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How do you find the standard deviation of a control chart?

Published in Control Chart Stats 3 mins read

The standard deviation of a control chart, often represented as sigma (σ), is found by estimating the process variation. This is critical for setting the control limits. The method used depends on the subgroup size.

Estimating Standard Deviation

The standard deviation isn't directly calculated but rather estimated using different approaches based on the size of your subgroups (samples) used in the control chart:

  • For Subgroups of 2-10:

    • You calculate the range (R) within each subgroup, which is the difference between the largest and smallest observation within that subgroup.
    • Then, you compute the average range (R̄), which is the mean of all the subgroup ranges.
    • Finally, you estimate the standard deviation using this formula: σ = R̄ / d₂ where d₂ is a constant that depends on the subgroup size. This d₂ value can be found in statistical tables and is different for each subgroup size. The constant d₂ is available in statistical tables.
  • For Subgroups Greater Than 10

    • You calculate the standard deviation (S) for each subgroup.
    • Then, calculate the average standard deviation (S̄) across all subgroups.
    • The standard deviation is then estimated by: σ = S̄ / c₄ where c₄ is a constant that depends on the subgroup size. The constant c4 is available in statistical tables.

Here's a summary in table format:

Subgroup Size (n) Calculation Method Formula
2 ≤ n ≤ 10 Average Range (R̄) R̄ / d₂
n > 10 Average Standard Deviation (S̄) S̄ / c₄

Example

  • Suppose you have a control chart using subgroups of 5. The average range (R̄) across all subgroups is 2.3. From statistical tables, we know that d₂ for subgroup size of 5 is 2.326. The estimated standard deviation would then be: σ = 2.3 / 2.326 = 0.989.
  • Suppose you have a control chart using subgroups of 15. You calculate the standard deviation (S) for each subgroup, and find that the average standard deviation (S̄) is 3.5. From statistical tables, we know that c₄ for a subgroup size of 15 is 0.9727. The estimated standard deviation would then be: σ = 3.5 / 0.9727 = 3.6.

Key Points

  • The use of d₂ or c₄ is essential. These constants account for bias when estimating the population standard deviation from sample data.
  • The choice of method depends entirely on your subgroup size.
  • Standard deviation is crucial for calculating control limits, thus, ensuring the chart's effectiveness at identifying out-of-control points.

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