3.1Averages and Variation

February 7, 2012

Three types of “average”:mean, median, and mode

Mode: the value that occurs most frequently

Find the mode:

6, 7, 2, 3, 4, 6, 2, 6

Find the mode:

6, 7, 2, 3, 4, 5, 9, 8

There is no mode for this data.

The Median

the central value of an ordered distribution

To find the median of raw data:

•Oorder the data from smallest to largest.•Ffor an odd number of data values, the median is the middle value.•Ffor an even number of data values, the median is found by dividing the sum of the two middle values by two.

Find the median:

Data:5, 2, 7, 1, 4, 3, 2

Rearrange:1, 2, 2, 3, 4, 5, 7

The median is 3.

Find the median:

Data: 31, 57, 12, 22, 43, 50

Rearrange: 12, 22, 31, 43, 50, 57

The median is the average of the middle two values =

372

4331

The Mean

The mean of a collection of data is found by:•ssumming all the entries•ddividing by the number of entries

entriesofnumber

entriesallofsummean

Find the mean:

6, 7, 2, 3, 4, 5, 2, 8

6.4625.48

37

8

82543276

mean

Sigma Notation

•Tthe symbol means “sum the following.”

• is the Greek letter (capital) sigma.

Notations for mean

Sample mean “x bar”

Population mean

Greek letter (mu)

x

Number of entries in a set of data

•Iif the data represents a sample, the number of entries = n.•Iif the data represents an entire population, the number of entries = N.

Sample mean

n

xx

Population mean

N

x

Resistant Measure

a measure that is not influenced by extremely high or low data values

Which is less resistant?

•Mmean•Mmedian

The mean is less resistant. It can be made arbitrarily large by increasing the size of one value.

Trimmed Mean

a measure of center that is more resistant than the mean but is still sensitive to specific data values

To calculate a (5 or 10%) trimmed mean

•Oorder the data from smallest to largest.•Ddelete the bottom 5 or 10% of the data.•Ddelete the same percent from the top of the data.•Ccompute the mean of the remaining 80 or 90% of the data.

Compute a 10% trimmed mean:15, 17, 18, 20, 20, 25, 30, 32, 36, 60

•Ddelete the top and bottom 10%•Nnew data list:17, 18, 20, 20, 25, 30, 32, 36•110% trimmed mean =

8.248

198

n

x

Pg. 96-98

2,4,8,11