The key difference between overall and within standard deviation lies in what variation they account for: overall standard deviation considers all variation, while within standard deviation attempts to isolate variation within subgroups, removing variation between subgroups.
Here's a more detailed breakdown:
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Overall Standard Deviation (σ): This is the standard deviation calculated using all the data points in a dataset, treating them as a single group. It reflects the total variability present, including both the variation within subgroups and the variation between them. This is your "ordinary" or "classical" standard deviation calculation. It estimates the population standard deviation.
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Within Standard Deviation (σw): This standard deviation focuses on the variation within subgroups or samples. The goal is to remove or minimize the impact of variation between these subgroups, giving a more precise measure of the inherent variability of the process or population if the subgroups were more homogenous. This is particularly useful in statistical process control (SPC) and when analyzing data from experiments with multiple groups. Within standard deviation is often calculated by averaging the standard deviations of the individual subgroups, pooling their variances.
To illustrate, consider a manufacturing process where parts are produced in batches each day.
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Overall Standard Deviation: This would measure the variation in the part's dimension across all parts produced over several days. It would include variations due to normal process fluctuations, as well as any shifts in the process from day to day (e.g., slight machine adjustments).
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Within Standard Deviation: This would measure the variation in the part's dimension within each day's batch. It attempts to eliminate or minimize the variation caused by shifts in the process between days, focusing only on the variability that occurs during a single day's production.
When to Use Which:
- Use the Overall Standard Deviation when you want to understand the total variation in your data, without regard for any subgrouping. It is used when you want to estimate the population standard deviation from your sample.
- Use the Within Standard Deviation when you want to isolate and understand the variation that is inherent within each subgroup, minimizing the influence of differences between subgroups. This is useful for identifying and addressing sources of variation within a controlled environment.
- The "Within" standard deviation is useful for removing variation due to "special cause" variations.
Example Scenario:
Imagine you're measuring the height of students in three different schools.
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Calculating the overall standard deviation of all students' heights would give you a general sense of the variability in height across the entire student population from all three schools.
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Calculating the within standard deviation would involve calculating the standard deviation of heights within each school and then combining those values. This would give you a sense of how much heights vary within each individual school, removing the influence of any average height differences between the schools.
In summary, the "within" standard deviation is useful when you want to understand the variation within groups, excluding the variation between groups, offering a more accurate representation of the inherent variability within each group. The "overall" standard deviation provides a broader measure of variability across the entire dataset.
