale of 1 to 5.

The means of the ratings given by R1, R2, R3, R4 and R5 were 3.4, 2.2, 3.8, 2.8 and 3.4 respectively.

The summary statistics of these ratings for the five workers is given below.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743421215/aptidude_questions/qmwcs5gcs9gohdshvlil.png)

* Range of ratings is defined as the difference between the maximum and minimum ratings awarded to a worker.

The following is partial information about ratings of 1 and 5 awarded by the restaurants to the workers.
(a) R1 awarded a rating of 5 to Waman, as did R2 to Xavier, R3 to Waman and Xavier, and R5 to Vasu.
(b) R1 awarded a rating of 1 to Ullas, as did R2 to Waman and Yusuf, and R3 to Yusuf.

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What is the median of the ratings given by R3 to the five workers?

Question

Five restaurants, coded R1, R2, R3, R4 and R5 gave integer ratings to five gig workers – Ullas, Vasu, Waman, Xavier and Yusuf, on a scale of 1 to 5.

The means of the ratings given by R1, R2, R3, R4 and R5 were 3.4, 2.2, 3.8, 2.8 and 3.4 respectively.

The summary statistics of these ratings for the five workers is given below.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743421215/aptidude_questions/qmwcs5gcs9gohdshvlil.png)

* Range of ratings is defined as the difference between the maximum and minimum ratings awarded to a worker.

The following is partial information about ratings of 1 and 5 awarded by the restaurants to the workers.
(a) R1 awarded a rating of 5 to Waman, as did R2 to Xavier, R3 to Waman and Xavier, and R5 to Vasu.
(b) R1 awarded a rating of 1 to Ullas, as did R2 to Waman and Yusuf, and R3 to Yusuf.

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What is the median of the ratings given by R3 to the five workers?

Accepted Answer

#### **Key Information:**
1. **Mean Ratings by R3:** 3.8  
   - Total ratings given by R3 across 5 workers = $$ 3.8 	imes 5 = 19 $$.  
   - Ratings are integers between 1 and 5.

2. **Partial Information About R3’s Ratings:**  
   - R3 awarded **5 to Waman and Xavier** (from partial information).  
   - R3 awarded **1 to Yusuf** (from partial information).

3. **Constraints from Summary Statistics for Workers:**  
   - Median rating for Ullas: **2**, Vasu: **4**, Waman: **4**, Xavier: **4**, Yusuf: **3**.  
   - Modal rating for each worker provides additional constraints:  
     - Ullas: Mode is **2**, meaning $$ 2 $$ occurs most frequently.  
     - Vasu: Mode is **4**, meaning $$ 4 $$ occurs most frequently.  
     - Waman and Xavier: Mode is **5**, meaning $$ 5 $$ occurs most frequently.  
     - Yusuf: Modes are **1 and 4**, meaning both occur frequently.

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#### **Step-by-Step Deduction:**
1. **Known Ratings by R3:**  
   - $$ 5 $$ for Waman and Xavier.  
   - $$ 1 $$ for Yusuf.

2. **Remaining Ratings to Assign (Total = 19):**
   - Sum of known ratings = $$ 5 + 5 + 1 = 11 $$.  
   - Remaining sum for Ullas and Vasu = $$ 19 - 11 = 8 $$.

3. **Assigning Ratings to Ullas and Vasu (Based on Median and Mode):**
   - Ullas’s median is $$ 2 $$, mode is $$ 2 $$. Assign $$ 4 $$ to Ullas.  
   - Vasu’s median is $$ 4 $$, mode is $$ 4 $$. Assign $$ 4 $$ to Vasu.

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#### **Ratings by R3 Across Workers:**
- Ullas: $$ 4 $$  
- Vasu: $$ 4 $$  
- Waman: $$ 5 $$  
- Xavier: $$ 5 $$  
- Yusuf: $$ 1 $$

When sorted in ascending order, the ratings are: $$ [1, 4, 4,[5][5] $$.

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#### **Median Calculation:**
The median is the middle value in the sorted list, which is **4**.

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### **Conclusion:**
The median of the ratings given by R3 to the five workers is **4**.

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