1. Measures of Central Tendency
Monroe and Levin and Fox
Elementary Statistics In Social Research
Chapter 3

2. 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.

3. 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.

4. 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.

5. The Mean The Mean: Here is formula for calculating the 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

6. Measures of Central Tendency
Mode: (Applies to Nominal)
It refers to the “most frequently occurring value or category.”
Median: (Applies to Ordinal and Interval)
Median is the middle: “half cases have higher values and half have lower values.” Often used to calculate income.
Mean: (Applies to Interval)
The mean is average: is calculated by adding up all of the individual values and dividing by the number of cases. Can only be computed for Interval Data.

7. Measures of Central Tendency
The Mode
The Median
The Mean (average)
What follows is a detailed review of the calculations involved with measure of Central Tendency.

8. Look at the figure below and identity the mode.
Let’s Practice!
Look at the figure below and identity the mode. 4%

9. A Review of Mode
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!

10. A Review of Mode
Another 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:

11. Ten Most Common Foreign Languages Spoken in the United States, 1990.
Number of Speakers
Spanish
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
Source: U.S. Bureau of the Census, Statistical Abstract of the United States, 2000, Table 51.

12. A Review of Mode Language Number of Speakers Is the mode 17,339,000?
NO!
Recall: The mode is the category or score, not the frequency!!
Thus, the mode is Spanish.
Language
Number of Speakers
Spanish
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

13. 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.

14. Example of a Bimodal Frequency Distribution

15. 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.

16. The Median
The Median (Mdn): Examples Odd Number of Cases: Median exactly in the middle 11, 12, 13, 16, 17, 20, 25 N = 7 (N + 1) ÷ 2 = (7 + 1) ÷ 2 = , 12, 13, 16, 17, 20, 25, 26 Mdn = 16

17. The Median
The 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 = Mdn = 16.5

18. So, now let’s look at ways we can find the median in sorted data:
Type of Data: Ordinal
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
Luis
Excellent
Karen
Total (N) 5 A
Poor
Jim
Luis
Only Fair
Bob
Good
Sue
Excellent
Karen
Total (N) 5 B

19. Calculating the median:
So, now let’s look at ways we can find the median in sorted data:
Type of Data: Ordinal
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.”
Jim
Poor
Luis
Bob
Only Fair
Sue
Good
Karen
Excellent 1 2 3 4 5

20. Calculating the median:
Another example:
The following is a list of the number of hate crimes reported in the nine largest U.S. states for 1997.
Type of Data: Interval
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

21. Calculating the median:
Type of Data: Interval
Finding the Median State for Hate Crimes
Order the cases from lowest to highest.
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

22. Finding the Median State for Hate Crimes out of Eight States
Order the cases from lowest to highest.
The median is always that point above which 50% of cases fall and below which 50% of cases fall.
For an even number of cases, there will be two middle cases.
In this instance, the median falls halfway between both cases.
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

23. The Mean The Mean: Here is formula for calculating the 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

24. Communicable Diseases: Tuberculosis
Finding the Mean
Communicable 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
N = 11
Total:
751
© World Health Organization, All rights reserved

25. Finding the Mean 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) =

26. 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. L R L R

27. 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. …

28. Contingency Tables