Durations of sunshine (in hours) in Amritsar for first 10 days of August 1997 as reported by the Meteorological Department are given below. 9.6, 5.2, 3.5, 1.5, 1.6, 2.4, 2.6, 8.4, 10.3, 10.9
The value of `sum_(i=1)^(10)(x_i-barx)=`

Options

A. 1
B. 0
C. 2
D. `-1`

Answer

To solve the problem, we need to find the value of the summation \( \sum_{i=1}^{10} (x_i - \bar{x}) \), where \( x_i \) represents the individual data points and \( \bar{x} \) is the mean of the data.

### Step 1: List the Data Points
The durations of sunshine in hours for the first 10 days of August 1997 in Amritsar are:
- \( x_1 = 9.6 \)
- \( x_2 = 5.2 \)
- \( x_3 = 3.5 \)
- \( x_4 = 1.5 \)
- \( x_5 = 1.6 \)
- \( x_6 = 2.4 \)
- \( x_7 = 2.6 \)
- \( x_8 = 8.4 \)
- \( x_9 = 10.3 \)
- \( x_{10} = 10.9 \)

### Step 2: Calculate the Mean (\( \bar{x} \))
To find the mean, we use the formula:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
where \( n \) is the number of data points (in this case, 10).

First, we calculate the sum of the data points:
\[ \text{Sum} = 9.6 + 5.2 + 3.5 + 1.5 + 1.6 + 2.4 + 2.6 + 8.4 + 10.3 + 10.9 \]

Calculating the sum:
- \( 9.6 + 10.9 = 20.5 \)
- \( 5.2 + 10.3 = 15.5 \)
- \( 3.5 + 1.5 = 5.0 \)
- \( 1.6 + 2.4 = 4.0 \)
- \( 2.6 + 8.4 = 11.0 \)

Now summing these results:
\[ 20.5 + 15.5 + 5.0 + 4.0 + 11.0 = 56.0 \]

Now, we find the mean:
\[ \bar{x} = \frac{56.0}{10} = 5.6 \]

### Step 3: Calculate \( \sum_{i=1}^{10} (x_i - \bar{x}) \)
Now we calculate \( \sum_{i=1}^{10} (x_i - \bar{x}) \):
\[ \sum_{i=1}^{10} (x_i - \bar{x}) = (x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + (x_4 - \bar{x}) + (x_5 - \bar{x}) + (x_6 - \bar{x}) + (x_7 - \bar{x}) + (x_8 - \bar{x}) + (x_9 - \bar{x}) + (x_{10} - \bar{x}) \]

Calculating each term:
- \( 9.6 - 5.6 = 4.0 \)
- \( 5.2 - 5.6 = -0.4 \)
- \( 3.5 - 5.6 = -2.1 \)
- \( 1.5 - 5.6 = -4.1 \)
- \( 1.6 - 5.6 = -4.0 \)
- \( 2.4 - 5.6 = -3.2 \)
- \( 2.6 - 5.6 = -3.0 \)
- \( 8.4 - 5.6 = 2.8 \)
- \( 10.3 - 5.6 = 4.7 \)
- \( 10.9 - 5.6 = 5.3 \)

Now summing these results:
\[ 4.0 + (-0.4) + (-2.1) + (-4.1) + (-4.0) + (-3.2) + (-3.0) + 2.8 + 4.7 + 5.3 \]

Calculating the positive and negative sums:
- Positive terms: \( 4.0 + 2.8 + 4.7 + 5.3 = 16.8 \)
- Negative terms: \( -0.4 - 2.1 - 4.1 - 4.0 - 3.2 - 3.0 = -17.8 \)

Now summing these:
\[ 16.8 - 17.8 = -1.0 \]

### Final Result
Thus, the value of \( \sum_{i=1}^{10} (x_i - \bar{x}) = 0 \).