Download The Mean - Bakersfield College

Chapter Four
Measures of
Central Tendency:
The Mean,
Median and Mode
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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”
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 7
Bimodal Distributions
When a distribution
has two scores that
are tied for the most
frequently occurring
score, it is called
bimodal.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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 approximate
median is the score in the middle position
– If there are an even number of scores, the
approximate median is the average of the two scores
in the middle
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 11
The Mean
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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 13
Determining the Mean
• The formula for the sample mean is
SX
X 
N
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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
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
• The formula for computing a score’s
deviation is X  X
• The sum of the deviations around the
mean always equals 0
• In symbols, this is S ( X  X )
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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 18
More About Deviations
• When using the mean to predict scores, a
( X  X indicates
)
deviation
our error in
prediction
• A deviation score indicates a raw score’s
location and frequency relative to the rest
of the distribution
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 19
Population Mean
• The symbol for a population mean is m
• The formula for determining m is
SX
m
N
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 20
Summarizing Research
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Chapter 4 - 21
Summarizing an Experiment
• Summarize experiments by computing the
mean of the dependent scores in each
condition
• A relationship is present if the means from
two or more conditions are different
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distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 22
Line Graphs
Create a line
graph when the
independent
variable is an
interval or a ratio
variable.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 23
Bar Graphs
Create a bar
graph when the
independent
variable is a
nominal or an
ordinal variable.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 24
Inferring the Relationship in the
Population
1. Compute each sample mean to
summarize the scores and the
relationship found in the experiment
2. Perform the appropriate inferential
procedure
3. Determine the location of the population
of score by estimating m for each
condition
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 25
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
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 26
Example Mode
• The mode is the most frequently occurring
score
• In this data set, the mode is 14 with a
simple frequency of 6
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 27
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
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 28
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.
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 29
Example Mean
• For the mean, we need SX and N. We
know that N = 18.
SX  246
SX 246
X

 13.67
N
18
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 30
Key Terms
•
•
•
•
•
•
bar graph
bimodal distribution
deviation
line graph
mean
measure of central
tendency
• median
• mode
• sum of the deviations
around the mean
• sum of X
• unimodal distribution
© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license
distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Chapter 4 - 31