There are three measures of central tendency:
the mean, the median, and the mode.
Now we get to talk about the most commonly used measure, the mean.
The mean is the score located at the mathematical center of the distribution.
It's most commonly used with interval and ratio level data.
And it is the most likely to be called the average,
although the mode or the median could truthfully be called an average.
When the scores in the data set form a normal distribution,
the mean equals the median equals the mode.
in order to calculate the mean,
you add up all of the scores in your data set and then divide by the total number of scores.
This is the formula. In this formula for mean (or M) the summation symbol is
sigma.
Remember that sigma means to add up the scores that follow.
Each raw score in the data set is an X.
ΣX means add up all of the X scores and create the sum of X, and
N is the number of scores for people or cases.
"N" is our term for sample size.
In this formula, the mean symbol is M, and here
I'm using
M to represent the mean. Now, this is APA style 6th edition,
but another way that you will often see the mean represented is as X̄, in
fact, any letter with a line over it is a mean. So Ȳ is the mean of Y.
z̅ is the mean of z.
Now to calculate the mean you add up all of the scores in the data set (sum of X), and
then divide by the total number of scores (N).
The measure of central tendency that you choose to report depends upon your data,
so it is important to know when we should choose the mean instead of the median or the mode.
The mean has many advantages when we do statistical analysis,
but the mean can only be used with interval or ratio data that are not excessively skewed.
In order to determine if the skewness is excessive, we can divide the skewness value by the
standard error of the skewness values that we get from SPSS.
Values greater than +/-1.96 are
excessive, and you can use this same technique for kurtosis and
standard error of the kurtosis as well.
The mean has some disadvantages that are important to understand. For instance, the mean may not actually exist in the sample,
such as the average family having 2.3 children.
But the mean can still be useful for planning, such as determining
how many new students will be entering a public school.
Based upon the number of families moving into a community, each family having an average of 2.3 children.
Now the mean is more susceptible to outliers than any other measure of central tendency,
but it takes quite a few outliers or especially large outliers to really shift the mean dramatically.
Skewed distributions can also create a misleading mean.
For example, the median income in the United states is $42,000.
Half of Americans make $42,000 a year or less,
but the mean income in the United States is
$60,000 nearly half again higher.
The mean income data are skewed by a relatively small number of millionaires and billionaires
whose excessive wealth shifts the income figures upwards and makes the income mean
non-representative of the typical American. Now, on the other hand,
ehen the outliers are not extreme or the outliers are balanced on both ends of the distribution,
the mean is still relatively stable and can be used for analysis.
The mean is the measure of choice for central tendency.
You should report the mean when the variables are at the scale level and not highly skewed. In other words, the
outliers are on only one end of the distribution, creating that positive or negative skew.
You should also report the mean when you plan on doing additional parametric statistical analyses such as t-tests, ANOVA,
correlation, or regression.
Parametric statistics require the use of the mean.