What Is Standard Deviation ?

Standard deviation, often represented by the symbol σ (sigma), is a measure of how spread out the values in a data set are around the mean (average). In other words, it tells us the average distance of each data point from the mean. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation shows that the data points are more spread out.

History

Key Points About Standard Deviation:

Calculating σ: To find standard deviation:
- Find the mean of the data set.
- Calculate the squared differences between each data point and the mean.
- Find the average of these squared differences (this is the variance).
- Take the square root of the variance to get the standard deviation.
How to calculate Variance ?
Definition: The average of the squared differences from the Mean.
Steps to Calculate:
1. Find the Mean (average of the numbers).
2. For each number, subtract the Mean and square the result (squared difference).
3. Calculate the average of these squared differences.

Interpretation:
- σ provides insight into variability within the data. For example, in manufacturing, if the standard deviation is low, it means the process produces items that are consistently similar, which is desirable for quality control.
- In a normal distribution (bell curve), approximately 68% of values lie within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ.

Use in Quality Control:
- In control charts, sigma levels help set control limits (usually ±3σ), helping to identify when a process may be out of control due to excessive variability.

Overall, standard deviation is a fundamental tool for understanding data variability and maintaining quality in many industries.