Describing Distributions With Numbers

Describing distributions with Numbers | SHUBLEKA     

Distribution: a brief description of a distribution must include its shape and numbers describing its center and spread. Measuring the Center: Mean Vs Median • •

Mean = “average value” Median = “middle value” Mean = x =

x1 + x2 +...+ xn n

=

∑ xi n

Example: Fuel economy MPG for 2004 vehicles (outlier: Honda Insight) ¾ The mean is sensitive to the influence of a few extreme observations that “pull it” towards the longer tail. The mean is not a resistant measure of the center of a distribution. ¾ We can make the mean as extreme as we want by significantly changing one single observation. Median = M = the midpoint of a distribution, the number such that half the observations are smaller and the other half are larger. The position of the median is

n+1 2

; if odd number of observations, the

middle observation is the median; if even number of observations, the median M is the average of the middle two observations. Demonstration: Mean and Median Applet Æ www.whfreeman.com/tps3e ¾ ¾ ¾ ¾

The median and mean of a symmetric distribution are close together If perfectly symmetric, the two measures are the same In a skewed distribution, the mean is farther out in the long tail than is the median Do not confuse “average” with “typical”

Five number summary: Min – Q1 – M – Q3 – Max ¾ The scope, quartiles, and median offer a reasonably complete description of center and spread. Example: Highway Gas Mileage: 13 18 23 27 32 ¾ Boxplot • Central box spans quartiles • A line in the box marks the median M • Lines extend from the box to the Min and Max values ¾ Interquartile Range – IQR = distance between quartiles = the range of the center half of the data = Q3 – Q1 ¾ Suspected outlier Ù it falls more than 1.5 IQR above the third quartile or below the first quartile One-Variable Statistics with the graphing calculator