Arithmetic mean of 10 observations is 60 and sum of squares of deviations from 50 is 5000, what is the standard deviation of the observation

Options

A. 20
B. 21
C. 22.36
D. 24.7

Answer

To solve the problem step by step, we need to determine the standard deviation of the observations based on the given information.

### Step 1: Understand the Given Information
- The arithmetic mean (average) of 10 observations is 60.
- The sum of squares of deviations from 50 is 5000.

### Step 2: Calculate the Total Sum of Observations
The arithmetic mean (μ) is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{N} \]
Where \(N\) is the number of observations. Rearranging gives us:
\[ \text{Sum of Observations} = \text{Mean} \times N \]
Substituting the values:
\[ \text{Sum of Observations} = 60 \times 10 = 600 \]

### Step 3: Use the Sum of Squares of Deviations
We know that the sum of squares of deviations from a point (in this case, 50) is given as 5000. This can be expressed as:
\[ \sum (x_i - 50)^2 = 5000 \]

### Step 4: Expand the Sum of Squares
Using the formula for the sum of squares of deviations:
\[ \sum (x_i - 50)^2 = \sum (x_i^2 - 100x_i + 2500) \]
This can be rewritten as:
\[ \sum x_i^2 - 100 \sum x_i + 2500N = 5000 \]
Substituting \(\sum x_i = 600\) and \(N = 10\):
\[ \sum x_i^2 - 100 \times 600 + 2500 \times 10 = 5000 \]
\[ \sum x_i^2 - 60000 + 25000 = 5000 \]
\[ \sum x_i^2 - 35000 = 5000 \]
\[ \sum x_i^2 = 40000 \]

### Step 5: Calculate the Variance
The variance (\(σ^2\)) is calculated using the formula:
\[ \sigma^2 = \frac{1}{N} \sum (x_i - \mu)^2 \]
Where \(\mu\) is the mean. We can express this as:
\[ \sigma^2 = \frac{1}{N} \left( \sum x_i^2 - 2\mu \sum x_i + N\mu^2 \right) \]
Substituting the known values:
\[ \sigma^2 = \frac{1}{10} \left( 40000 - 2 \times 60 \times 600 + 10 \times 60^2 \right) \]
Calculating each term:
- \(2 \times 60 \times 600 = 72000\)
- \(10 \times 60^2 = 36000\)

Now substituting these values back:
\[ \sigma^2 = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) \]
\[ \sigma^2 = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) = \frac{1}{10} \left( 40000 - 72000 + 36000 \right) \]

### Step 6: Calculate the Standard Deviation
The standard deviation (\(σ\)) is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} \]
Substituting the variance we calculated:
\[ \sigma = \sqrt{2500} = 50 \]

### Final Answer
The standard deviation of the observations is **50**.