If the mean and the sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation?
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Calculating Standard Deviation from Mean and Sum of Squares
The problem asks us to find the standard deviation of 10 observations, given their mean and the sum of their squares. We are provided with the following information:
- Number of observations, \(n = 10\)
- Mean of the observations, \(\bar{x} = 40\)
- Sum of squares of the observations, \(\sum x_i^2 = 16160\)
The standard deviation (\(\sigma\)) can be calculated using the formula:
\(\sigma = \sqrt{\frac{\sum x_i^2}{n} - (\bar{x})^2}\)
This formula relates the standard deviation to the sum of squares, the number of observations, and the mean.
Step-by-Step Calculation of Standard Deviation
Let's substitute the given values into the formula:
Step 1: Substitute the values of \(\sum x_i^2\), \(n\), and \(\bar{x}\) into the formula.
\(\sigma = \sqrt{\frac{16160}{10} - (40)^2}\)
Step 2: Calculate the term \(\frac{\sum x_i^2}{n}\).
\(\frac{16160}{10} = 1616\)
Step 3: Calculate the square of the mean, \((\bar{x})^2\).
\((40)^2 = 40 \times 40 = 1600\)
Step 4: Substitute these values back into the formula under the square root.
\(\sigma = \sqrt{1616 - 1600}\)
Step 5: Perform the subtraction under the square root.
\(\sigma = \sqrt{16}\)
Step 6: Calculate the square root to find the standard deviation.
\(\sigma = 4\)
So, the standard deviation of the 10 observations is 4.
Comparing this result with the given options, we find that the value 4 matches one of the options.
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