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.)