Find the variance for the following data : 65,58,68,44,48,45,60,62,60,50

Answer

To find the variance of the given data set: 65, 58, 68, 44, 48, 45, 60, 62, 60, 50, we will follow these steps:

### Step 1: Calculate the Mean (x̄)
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( n \) is the number of observations.

1. **Sum the data**:
\[ 65 + 58 + 68 + 44 + 48 + 45 + 60 + 62 + 60 + 50 = 560 \]

2. **Count the number of observations**:
\[ n = 10 \]

3. **Calculate the mean**:
\[ \bar{x} = \frac{560}{10} = 56 \]

### Step 2: Calculate \( x_i - \bar{x} \)
Now, we will calculate the difference between each observation and the mean.

- For \( x_1 = 65 \): \( 65 - 56 = 9 \)
- For \( x_2 = 58 \): \( 58 - 56 = 2 \)
- For \( x_3 = 68 \): \( 68 - 56 = 12 \)
- For \( x_4 = 44 \): \( 44 - 56 = -12 \)
- For \( x_5 = 48 \): \( 48 - 56 = -8 \)
- For \( x_6 = 45 \): \( 45 - 56 = -11 \)
- For \( x_7 = 60 \): \( 60 - 56 = 4 \)
- For \( x_8 = 62 \): \( 62 - 56 = 6 \)
- For \( x_9 = 60 \): \( 60 - 56 = 4 \)
- For \( x_{10} = 50 \): \( 50 - 56 = -6 \)

### Step 3: Calculate \( (x_i - \bar{x})^2 \)
Next, we will square each of the differences calculated in the previous step.

- For \( 9^2 = 81 \)
- For \( 2^2 = 4 \)
- For \( 12^2 = 144 \)
- For \( (-12)^2 = 144 \)
- For \( (-8)^2 = 64 \)
- For \( (-11)^2 = 121 \)
- For \( 4^2 = 16 \)
- For \( 6^2 = 36 \)
- For \( 4^2 = 16 \)
- For \( (-6)^2 = 36 \)

### Step 4: Sum of Squared Differences
Now, we will sum all the squared differences:
\[ \sum (x_i - \bar{x})^2 = 81 + 4 + 144 + 144 + 64 + 121 + 16 + 36 + 16 + 36 = 546 \]

### Step 5: Calculate the Variance
Finally, we will calculate the variance using the formula:
\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]
\[ \text{Variance} = \frac{546}{10} = 54.6 \]

### Final Answer
The variance of the given data set is **54.6**.

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