Chapter 4 Measures of Center STAT 1301 Chapter 4 Measures of Center

It is often difficult to work with complete distributions. So, we SUMMARIZE Descriptive Measures of Center Spread Today, we will concentrate on measures of “center”

Histogram Families by Size in 1988 Distribution of Families by size in 1988 Familty Size Family Size Source: Population Survey data tape

Gas Mileage for Compact Cars % per unit mpg Miles per Gallon

Schematic Representations of Histogram Symmetric Long Right Tail Long Left Tail (skewed to the right) (skewed to the left)

number of observations Measures of Center Average - arithmetic mean AVG = Median - middle observation from ordered data - middle value for an odd number of observations - average of 2 middle values for even # of obs. Mode - most frequently occurring observation not necessarily unique does not always exist sum of observations number of observations

WARNING ! Averages are sensitive to extreme values.

Salary Data Employee Hourly Wage 110-15-2436 5.00 109-16-4134 5.00 110-15-2436 5.00 109-16-4134 5.00 015-16-4134 5.00 101-45-1362 5.00 515-60-4142 5.00 612-45-3627 6.00 413-21-6561 6.00 218-35-4425 7.00 806-56-7132 8.00 Mr. Pearson 35.00

Examples 1995 - Duke Univ. graduates of Dept. of Communications had an average starting salary of $418,000 - Grant Hill (NBA player) Data on Household Income - which should be larger - AVG or median? 2002 – US household income data - AVG $57,208 - Median $43,057

“Center” of Histogram Average - histogram balances Median - divides histogram into 2 equal parts based on area Mode - modal class is the class interval with the highest bar

“Center” of Histogram

Root Mean Square (RMS) RMS size of a list: (S) square values in list (M) sum squared values and divide by total # of values in list (R) take square root sum of squared values RMS = # of values

RMS measures size of values in list ignoring signs “sort of like average ignoring sign”