If 69.5 is the mean of 72, 70, x, 62, 50, 71, 90, 64, 58 and 82, find the value of x.

Options

A. `76`
B. `72`
C. `56`
D. None

Answer

To find the value of \( x \) in the given dataset where the mean is 69.5, we can follow these steps:

### Step 1: Understand the formula for the mean
The mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

### Step 2: Identify the number of observations
In this case, we have the following observations: 72, 70, \( x \), 62, 50, 71, 90, 64, 58, and 82. Thus, the total number of observations is 10.

### Step 3: Set up the equation for the mean
Given that the mean is 69.5, we can set up the equation:
\[ 69.5 = \frac{72 + 70 + x + 62 + 50 + 71 + 90 + 64 + 58 + 82}{10} \]

### Step 4: Calculate the sum of the known observations
First, we need to calculate the sum of the known values:
\[ 72 + 70 + 62 + 50 + 71 + 90 + 64 + 58 + 82 = 619 \]

### Step 5: Substitute the sum into the equation
Now, we can substitute this sum into our mean equation:
\[ 69.5 = \frac{619 + x}{10} \]

### Step 6: Multiply both sides by 10 to eliminate the fraction
To eliminate the fraction, multiply both sides by 10:
\[ 69.5 \times 10 = 619 + x \]
\[ 695 = 619 + x \]

### Step 7: Solve for \( x \)
Now, isolate \( x \) by subtracting 619 from both sides:
\[ x = 695 - 619 \]
\[ x = 76 \]

### Final Answer
Thus, the value of \( x \) is \( 76 \).

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