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Chapter 12: Problem 41

Describe how the standard deviation is computed.

Short Answer

Expert verified
The standard deviation is computed using the formula \( \sigma = \sqrt{\dfrac{1}{N-1} \sum_{i=1}^{N} (x_i - \overline{x})^2 }\). It involves calculating the mean, computing the deviation of each data point from this mean, squaring those deviations, dividing the sum of these squares by the degrees of freedom (N-1), and then taking the square root of that result.

Step by step solution

01

Understanding Standard Deviation

Standard deviation is a measure of how spread out numbers are. It gives an idea of how much the individual data points in a dataset are likely to differ from the mean (average) value of the set.
02

Introduce the Formula

The formula for the standard deviation (\( \sigma \)) of a sample is given by: \[ \sigma = \sqrt{\dfrac{1}{N-1} \sum_{i=1}^{N} (x_i - \overline{x})^2 }\] where \(N\) is the number of data points, \(\overline{x}\) is the mean of the data points and \(x_i\) is each individual data point.
03

Explain Each Component

Let's break down each component of this formula: - \(N-1\) is the degrees of freedom. We use \(N-1\) instead of \(N\) to make our estimate of the population standard deviation more accurate. - \(\overline{x}\) is the mean of the data - the sum of all data points divided by the number of data points. - The sum symbol \(\sum\) adds up the results for each data point. - \(x_i - \overline{x}\) gives the deviation of each data point from the mean. - \((x_i - \overline{x})^2\) squares the deviation, which removes any negative values. - \(\sqrt{\dfrac{1}{N-1} \sum_{i=1}^{N} (x_i - \overline{x})^2 }\) is finally the square root of the mean of these squared deviations, giving the standard deviation.
04

Calculation Example

Assume a simple data set: 1, 2, 3. - First, calculate the mean: (1 + 2 + 3) / 3 = 2. - Then calculate each data point's deviation from the mean: (1-2, 2-2, 3-2) = (-1, 0, 1). - Next, square each of those:((-1)^2, 0^2,1^2) = (1, 0, 1). - Now, sum up these squares: 1 + 0 + 1 = 2. - Divide this sum by the degrees of freedom (N-1): 2 / (3-1) = 1. - Finally, take the square root of this quotient: \(\sqrt{1} = 1\). Therefore, the standard deviation of this data set is 1.

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