Standard Deviation and Variation

• Population parameter:

  Mean: µ (mew)

  Standard deviation: σ (sigma)

  Variance: σ^2

• Sample statistic

  Mean: x̄ (x bar)

  Standard deviation: s

  Variance: s^2

• Standard deviation: typical distance between mean and values in a distribution

• If a value is less than the mean is added to the data set, the distribution decreases

• Once observations become more spread out, SD (standard deviation) increases

• Either 0 or any positive number

• Not resistant to outliers

• Standard deviation: units

• Variation: units^2

Interquartile Range (IQR)

Percentile

• The pth percentile is defined as the value that P percent of values are less than or equal to it

Quartiles

• First quartile (Q1) (lower quartile) (25th percentile)

• Median (Q2) (50th percentile)

• Third quartile (Q3) (75th percentile)

• Order values in increasing order, split at median

  • Everything to the left is data for Q1, and to the right is Q3

IQR

The difference of Q1 and Q3

• Resistant to outliers, usable to show center when data isn’t symmetrical

• Identifying outliers

  • Value is an outlier if less than Q1-(1.5 x IQR) or greater than Q2-(1.5 x IQR)

Five Number Summary

1. Minimum

2. Lower Quartile (Q1)

3. Median (Q2)

4. Higher Quartile (Q3)

5. Maximum

Additive Transformation

• When a constant is added or subtracted from the original values

• Shifts center by that amount (mean/median)

• Doesn’t affect spread or shape (IQR, Range, SD)

Multiplicative Transformation

• When a constant multiplies or divides all original values

• Affects center (mean/median)

• Affects spread (IQR, Range, SD)

• Doesn’t affect spread