Measures of Central
Tendency
New Symbols and Terminology
• The symbol
S is the Greek letter “S” and is called sigma
• This symbol means to sum (add)
• You will see it used in notations such as SX. This is pronounced as the “sum
of X” and means to find the sum of the X scores
• x̅ (X-bar) is the symbol for the mean
• Beginning with this module, we will be using common statistical formulas.
In them, we use X as the generic symbol for a score. When a formula says
to do something to X, it means to do it to all of the scores you are calling X.
3-2
Central Tendency
• Measures of central tendency answer the question:
• “Are the scores generally high scores or generally low scores?”
3-3
What is Central Tendency?
• A measure of central tendency is a statistic summarizing the
location of a distribution on a variable
• It indicates where the center of the distribution tends to be located
3-4
The Mode
• The mode is the score having the highest frequency in the data
• The mode is used to describe central tendency when the scores
reflect a nominal scale of measurement
3-5
Unimodal Distributions
When a polygon
has one hump (such
as on the normal
curve) the
distribution is called
unimodal.
3-6
Bimodal Distributions
When a distribution
has two scores
tied for the most
frequently occurring
score, it is called
bimodal.
3-7
The Median
• The median is the score at the 50th percentile
• The median is used to summarize ordinal or highly skewed interval or
ratio scores
3-8
Determining the Median
• When data are normally distributed, the median is the same score as
the mode.
• The symbol for the median is Mdn.
• Use the median for ordinal data or when you have interval or ratio
scores in a very skewed distribution.
3-9
Determining the Median
When data are normally distributed, the median is the same score as
the mode. When data are not normally distributed:
1.
2.
3.
4.
Arrange the scores from lowest to highest.
Determine N.
If N is an odd number, the median is the score in the middle position.
If N is an even number, the median is the average of the two scores in the
middle.
3 - 10
The Mean
• The mean is the score located at the exact mathematical center of a
distribution
• The mean is used to summarize interval or ratio data in situations
when the distribution is symmetrical and unimodal
3 - 11
Computing a Sample Mean
The formula for the sample mean is
SX
X 
N
3 - 12
Comparing the Mean,
Median, and Mode
On a perfect normal distribution, all three
measures of central tendency are
located at the same score.
3 - 13
Central Tendency and
Skewed Distributions
3 - 14