Chapter 3: Central Tendency

When reporting central tendency in research reports, the means are displayed by either

a bar or a point above each of the groups.

Central tendency

a statistical measure that determines a single value that accurately describes the center of the distribution and this single value is the best representative to summarize the entire set of data.

By identifying the "average score," central tendency allows

researchers to summarize a large set of data into a single value.

In a skewed distribution, the mode will be located at the ______ on one side and the mean usually will be displaced toward the ______ on the other side. The median is usually located

1. peak 2. tail 3. between the mean and the mode.

In addition to possible to easily compare two (or more) sets of data by

simply comparing the average score (central tendency) for one set versus the average score for another set.

Therefore, researchers have developed three commonly used techniques for measuring central tendency:

1 The mean 2. The median 3. The mode

What can change the mean?

1) Because the calculation of the mean involves every score in the distribution, changing the value of any score (except a mean value) will change the value of the mean. 2) Modifying a distribution by discarding scores or by adding new scores will usually change the value of the mean.

Give the situations where the mean does not provide a good, representative value for central tendency

1) When a distribution contains a few extreme scores (or is very skewed), the mean will be pulled toward the extremes or outliers (displaced toward the tail). In this case, the mean will not provide appropriate information for central tendency. 2) With data from a nominal scale (like 10 for male, 20 for female) it is impossible to compute a mean 3) when data are measured on an ordinal scale (ranks), it is usually inappropriate to compute a mean, while we use it to estimate mean

To determine how the mean will be affected for any specific situation you must consider two things:

1) how the number of scores is affected 2) how the sum of the scores is affected.

Conceptually, the mean can also be defined in the following two ways:

1. The mean is the amount that each individual receives when the total (ΣX) is divided equally among all N individuals. 2. The mean is the balance point of the distribution because the sum of the distances below the mean is exactly equal to the sum of the distances above the mean.

The median divides the scores so that

50% of the scores in the distribution have values that are equal to or less than the median.

Single value example

90 is a single value, 92 is a single value, 84 is a single value, etc

Is there a single procedure that always produces a good, representative value for a central tendency?

No

When will the mode, mean, and median have all the same value?

If a symmetrical distribution has only one mode

Example of adding the constant value

If you add 2 to every number, then the average increases by 2

Is the mean a real value, hypothetical value, or either one?

It can either one depending on if the average is actually distributed in the data!

When reporting central tendency in research reports, what is reported along the horizontal axis?

The different groups or treatment conditions

What is the standardized notation for reporting the median?

There is none

What is the standardized notation for reporting the mode?

There is none

For median, scores are measured on

an ordinal, interval, or ratio scale

The mode can be determined for data measured on

any scale of measurement: nominal, ordinal, interval, or ratio.

The mean is obtained by

computing the sum, or total, for the entire set of scores, then dividing this sum by the number of scores:

Usually, the median can found by a simple ________ procedure

counting

Thus, central tendency serves as a ________ statistic, because

descriptive it allows researchers to describe a set of data in a very simplified, concise form.

Computation of the man requires that are numerical values measured on an _______ or ______ scale

interval, ratio

Sometimes ________ is not the best method to use

mean

The most commonly used measure of central tendency

mean

If the scores in a distribution are listed in order from smallest to largest, the median is defined as the

midpoint of the list.

The mode is defined as the

most frequently occurring category or score in the distribution.

In a frequency distribution graph, the mode is the category or score corresponding to the

peak or high point of the distribution.

In research situations where several means are obtained for different groups or for different treatment conditions, it is common to

present all of the means in a single graph.

Computation of the median requires scores that can be placed in

rank order (smallest to largest)

In manuscripts and in published research reports, the sample mean is identified with

the letter M (or _𝑋_).

If every score is multiplied by a constant value, then

the mean is also multiplied by the same constant value.

If a constant value is added to every score in a distribution, then

the same constant value is added to the mean.

The height of the bar (or point) indicates the

value of the mean for each group