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{12 + 24 + 38 + 412 + 512 + 62}{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.
Sources
![How to Calculate a Sample Standard Deviation](https://www.thoughtco.com/thmb/qmSnmY_b1ztjvvdp-QhdT4-DA8c=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/calculate-a-sample-standard-deviation-3126345-v4-CS-01-5b76f58f46e0fb0050bb4ab2.png)
![What Is Sample Standard Deviation Formula? Examples](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/sample-standard-deviation-formula-1612256392.png)
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