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Transcript
Chapter 3
SUMMARIZING SCORES WITH
MEASURES OF CENTRAL TENDENCY
Moving Forward
Your goals in this chapter are to learn:
• What central tendency is
• What the mean, median, and mode indicate
and when each is appropriate
• The uses of the mean
• What deviations around the mean are
• How to interpret and graph the results of an
experiment
New Symbols and Procedures
• The symbol S is the Greek letter “S” and is
called sigma.
• This symbol means to sum (add).
• You will see it used with a symbol for scores
such as SX. This is pronounced as the “sum of
X” and means to find the sum of the X scores.
What is Central Tendency?
Central Tendency
• Measures of central tendency answer the
question:
– “Are the scores generally high scores or generally
low scores?”
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
Computing the Mean, Median,
and Mode
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
Unimodal Distributions
When a polygon
has one hump
(such as on the
normal curve) the
distribution is
called unimodal.
Bimodal Distributions
When a distribution
has two scores
tied for the most
frequently occurring
score, it is called
bimodal.
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
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
Determining the Median
When data are normally distributed, the median
is the same score as the mode. When data are
only approximately normally distributed:
1. Arrange the scores from lowest to highest
2. Determine N
3. If N is an odd number, the median is the score in
the middle position
4. If N is an even number, the median is the
average of the two scores in the middle
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
Computing a Sample Mean
• The formula for the sample mean is
SX
X 
N
Comparing the Mean,
Median, and Mode
• All three measures of central tendency are
located at the same score on a perfectly
normal distribution
• In a roughly normal distribution, the mean,
median, and mode will be close to the same
score
• The mean inaccurately describes a skewed
distribution
Central Tendency and
Skewed Distributions
Applying the Mean to Research
Deviations
• A score’s deviation is equal to the score minus
the mean
• In symbols, this is X  X
• The sum of the deviations around the mean is
the sum of all the differences between the
scores and the mean
• It is symbolized by S X  X


Summarizing Research
• Scores are from the dependent variable
• Choose the mean, median, or mode based on
1. The scale of measurement used on the
dependent variable and
2. The shape of the distribution, if you have
interval or ratio scores
Graphing the Results
of an Experiment
• Plot the independent variable on the X
(horizontal) axis and the dependent variable
on the Y (vertical) axis
• Create a line graph when the independent
variable is an interval or a ratio variable
• Create a bar graph when the independent
variable is a nominal or ordinal variable
Line Graphs
A line graph uses straight lines to
connect adjacent data points
Bar Graphs
The bar above
each condition
on the X axis is
placed to the
height on the Y
axis that
corresponds
to the mean
score for that
condition.
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
Example—Mode
• The mode is the most frequently occurring
score
• In this data set, the mode is 14 with a
frequency of 6
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
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 formula:
14  14
 14
2
• The median is 14
Example—Mean
• For the mean, we need SX and N
• We know N = 18
SX  246
SX 246
X

 13.67
N
18