## Forty Randomly Selected Students Were Asked The Number Of Pairs

The standard deviation is a measure of the dispersion of a set of data. It is equal to the square root of the variance and the calculations differ depending on the type of data, whether it is a population or a sample. The formula for population standard deviation is the square root of 1/N times the sum from i=1 to N of (x_i - μ)^2, while for sample standard deviation it is the square root of 1/(N-1) times the sum from i= 1 to N of (x_i - x_bar)^2. To calculate the standard deviation, one must find the mean, the deviation of each score from the mean, and then take the square root. A sample standard deviation is a statistic calculated from a smaller sample size, while a population standard deviation is based on the entire population.

To find the sample standard deviation, we can use the following formula:

[ s = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N - 1}} ]

Where:

- ( x_i ) = each individual value
- ( \bar{x} ) = mean of the sample
- ( N ) = total number of samples
- ( s ) = sample standard deviation

Given the frequency distribution: [ X = [1, 2, 3, 4, 5, 6] ] [ \text{Frequency} = [2, 4, 8, 12, 12, 2] ]

We can calculate the sample standard deviation using the above formula:

Step 1: Calculate the mean ( \bar{x} )

[ \bar{x} = \frac{1*2 + 2*4 + 3*8 + 4*12 + 5*12 + 6*2}{2+4+8+12+12+2} ]

Step 2: Calculate the deviation of each score from the mean and its squared value:

[ (x_i - \bar{x})^2 ]

Step 3: Sum the squared deviations:

[ \sum_{i=1}^{N} (x_i - \bar{x})^2 ]

Step 4: Plug the values into the sample standard deviation formula and calculate ( s ). After that take the square root of it to get the final value.

I will go ahead and calculate ( \bar{x} ) and then proceed with the rest of the steps.

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