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Measures of Central Tendency
Monroe and Levin and FoxElementary Statistics In Social Research
Chapter 3
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Measures of central tendency:
Measures of central tendency: Measures of central tendency are numbers that describe what is average or
typical in a distribution
We will focus on three measures of central tendency:– The Mode– The Median– The Mean (average)
Our choice of an appropriate measure of central tendency depends on three factors: (a) the level of measurement, (b) the shape of the distribution, (c) the purpose of the research.
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The Mode
The Mode: The mode is the most frequent, most typical or most common value or category
in a distribution.
Example: There are more protestants in the US than people of any other religion.
The mode is always a category or score, not a frequency. The mode is the only measure of available to nominal-level variables, but can be used to describe the most common score in a distribution regardless of the level of measurement.
The mode is not necessarily the category with the majority (that is, 50% or more) of cases. It is simply the category in which the largest number (or proportion) of cases falls.
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Look at the figure below and identity the mode.
4%
Let’s Practice!
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The pie chart shows answers of 1998 GSS respondents to the question, “Would you say your own health, in general, is excellent, good, fair, or
poor?”
Note that the highest percentage (49%) of respondents is associated with the answer “good.”
The answer “good” is the mode.
Remember: The mode is used to describe nominal variables!
A Review of Mode
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A Review of ModeAnother Mode Example:
Our question is the following: “What is the most common foreign language spoken in the United States today,
as determined by the mode?”
To answer this question, let’s look at a list of the ten most commonly spoken foreign languages in the United States and the number of people who speak each foreign language:
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Language Number of SpeakersSpanish 17,339,000
French 1,702,000
German 1,547,000
Italian 1,309,000
Chinese 1,249,000
Tagalog 843,000
Polish 723,000
Korean 626,000
Vietnamese 507,000
Portuguese 430,000
Ten Most Common Foreign Languages Spoken in the United States, 1990.
Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 2000, Table 51.
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Is the mode 17,339,000?
NO!
Recall: The mode is the category or score, not the frequency!!
Thus, the mode is Spanish.
A Review of Mode
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Language Number of SpeakersSpanish 17,339,000
French 1,702,000
German 1,547,000
Italian 1,309,000
Chinese 1,249,000
Tagalog 843,000
Polish 723,000
Korean 626,000
Vietnamese 507,000
Portuguese 430,000
The Mode
Some additional points to consider about modes:Some distributions have two modes where two response categories have the
highest frequencies.
Such distributions are said to be bimodal.
NOTE: When two scores or categories have the highest frequencies that are quite close, but not identical, in frequency, the distribution is still “essentially” bimodal. In these instances report both the “true” mode and the highest frequency categories.
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Example of a Bimodal Frequency Distribution
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The Median
The Median (Mdn):The median is the score that divides the distribution into two equal parts so
that half of the cases are above it and half are below it.
The median can be calculated for both ordinal and interval levels of measurement, but not for nominal data.
It must be emphasized that the median is the exact middle of a distribution.
Calculating the Median: (N + 1) ÷ 2.
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The Median
The Median (Mdn): Examples
Odd Number of Cases: Median exactly in the middle11, 12, 13, 16, 17, 20, 25 N = 7(N + 1) ÷ 2 = (7 + 1) ÷ 2 = 4
1 2 3 4 11, 12, 13, 16, 17, 20, 25, 26Mdn = 16
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The MedianThe Median (Mdn): Examples
Even Number of Cases: Median is the point above and below which 50% of the cases fall:
11, 12, 13, 16, 17, 20, 25, 26 N = 8 (N + 1) ÷ 2 = (8 + 1) ÷ 2 = 4.5
1 2 3 4 4.511, 12, 13, 16, 17, 20, 25, 26Mdn = 16.5
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So, now let’s look at ways we can find the median in sorted data:
Let’s look at the responses (A) to the question: “Think about the economy, how would you rate economic conditions in the country today?”
First, we arrange the responses (B) in order from lowest to highest (or highest to lowest).
Since we have an odd number (5)of cases, let’s find the middle case.
Poor Jim
Good Sue
Only Fair Bob
Poor Luis
Excellent Karen
Total (N) 5
Poor Jim
Poor Luis
Only Fair Bob
Good Sue
Excellent Karen
Total (N) 5
A
B
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Type of Data: Ordinal
Calculating the median:
Jim Poor
Luis Poor
Bob Only Fair
Sue Good
Karen Excellent
We can find the median through visual inspection and through calculation.
For example, we can also find the middle case (once it has been ordered) by adding 1 to N and dividing by 2: (N + 1) ÷2.
Since N is 5, you calculate (5 + 1) ÷ 2 = 3.
The middle case is, thus, the third case (Bob), the median response is “Only Fair.”
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Type of Data: Ordinal
12345
So, now let’s look at ways we can find the median in sorted data:
Calculating the median:
State Number
California 1831
Florida 93
Virginia 105
New Jersey 694
New York 853
Ohio 265
Pennsylvania 168
Texas 333
North Carolina 42
TOTAL N = 9
Another example:The following is a list of the number of hate crimes reported in the nine
largest U.S. states for 1997.
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Type of Data: Interval
Calculating the median:
Finding the Median State for Hate Crimes
1. Order the cases from lowest to highest.
2. In this situation, we need the 5th case:
(9 + 1) ÷ 2 = 5
Which is Ohio
Remember: (N + 1) ÷2.
State Number
1. North Carolina 42
2. Florida 93
3. Virginia 105
4. Pennsylvania 168
5. Ohio 265
6. Texas 333
7. New Jersey 694
8. New York 853
9. California 1831
N = 9
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Type of Data: Interval
Finding the Median State for Hate Crimes out of Eight States
1. Order the cases from lowest to highest.
2. The median is always that point above which 50% of cases fall and below which 50% of cases fall.
3. For an even number of cases, there will be two middle cases.
4. In this instance, the median falls halfway between both cases.
5. However, the circumstances being explained should determine if you use the two middle cases (nominal) or the point (interval)halfway between both cases for your explanation.
