Standard Deviation

Standard Deviation

Introduction

  • Standard deviation is a measure of how spread out a set of numbers is.
  • A small standard deviation indicates that the numbers are clustered closely together.
  • A large standard deviation indicates that the numbers are more spread out.
  • The standard deviation helps statisticians quickly understand the distribution of data.

Calculating Standard Deviation

Example 1: Small Data Set
  • Consider the numbers: 1, 2, 3, 4, 5.
  • Step 1: Calculate the mean xˉ\bar{x}
    • Add the numbers: 1+2+3+4+5=151 + 2 + 3 + 4 + 5 = 15
    • Divide by the number of values: 155=3\frac{15}{5} = 3. Therefore, xˉ=3\bar{x} = 3
  • Step 2: Determine the distance from the mean
    • Subtract the mean from each number:
      • 1−3=−21 - 3 = -2
      • 2−3=−12 - 3 = -1
      • 3−3=03 - 3 = 0
      • 4−3=14 - 3 = 1
      • 5−3=25 - 3 = 2
    • Note: These distances always sum to zero.
  • Step 3: Square the distances
    • This makes all values positive.
    • (−2)2=4(-2)^2 = 4
    • (−1)2=1(-1)^2 = 1
    • 02=00^2 = 0
    • 12=11^2 = 1
    • 22=42^2 = 4
  • Step 4: Calculate the Variance
    • Variance is the average of the squared distances from the mean.
    • Add the squared distances: 4+1+0+1+4=104 + 1 + 0 + 1 + 4 = 10
    • Divide by the number of values: 105=2\frac{10}{5} = 2
    • The variance is 2.
  • Step 5: Calculate the Standard Deviation σ\sigma
    • The standard deviation is the square root of the variance.
    • σ=2≈1.414\sigma = \sqrt{2} \approx 1.414
Example 2: ACT Scores
  • Consider the ACT scores: 26, 24, 21, 19.
  • Step 1: Calculate the mean xˉ\bar{x}
    • Add the scores: 26+24+21+19=9026 + 24 + 21 + 19 = 90
    • Divide by the number of scores: 904=22.5\frac{90}{4} = 22.5. Therefore, xˉ=22.5\bar{x} = 22.5
  • Step 2: Determine the distance from the mean
    • Subtract the mean from each score:
      • 26−22.5=3.526 - 22.5 = 3.5
      • 24−22.5=1.524 - 22.5 = 1.5
      • 21−22.5=−1.521 - 22.5 = -1.5
      • 19−22.5=−3.519 - 22.5 = -3.5
  • Step 3: Square the distances
    • (3.5)2=12.25(3.5)^2 = 12.25
    • (1.5)2=2.25(1.5)^2 = 2.25
    • (−1.5)2=2.25(-1.5)^2 = 2.25
    • (−3.5)2=12.25(-3.5)^2 = 12.25
  • Step 4: Calculate the Variance
    • Add the squared distances: 12.25+2.25+2.25+12.25=2912.25 + 2.25 + 2.25 + 12.25 = 29
    • Divide by the number of values: 294=7.25\frac{29}{4} = 7.25
    • The variance is 7.25.
  • Step 5: Calculate the Standard Deviation σ\sigma
    • The standard deviation is the square root of the variance.
    • σ=7.25≈2.69\sigma = \sqrt{7.25} \approx 2.69

Calculator Use

  • For large datasets, calculators or software are preferred.
  • Enter the data into a list or spreadsheet.
  • Use the calculator's statistical functions to compute one-variable statistics.
  • The calculator directly provides the standard deviation σ\sigma.
  • Note that the calculator may not directly display the variance, but it can be easily calculated by squaring the standard deviation obtained from the calculator.

Key Takeaways

  • Standard deviation measures the spread of data.
  • Variance is the square of the standard deviation.
  • Calculators simplify the calculation of standard deviation for large datasets.