Calculating Standard Deviation from Deviations and Mean

The problem asks us to find the standard deviation of 10 observations. We are given the arithmetic mean and the sum of squares of deviations from a value different from the mean.

Understanding the Given Information

• Number of observations (n): \(n = 10\)
• Arithmetic mean (\(\bar{x}\)): \(\bar{x} = 60\)
• Sum of squares of deviations from 50: \(\sum_{i=1}^{n} (x_i - 50)^2 = 5000\)

We need to find the standard deviation (\(\sigma\)). The standard deviation is the square root of the variance (\(\sigma^2\)). The variance is typically calculated using the sum of squares of deviations from the mean.

Relating Deviations from an Arbitrary Point to Deviations from the Mean

We are given \(\sum (x_i - 50)^2\), but we need \(\sum (x_i - \bar{x})^2 = \sum (x_i - 60)^2\) to calculate the variance directly. There is a formula that relates the sum of squares of deviations from an arbitrary point (let's call it \(A\)) to the sum of squares of deviations from the mean (\(\bar{x}\)).

The formula is:

\(\sum_{i=1}^{n} (x_i - A)^2 = \sum_{i=1}^{n} (x_i - \bar{x})^2 + n(\bar{x} - A)^2\)

In this problem, \(A = 50\), \(\bar{x} = 60\), and \(n = 10\). We are given \(\sum (x_i - 50)^2 = 5000\). Let's plug these values into the formula:

\(5000 = \sum (x_i - 60)^2 + 10(60 - 50)^2\)

Step-by-Step Calculation

Step 1: Substitute the values into the formula.

\(5000 = \sum (x_i - 60)^2 + 10(60 - 50)^2\)

Step 2: Calculate the term \((\bar{x} - A)^2\).

\((60 - 50)^2 = (10)^2 = 100\)

Step 3: Calculate the term \(n(\bar{x} - A)^2\).

\(10 \times 100 = 1000\)

Step 4: Substitute this back into the main equation.

\(5000 = \sum (x_i - 60)^2 + 1000\)

Step 5: Solve for the sum of squares of deviations from the mean, \(\sum (x_i - 60)^2\).

\(\sum (x_i - 60)^2 = 5000 - 1000 = 4000\)

So, the sum of squares of deviations from the mean is 4000.

Step 6: Calculate the variance (\(\sigma^2\)).

The variance is the average of the sum of squares of deviations from the mean:

\(\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}\)

\(\sigma^2 = \frac{4000}{10} = 400\)

Step 7: Calculate the standard deviation (\(\sigma\)).

The standard deviation is the square root of the variance:

\(\sigma = \sqrt{\sigma^2}\)

\(\sigma = \sqrt{400}\)

\(\sigma = 20\)

The standard deviation of the observations is 20.

Revision Table: Key Statistical Measures

Additional Information on Standard Deviation Calculation

• The standard deviation provides a measure of the dispersion or spread of a dataset. A low standard deviation indicates that data points are clustered closely around the mean, while a high standard deviation indicates that data points are more spread out.
• Calculating the sum of squares of deviations from the mean is a crucial step in finding variance and standard deviation.
• The formula \(\sum (x_i - A)^2 = \sum (x_i - \bar{x})^2 + n(\bar{x} - A)^2\) is very useful when the sum of squares is given from a point other than the mean. It allows us to easily find the sum of squares from the mean without needing the individual data points (\(x_i\)).
• This relationship holds true because the sum of deviations from the mean is always zero (\(\sum (x_i - \bar{x}) = 0\)).

In this problem, using the relationship allowed us to convert the sum of squares from 50 to the sum of squares from the mean (60), which was necessary to compute the variance and subsequently the standard deviation.