We are generally interested in studying three important characteristics of data:

 

·        The shape of the distribution of the data (this could be uniform, bell-shaped, or skewed).

·        A representative value, such as an average.

·        A measure of scattering or variation.

 

 

Measures of Center

 

 

A measure of center is a value at the center or middle of a data set. There are several ways of determining center.

 

 

The arithmetic mean (or “average”) of a set of scores is obtained by adding the scores and then dividing this sum by the total number of scores.

 

 

Some Possibly Unpleasant Notation Related to Measures of Center

 

 

 

 

 

The median of a set of scores is the middle value when the scores are arranged in ascending order. If there are an even number of scores, the median is the mean of the two middle scores. (Notation:  )

 

 

The mode of a set of data is the value that occurs most frequently. (Notation for the mode: M)

 

 

The TI-83 can find the mean and median, but not the mode (directly). Look for these in the <LIST><MATH> menu.

 

 

A Round-Off Rule

 

How far do I round off my answer? Carry one more decimal place than is present in the original set of data.

 

 

Comparison of Measures of Central Tendency

 

Which measure is best? Which are sensitive to extreme values? Which takes every score into account?

 

Note that it does not make sense to perform numerical calculations with data at the nominal level of measurement.

 

 

As another example, for each of the 50 states, a researcher obtains the mean salary of secondary school teachers (data from the National Education Association):

 

$37,200           $49,400           $40,000           …        $37,800

 

The mean of the 50 amounts is $42,210, but is this the mean salary of all secondary school teachers in the United States? Why or why not?

 

 

Computing a Mean From a Frequency Table

 

 

We must assume that all scores within a given class have a value equal to the class mark.

 

 

In this formula, x denotes the class mark, f denotes the frequency of that class, and  is simply the sample size n.

 

 

Sometimes we wish to compute a mean of different scores, each of which carries a different weight (such as your final grade for this course). We can compute the weighted mean of such scores from the following formula:

 

 

 

 

Skewness of Data

 

 

A distribution of data is skewed if it is not symmetric and extends more to one side than to the other. (A distribution is symmetric if the left half of its histogram is roughly a mirror image of the right half.)