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How do you tabulate standard deviation?

Published in Statistics 3 mins read

You tabulate standard deviation by systematically organizing the data and calculations in a table to make the process clear and less error-prone.

Here's a breakdown of how to tabulate standard deviation:

Steps to Tabulate Standard Deviation

Here's a step-by-step guide, suitable for manual calculation, demonstrating how to tabulate standard deviation, often denoted by the symbol σ (sigma) for population standard deviation or s for sample standard deviation.

  1. Create a Table: Design a table with the following columns:

    • Column 1 (X): The original data values.
    • Column 2 (X - Mean): The deviation of each data point from the mean.
    • Column 3 (X - Mean)²: The square of each deviation from the mean.

  2. Calculate the Mean (Average): Sum all the data values (X) and divide by the number of data values (n). This is the average, often represented as μ (mu) for a population or x̄ for a sample.

    • Formula: Mean (μ or x̄ ) = ΣX / n

  3. Calculate Deviations: For each data point, subtract the mean from the data point. Record this value in the "X - Mean" column.

  4. Square the Deviations: Square each of the deviation values (from the "X - Mean" column) and record these values in the " (X - Mean)² " column. Squaring removes negative signs.

  5. Sum the Squared Deviations: Add up all the values in the " (X - Mean)² " column. This is the sum of squares (SS).

  6. Calculate the Variance:

    • Population Variance (σ²): Divide the sum of squares (SS) by the number of data values (n).
      • Formula: σ² = Σ(X - μ)² / n
    • Sample Variance (s²): Divide the sum of squares (SS) by (n-1), where n is the number of data values. Using (n-1) provides a more unbiased estimate of the population variance when working with a sample.
      • Formula: s² = Σ(X - x̄)² / (n-1)

  7. Calculate the Standard Deviation: Take the square root of the variance.

    • Population Standard Deviation (σ): The square root of the population variance.
      • Formula: σ = √σ² = √[Σ(X - μ)² / n]
    • Sample Standard Deviation (s): The square root of the sample variance.
      • Formula: s = √s² = √[Σ(X - x̄)² / (n-1)]

Example

Let's say we have the following data set: 4, 8, 6, 5, 3

  1. Table Setup:

    X X - Mean (X - Mean)²

  2. Calculate the Mean:

    • Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2

  3. Complete the Table:

    X X - Mean (X - Mean)²
    4 4 - 5.2 = -1.2 (-1.2)² = 1.44
    8 8 - 5.2 = 2.8 (2.8)² = 7.84
    6 6 - 5.2 = 0.8 (0.8)² = 0.64
    5 5 - 5.2 = -0.2 (-0.2)² = 0.04
    3 3 - 5.2 = -2.2 (-2.2)² = 4.84

  4. Sum of Squares: Σ(X - Mean)² = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8

  5. Calculate Variance (assuming a sample):

    • Sample Variance (s²) = 14.8 / (5 - 1) = 14.8 / 4 = 3.7

  6. Calculate Standard Deviation (assuming a sample):

    • Sample Standard Deviation (s) = √3.7 ≈ 1.92

Therefore, the sample standard deviation of the dataset 4, 8, 6, 5, 3 is approximately 1.92. The table provides a structured approach to these calculations, reducing the likelihood of errors.

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