Following is a data sample of 8 observations in ascending order. If the mean of this data is 31 and the mode is 35. Find the missing observations.
20 24 x 30 31 35 y 45

Answer

To solve the problem step by step, we need to find the missing observations \(x\) and \(y\) in the data set:

20, 24, \(x\), 30, 31, 35, \(y\), 45.

Given:
- Mean = 31
- Mode = 35

### Step 1: Understand the Mode
The mode is the value that appears most frequently in a data set. Since the mode is 35, it must appear more than once. In the given data, we can see that 35 appears only once. Therefore, for the mode to be 35, \(y\) must also be 35.

**Conclusion:**
\[ y = 35 \]

### Step 2: Calculate the Mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the number of observations is 8. So, we can express the mean as:
\[ 31 = \frac{20 + 24 + x + 30 + 31 + 35 + y + 45}{8} \]

### Step 3: Substitute \(y\) in the Mean Formula
Now, substituting \(y = 35\) into the equation:
\[ 31 = \frac{20 + 24 + x + 30 + 31 + 35 + 35 + 45}{8} \]

### Step 4: Calculate the Sum of Known Values
Calculating the sum of the known values:
\[ 20 + 24 + 30 + 31 + 35 + 35 + 45 = 220 \]
So, we can rewrite the equation:
\[ 31 = \frac{220 + x}{8} \]

### Step 5: Multiply Both Sides by 8
To eliminate the fraction, multiply both sides by 8:
\[ 31 \times 8 = 220 + x \]
Calculating \(31 \times 8\):
\[ 248 = 220 + x \]

### Step 6: Solve for \(x\)
Now, isolate \(x\):
\[ x = 248 - 220 \]
Calculating the right side:
\[ x = 28 \]

### Final Conclusion
Thus, the missing observations are:
\[ x = 28 \quad \text{and} \quad y = 35 \]

### Summary of Results
The final values are:
- \(x = 28\)
- \(y = 35\)

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