Download Chapter Four - Bakersfield College

Chapter Four
Measures of
Central Tendency:
The Mean,
Median, and
Mode
New Statistical Notation
• An important symbol is S, it is the Greek
letter S and is called sigma
• The symbol SX means to sum (add) the
X scores
• The symbol SX is pronounced “sum of
X”
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Chapter 4 - 2
What Is Central Tendency?
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Chapter 4 - 3
What Is Central Tendency?
• A measure of central tendency is a
score that summarizes the location of a
distribution on a variable
• It is a score that indicates where the
center of the distribution tends to be
located
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Chapter 4 - 4
The Mode
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Chapter 4 - 5
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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Chapter 4 - 6
Unimodal Distributions
When a polygon
has one hump
(such as on the
normal curve) the
distribution is
called unimodal.
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Chapter 4 - 7
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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Chapter 4 - 8
The Median
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Chapter 4 - 9
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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Chapter 4 - 10
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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Chapter 4 - 11
The Mean
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Chapter 4 - 12
The Mean
• The mean is the score located at the
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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Chapter 4 - 13
Determining the Mean
• The formula for the sample mean is
SX
X 
N
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Chapter 4 - 14
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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Chapter 4 - 15
Central Tendency and
Skewed Distributions
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Chapter 4 - 16
Deviations Around
the Mean
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Chapter 4 - 17
Deviations
• A score’s deviation is the distance
separate the score from the mean
• In symbols, this is S ( X  X )
• The sum of the deviations around the
mean always equals 0.
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Chapter 4 - 18
More About Deviations
• When using the mean to predict scores,
a deviation ( X  X ) indicates our error
in prediction.
• A deviation score indicates a raw
score’s location and frequency relative
to the rest of the distribution.
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Chapter 4 - 19
Example
• For the following data set, find the
mode, the median, and the mean
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
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Chapter 4 - 20
Example Mode
• The mode is the most frequently
occurring score.
• In this data set, the mode is 14 with a
simple frequency of 6.
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Chapter 4 - 21
Example Median
• The median is the score at the 50th
percentile. To find it, we must first place
the scores in order from smallest to
largest.
10
11
12
13
13
13
13
14
14
14
14
14
14
15
15
15
15
17
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Chapter 4 - 22
Example Median
• Since this data set has 18 observations, the
median will be half-way between the 9th and
10th score in the ordered dataset.
• The 9th score is 14 and the
10th score also is 14. To find
the midpoint, we use the
following formula.
14  14
 14
20
• The median, then is 14.
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Chapter 4 - 23
Example Mean
• For the mean, we need SX and N. We
know that N = 18.
SX  246
SX 246
X

 13.67
N
18
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Chapter 4 - 24