Measures of Central Tendency

Monroe and Levin and Fox

Elementary Statistics In Social Research

Chapter 3

1

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.

2

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.

3

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.

4

The Mean

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

XX

N

where X mean (read as X bar)

sum (expressedas the Greek letter sigma)

X raw score in a set of scores

N total number of scoresin a set

5

Measures of Central Tendency

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.

Measures of Central Tendency

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.

Look at the figure below and identity the mode.

4%

Let’s Practice!

8

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

9

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:

10

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

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. 11

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

12

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

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.

13

Example of a Bimodal Frequency Distribution

14

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.

15

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 = 4

1 2 3 4

11, 12, 13, 16, 17, 20, 25, 26

Mdn = 16

16

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 = 4.5

1 2 3 4 4.5

11, 12, 13, 16, 17, 20, 25, 26

Mdn = 16.5

17

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

18

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.”

19

Type of Data: Ordinal

1

2

3

4

5

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.

20

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

21

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

22

The Mean

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

XX

N

where X mean (read as X bar)

sum (expressedas the Greek letter sigma)

X raw score in a set of scores

N total number of scoresin a set

23

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 71

N = 11 Total:

751

© World Health Organization, 2008. All rights reserved 24

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

XX

N

25

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 71

N = 11 Total:

751

© World Health Organization, 2008. All rights reserved 26

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:

XX

N

where X mean (read as X bar)

sum (expressedas the Greek letter sigma)

X raw score in a set of scores

N total number of scoresin a set

27

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

Deviation X X

ondistributitheofmeanX

28

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.

29

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

32

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.

33

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.

34

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.”

35

Review: The Mean

The 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

XX

N

where X mean (read as X bar)

sum (expressedas the Greek letter sigma)

X raw score in a set of scores

N total number of scoresin a set

36

Review: Measures of Central Tendency

Reasons Why Homeowners get a Home Equity Line of Credit.

Consolidate debts: 26

Invest in other real estate: 3

Home improvements/repairs: 45

Other purposes: 9

Purchase auto: 9

Pay for education or medical: 4

37

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

38

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

39

Review: Shape of the Distribution

Choosing a Measure of Central Tendency

The shape or form of a distribution can influence the researcher’s choice of a measure of tendency.

40

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.

41