Solves Standard Deviation of a Dataset

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\sigma=\sqrt{\frac{\sum_{ }^{ }\left(x_i-\overline{x}\right)^2}{n}}

Standard deviation is a measure of how spread out or how much variation there is in a set of numbers. It tells us how much the numbers in a data set differ from the average (mean) of the data set.

Let’s say we have a set of numbers: 5, 10, 15, 20, and 25.

1. First, find the mean of the numbers:

Mean = (5 + 10 + 15 + 20 + 25) / 5 = 15.

2. Next, find the difference between each number and the mean:

Differences from the mean: (-10, -5, 0, 5, 10)

3. Then, square each of these differences:

Squared differences: (100, 25, 0, 25, 100)

4. Add up all the squared differences:

Sum of squared differences = 100 + 25 + 0 + 25 + 100 = 250.

5. Divide this sum by the total number of numbers (n) minus 1, and then take the square root to find the standard deviation:

Standard deviation = √(250 / (5-1)) = √(250 / 4) = √62.5 ≈ 7.91.

So, the standard deviation of this set of numbers (5, 10, 15, 20, 25) is approximately 7.91. This tells us that the numbers are spread out or vary around the average of 15.

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