Standard deviation formula

Mathematics
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standard deviation formula

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Standard Deviation Formula

Answer:
Standard deviation is a statistical measure that quantifies the dispersion or spread of a set of data points. It indicates how much the values in a dataset deviate from the mean (average) value. Understanding standard deviation is crucial in fields such as finance, engineering, and social sciences, where data variability plays a significant role in analysis and decision-making.

Core Concept:

  1. Population Standard Deviation (\sigma):
    The formula for the population standard deviation is given by:

    \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}

    Where:

    • \sigma is the population standard deviation
    • N is the total number of data points in the population
    • x_i represents each data point
    • \mu is the population mean, calculated as \mu = \frac{1}{N} \sum_{i=1}^{N} x_i

  2. Sample Standard Deviation (s):
    The formula for the sample standard deviation is given by:

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

    Where:

    • s is the sample standard deviation
    • n is the total number of data points in the sample
    • x_i represents each data point
    • \bar{x} is the sample mean, calculated as \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Step-by-Step Calculation:

Example Calculation for Sample Standard Deviation:

  1. Step 1: Calculate the Mean

    • Suppose we have a sample dataset: [4, 8, 6, 5, 3].

    • Calculate the sample mean \bar{x}:

      \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{1}{5} (4 + 8 + 6 + 5 + 3) = \frac{26}{5} = 5.2

  2. Step 2: Calculate Each Data Point’s Deviation from the Mean

    • Determine the difference between each data point and the mean (5.2):

      \begin{align*} (4 - 5.2) &= -1.2 \\ (8 - 5.2) &= 2.8 \\ (6 - 5.2) &= 0.8 \\ (5 - 5.2) &= -0.2 \\ (3 - 5.2) &= -2.2 \\ \end{align*}

  3. Step 3: Square Each Deviation

    • Square each calculated deviation:

      \begin{align*} (-1.2)^2 &= 1.44 \\ (2.8)^2 &= 7.84 \\ (0.8)^2 &= 0.64 \\ (-0.2)^2 &= 0.04 \\ (-2.2)^2 &= 4.84 \\ \end{align*}

  4. Step 4: Calculate the Variance

    • Sum these squared deviations and divide by (n-1) to find the variance:

      \text{Variance } (s^2) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{4} (1.44 + 7.84 + 0.64 + 0.04 + 4.84) = \frac{14.8}{4} = 3.7

  5. Step 5: Calculate the Standard Deviation

    • Take the square root of the variance:

      s = \sqrt{3.7} \approx 1.92

Final Answer:
For the sample dataset [4, 8, 6, 5, 3], the sample standard deviation is approximately 1.92.

By following these steps, you can calculate the standard deviation for any given dataset, helping you understand the variability within your data.