Transcript Measures of Central Tendency (MCT) Describe how MCT describe data

Measures of Central Tendency
(MCT)
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Describe how MCT describe data
Explain mean, median & mode
Explain sample means
Explain “deviations around mean”
More Statistical Notation
An important symbol is ∑, it is the Greek
letter ∑
called sigma
This symbol means to sum (add)
You will see it used in notations such as ∑
X. This is pronounced as the “sum of X”
and means to find the sum of the X scores
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Why Is It Important to Know
about MCT?
Central Tendency
MCT answer the question:
– “Are the scores generally high scores or
generally low scores?”
What are they?
– A MCT is a score that summarizes the
location of a distribution on a variable
– It is the score that indicates where the center
of the distribution tends to be located
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The Mode
The most frequently occurring score is
called the mode
The mode is typically used to describe
central tendency when the scores reflect a
nominal scale of measurement
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Unimodal Distributions
When a polygon
has one hump (such
as on the normal
curve) the
distribution is called
unimodal.
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Bimodal Distributions
When a distribution
has two scores that
are tied for the most
frequently occurring
score, it is called
bimodal.
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The Median
The Median
The median (Mdn) is the score at the 50th
percentile
The median is used to summarize ordinal
or highly skewed interval or ratio scores
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Determining the Median
When data are normally distributed, the median
is the same score as the mode.
When data are not normally distributed, follow
the following procedure:
– Arrange the scores from lowest to highest.
– If there are an odd number of scores, the median is
the score in the middle position.
– If there are an even number of scores, the median is
the average of the two scores in the middle.
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The Mean
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
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Determining the Mean
The formula for the sample mean is
X
X 
N
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Sample Mean versus
Population Mean
X is the sample mean. It is a sample
statistic.
The mean of a population is a parameter.
It is symbolized by m (pronounced “mew”).
X is used to estimate the corresponding
population mean m.
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Central Tendency and
Normal Distributions
On a perfect normal distribution all three
measures of central tendency are located
at the same score.
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Central Tendency and
Skewed Distributions
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Deviations Around
the Mean
Deviations
A score’s deviation is equal to the score
minus the mean.
 (X  X )
In symbols, this is
The sum of the deviations around the
mean
X  X 
always equals 0.
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More About Deviations
When using the mean to predict scores, a
deviation
indicates
our
error
 (X  X )
in prediction.
A deviation score indicates a raw score’s
location and frequency relative to the rest
of the distribution.
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