1.2 Describing Distributions with NumbersAP Statistics

Fuel Economy for 2004 car models

2-seater Cars Compact cars

Model City HWY Model City Hwy

Acura NSX 17 24 Ashton Mar 12 19

Audi 20 28 Audi TT 21 29

BMW z4 20 28 BMW 325 19 27

Cadillac XLR 17 25 BMW 330 19 28

Corvette 18 25 BMWM3 16 23

Miata 22 28 Jaguar XK8 18 26

Viper 12 20 Jaguar XKR 16 23

Ferrari 360 11 16 Lexus SC 18 23

Ferrari M 10 16 Mini Cooper 25 32

Honda I 60 66 Mits Eclipse 23 31

Thunderbird 17 23 Porsche 911 14 22

Lotus 15 22 Mits Spyder 20 29

Mean and Median

Mean: The Average

Median: The Middle

Mean Highway mileage for two seaters

1 2 ... 51824.7

21 21nx x x

x

Caution!!!

The mean is sensitive to the influence of a few extreme observations. It is not a resistant measure of center.

Measuring Spread

The simplest useful numerical description of a distribution includes a description of both the center and the spread.

Range: simplest measure of spread. The highest value minus the lowest value.

Pth percentile of a distribution is the percent of observations that fall below it.

The Quartiles

Calculating Quartiles

The highway mileage of 20 2-seaters arranged in numerical order are:

13 15 16 16 17 19 20 22 23 23 |23 24 25 25 26 28 28 28 29 32

The median is marked by |.

The 1st Quartile (Q1) is the median of the 10 numbers to the left of the median.

The 3rd Quartile (Q1) is the median of the 10 numbers to the right of the median.

If there are an odd number of observations, the median is the middle number and it is excluded from calculations of Q1 and Q3.

The Five–Number Summary and Boxplots

The Five–Number Summary and Boxplots

Boxplots of the highway and city gas mileages for cars classified as two-seaters and minicompacts.

Outliers

Lower Bound: Q1 – 1.5(IQR)•If an observation falls below the lower bound it is considered an outlier.

Upper Bound: Q3 + 1.5(IQR)•If an observation falls above the upper bound it is considered an outlier.

Measuring Spread: Variance and Standard Deviation

Some FAQs Why do we square the deviations?

If not, they would all add to zero. Why do we emphasize the standard deviation rather

than the variance?The standard deviation measures spread about the mean

in the original scale. Why do we average by dividing by n-1 rather than n

in calculating the variance? Because the sum of the deviations is always zero, the last

deviation can be found once we know the other n-1. The number n-1 is called the degrees of freedom.

Choosing Measures of Center and Spread