Lecture Notes on Data Variation and Standard Deviation

Introduction to Variation

• Variation in a data set refers to how spread out the data is.
• It's important to understand that the mean is not the only measurement needed for understanding data.
• Example: Bank line wait times from three banks demonstrate different data spreads despite having the same mean wait time.

Measures of Variation

1. Range

   • Calculated as the highest value minus the lowest value.
   • Pros: Easiest to calculate.
   • Cons: Only considers two data points, doesn't represent the entire data set well.

2. Standard Deviation

   • More important and useful than range.
   • Measures the average distance of data values from the mean.
   • Properties:
     • Never negative.
     • Only zero if all values are the same.
     • Affected by outliers.
     • Symbol for sample standard deviation: S.

Calculating Standard Deviation

• Formula for sample standard deviation involves:
  • Finding the distance from each data value to the mean.
  • Squaring the distances to keep them positive.
  • Finding the average of these squared distances.
  • Taking the square root of the average.
• Two Formulas
  1. Using mean calculation.
  2. Without mean calculation (saves steps in large data sets).

Example Calculation

• Calculating standard deviation using examples from bank wait times.
• Step-by-step method:
  • List data values.
  • Calculate the mean.
  • Subtract mean from each value, square the result.
  • Sum the squared differences.
  • Divide by n-1 for samples, take the square root.

Population vs Sample Standard Deviation

• Population standard deviation uses the Greek letter σ and divides by N (total number of data points).
• Sample standard deviation uses S and divides by n-1.
• Different calculations for sample and population due to sample's estimation error.

Variance

• Variance is simply the square of the standard deviation.
• Symbols: S^2 for sample variance and σ^2 for population variance.

Using a Calculator

• Explanation on using statistical calculators to find standard deviation.
• Understanding outputs: S for sample, σ for population.

Empirical Rule

• Works with normally distributed data.
• 68-95-99.7 rule:
  • 68% of data falls within 1 standard deviation from the mean.
  • 95% within 2 standard deviations.
  • 99.7% within 3 standard deviations.
• Determines usual vs unusual data.
• Data value within two standard deviations is usual, outside is unusual.

Coefficient of Variation

• Used to compare variability between different data sets.
• Calculated as (Standard Deviation / Mean) * 100%.
• Provides a percentage to compare variability relative to their means.

Conclusion

• Understanding variation and standard deviation is fundamental in statistics.
• These concepts aid in hypothesis testing and data analysis.
• More advanced concepts like z-scores and normal distribution analysis will build on these foundations.