Measuring Center

• Median

  • The middle number

  • The median is resistant to outliers

• Mean

  • The average number

  

• Calculation

  • STAT + CALC + 1

  

• 2ND + 1 (L1)

  

• Enter

  

Measuring Position

• Percentiles

  • Percentage of observations your value is above

  • 30th percentile is the value below which 30 percent of the observations may be found

  • Take the average if there are two values

• Quartiles

  • Q1 = first quartile = 25th percentile

  • Q2 = median = 50 th percentile

  • Q3 = third quartile = 75th percentile

Measuring Spread

• Range

  • Highest - Lowest

• IQR (Interquartile Range)

  • Q3 - Q1

• Variance / Standard Deviation

  

  

• Use Sx in the TI-84 calculator

  

Outliers

• 1.5 * IQR rule

  • Q1 - 1.5 * IQR = Bottom fence

  • Q3 + 1.5 * IQR = Upper fence

  • Outside the "fence" = Outlier

• Example

Boxplots

• Graph of the "5-number Summary"

• Min Q1 Median Q3 Max

• Example

  • 18, 18, 17, 27, 22, 19, 20, 21, 18, 35, 32, 40

  • Min: 17

  • Q1: 18

  • Median: 20.5

  • Q3: 29.5

  • Max: 40

  

• Calculator

  • Boxplot

  

  

• Modified Boxplot

  

  

SOCS

• Shape

  • Skewed left/right

• Outlier

  • Q1 - 1.5 * IQR

  • Q3 + 1.5 * IQR

• Center

  • Mean or Median

• Spread

  • SD or IQR

• Example

  • 18, 18, 17, 21, 26, 40, 23, 27, 22, 19, 20, 21, 18, 35, 32

  • Skewed right

  • No outliers

  • Centered at a median of 20.5

  • A spread of IQR = 11.5

The Effect of Changing Units on Summary Measures

• Linear Transformations

  • X_new = A + B * X_old

  • A: only affect mean and median

  • B: affect all

Examples

• Construct a modified boxplot for this data set

  • Min = 56

  • Q1 = 66

  • Med = 72

  • Q3 = 77

  • Max = 100

  • IQR = Q3 - Q1 = 11

  • IQR * 1.5= 16.5

  • Bottom fence = Q1 - IQR * 1.5 = 49.5

  • Upper fence = Q3 + IQR * 1.5 = 93.5

• Describe the distribution of test scores

  • Symmetric

  • One outlier at 110

  • Centered at a median of 72

  • With a spread of IQR of 11