11.1 Find Measures of Central Tendency and Dispersion

Statistics

Numerical values used to summarize and compare sets of data

Measure of central tendency

A number used to represent the center or middle of a set of data values. This is represented by the mean, median, and mode.

Measure of dispersion

A statistic that tells you how dispersed, or spread out, data values are

Standard deviation

A measure that describes the typical difference (or deviation) between a data value and the mean

Outlier

A value that is much greater than or much less than most of the other values in a data set

MEASURES OF CENTRAL TENDENCY

· The mean, or __average__ , of n numbers is the __sum__ of the numbers __divided__ by n. The mean is denoted by , which is read as "x-bar." For the data set x₁,x₂,…x_n, the mean is

· The median of n numbers is the __middle__ number when the numbers are written in order. (If n is even, the median is the __mean__ of the two middle numbers.)

· The mode of n numbers is the number or numbers that occur __most frequently__. There may be _one_ mode, __no__ mode, or __more than one__ mode.

Example 1

Find measures of central tendency

Quiz Scores The data sets at the right give quiz scores for two different biology classes. Find the mean, median, and mode of each data set.

Class A	Class B
15, 17, 17, 17, 18, 19, 21, 22, 25	16, 18, 19, 21, 22, 22, 22, 24, 25

Class A: Mean:

Median: __18__ Mode: __17__

Class B: Mean:

Median: __22__ Mode: __22__

STANDARD DEVIATION OF A DATA SET

The standard deviation s (read as "sigma") of x₁, x₂,…x_n is:

s =

Example 2

Find the range and standard deviation

Find the range and standard deviation for the quiz scores in each data set from Example1.

Class A: Range = __25__ - __15__ = __10__

s =

» __2.9__

Class B: Range = __25__ - __16__ = ___9___

s =

» __2.7__

Because the range and standard deviation for Class __A__ are greater, its quiz scores are __more__ spread out.

Example 3

Examine the effect of an outlier

Soccer The winning scores for the first 9 games of the soccer season are: 3, 4, 2, 5, 3,1, 4, 3, 2.

a. Find the mean, median, mode, range, and standard deviation of the data set.

b. The winning score in the next game is an outlier, 9. Find the new mean, median, mode, range, and standard deviation.

c. Which measure of central tendency does the outlier affect the most? the least?

d. What effect does the outlier have on the range and standard deviation?

Solution

a. Mean:

Median: __3__ Mode: __3__ Range: __5 - 1__ = __4__

Std. Dev.: s =

» __1.2__

b. Mean: =

Median: _3_ Mode: _3_ Range: 9 - 1 = _8_

Std. Dev.:

s =

» __1.2__

c. The _mean_ is most affected by the outlier. The _median_ and _mode_ are not affected by the outlier.

d. The outlier caused both the range and standard deviation to _increase_.