State Number
North Carolina 42
Florida 93
Virginia 105
Pennsylvania 168
Ohio 265
Texas 333
New Jersey 694
New York 853
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The MeanThe Mean: The mean is what most people call the average. To find the mean of any distribution
simply add up all the scores and divide by the total number of scores.
Here is formula for calculating the mean
X XN
where X mean (read as X bar)
sum (expressed as the Greek letter sigma)
X raw score in a set of scoresN total number of scores in a set
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Finding the MeanCommunicable Diseases: Tuberculosis
2005
Bangladesh 37
Bhutan 44
Democratic People's Republic of Korea 103
India 58
Indonesia 47
Maldives 76
Myanmar 119
Nepal 64
Sri Lanka 71
Thailand 61
Timor-Leste 71
N = 11Total:751
© World Health Organization, 2008. All rights reserved 20
Finding the Mean:To identify the number of new tuberculosis cases found in 2005 by the WHO
in this region,
– Add up the cases for all of the countries in the region and– Divide the sum by the total number of cases.
Thus, the mean number of new tuberculosis cases found in 2005 is:
(751 ÷ 11) = 68.27.
Finding the Mean
X XN
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Finding the MeanCommunicable Diseases: Tuberculosis
2005
Bangladesh 37
Bhutan 44
Democratic People's Republic of Korea 103
India 58
Indonesia 47
Maldives 76
Myanmar 119
Nepal 64
Sri Lanka 71
Thailand 61
Timor-Leste 71
N = 11Total:751
© World Health Organization, 2008. All rights reserved 22
Using a formula to calculate the mean:The Usefulness of Formulas: The mean introduces the usefulness of a formula, which may be defined as a
is a shorthand way to explain what operations we need to follow to obtain a certain result.
Again, the formula that defines the mean is:
X XN
where X mean (read as X bar)
sum (expressed as the Greek letter sigma)
X raw score in a set of scoresN total number of scores in a set
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Deviation:
Deviation:The deviation indicates the distance and direction of any raw score from the
mean.
To find the deviation of a particular score, we simply subtract the mean from the score:
Where X = any raw score in the distribution
DeviationX X
ondistributitheofmeanX
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So what does this tell us?
The mode is the peak of the curve.
The mean is found closest to the tail, where the relatively few extreme cases will be found.
The median is found between the mode and mean or is aligned with them in a normal distribution.
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L R L R
The Concept of Relationships (92)
Contingency Tables: (Cross-Tabulation)This is a table showing the frequencies of each combination of categories on
the two variables. Can be used to present nominal (party, gender, age) and ordinal data.
Scattergrams: (Scatterplots)Relationships between two interval variables are shown in scatterplots.
Enables you to present interval data.…
Contingency Tables
Chapter Three: Review
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Review: The Mode
The Mode: The mode is the category with the largest frequency (or percentage) in the
distribution.
The mode is always a category or score, not a frequency.
The mode is not necessarily the category with the majority (that is, 50% or more) of cases.
It is simply the category in which the largest number (or proportion) of cases falls.
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Review: The Median
The Median:The median is the score that divides the distribution into two equal parts so
that half of the cases are above it and half are below it.
The median can be calculated for both ordinal and interval levels of measurement, but not for nominal data.
It must be emphasized that the median is the exact middle of a distribution.
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Review: The median:
Jim Poor
Jorge Poor
Bob Only Fair
Sue Good
Karen Excellent
Calculating the median:We can find the median through visual inspection
and through calculation.
We can also find the middle case when N is odd by adding 1 to N and dividing by 2: (N + 1) ÷2.
Since N is 5, you calculate (5 + 1) ÷ 2 = 3.
The middle case is, thus, the third case (Bob), the median response is “Only Fair.”
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Review: The MeanThe Mean: The mean is what most people call the average. It find the mean of any distribution
simply add up all the scores and divide by the total number of scores.
Here is formula for calculating the mean
X XN
where X mean (read as X bar)
sum (expressed as the Greek letter sigma)
X raw score in a set of scoresN total number of scores in a set
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Review: Measures of Central Tendency
Reasons Why Homeowners get a Home Equity Line of Credit.
Consolidate debts: 26Invest in other real estate: 3Home improvements/repairs: 45Other purposes: 9Purchase auto: 9Pay for education or medical: 4
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Review: Measures of Central Tendency
We want to know the mo, mdn, and
First, let’s arrange the scores from highest to lowest.
Home improvements/
repairs
45
Consolidate debts 26
Other purposes 9
Purchase auto 9
Pay for education or medical
4
Invest in other real estate
3
Total 96
X
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What’s the most frequent case (Mo)?– Other purposes and Purchase auto
because they both have the score of 9.
What is the middlemost score (Mdn)?– 9, because 9 + 9= 18 and if we divide
18 by 2, we get 9.
What is the mean ( )?– 16, because the sum of the scores is
96 and we divide this by 6 to get 16.
Home improvements/
repairs
45
Consolidate debts
26
Other purposes
9
Purchase auto 9
Pay for education or medical
4
Invest in other real estate
3
Total (N = 6) 96
X
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Review: Shape of the Distribution
Choosing a Measure of Central TendencyThe shape or form of a distribution can influence the researcher’s choice of a
measure of tendency.
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Review: Shape of the Distribution
The mode is the peak of the curve.
The mean is found closest to the tail, where the relatively few extreme cases will be found.
The median is found between the mode and mean or is aligned with them in a normal symmetrical/unimodal distribution.
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