Lecture Notes: Understanding Standard Deviation

Introduction to Standard Deviation

• Standard deviation is commonly used in finance and other fields to measure data spread.
• It helps determine the 'typical' value or middle of a dataset.
• Standard deviation measures how spread out data in a set is.

Why Know the Standard Deviation?

• Knowing the middle of a dataset (mean or median) does not provide complete information about data distribution.
• Standard deviation helps understand data variability and the overall shape of data.
• In finance, it measures price volatility; in opinion polling, it calculates the margins of error.

Examples of Standard Deviation

• Example 1: Two Companies
  • Company A: Employees earn $19 and $21; standard deviation = 1.
  • Company B: Employees earn $15, $24, and $26; standard deviation = 5.
  • Indicates how wages are spread around the average wage.

Calculating Standard Deviation

• Calculate the difference between each value and the mean.
• Square these differences.
• Take the average of these squared differences.
• Take the square root of this average to find the standard deviation.

Importance in Statistical Formulas

• Standard deviation is a key parameter in statistical formulas, notably the normal distribution.
• The normal distribution, or bell curve, is shaped by its mean and standard deviation.

Normal Distribution

• A normal distribution shows how likely it is for a random process to achieve a certain value.
• Mean determines the center of the curve; standard deviation determines its width.
• Approximately 68% of data falls within one standard deviation of the mean in a normal distribution.

Application Example: Standardized Tests

• If test scores are normally distributed with a mean of 100 and a standard deviation of 10:
  • About 66% of scores between 90 and 110.
  • About 95% between 80 and 120.
  • About 99.7% within three standard deviations.

Conclusion

• Understanding standard deviation is vital for interpreting data spread and variability.
• It provides insight into the consistency and reliability of data across various fields.