Measures of Central Tendency

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 measureof available to nominal-level variables.

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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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: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.

So, now let’s look at ways we can find the median in sorted data:

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In some cases, we can find the median by simple inspection.

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 of cases, let’s find the middle case.

Poor Jim

Good Sue

Only Fair Bob

Poor Jorge

Excellent Karen

Total (N) 5

Poor Jim

Poor Jorge

Only Fair Bob

Good Sue

Excellent Karen

Total (N) 5

A

B

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Calculating the median:

Jim Poor

Jorge Poor

Bob Only Fair

Sue Good

Karen Excellent

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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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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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

North Carolina 42

Florida 93

Virginia 105

Pennsylvania 168

Ohio 265

Texas 333

New Jersey 694

New York 853

California 1831

N = 9

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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 or the point 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 median in frequency distributions:

So now, let’s find the median in frequency distributions:Often the data are arranged in frequency distributions.

The procedure is a bit more involved:– We have to find the category associated with the observation located in

the middle of the distribution.– To do this, we construct a cumulative percentage distribution.

So, let’s take a look at a frequency distribution…

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Table: Political Views of GSS Respondents, 1988

Political Views

Frequency (f)

Cf Percentage C%

Extremely Liberal

32 32 2.4 2.4

Liberal 175 207 12.9 15.3

Slightly Liberal

189 396 13.9 29.2

Moderate 502 898 37.0 66.2

Slightly Conservative

211 1109 15.6 81.8

Conservative 203 1312 15.0 96.8

Extremely Conservative

44 1356 3.2 100.00

Total 1356 100.00

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Cumulative Percentage Distribution: We construct a cumulative percentage distribution to help locate the middle of

the distribution.

The observation located in the middle of the distribution is the one that has the cumulative percentage value equal to 50%.

Notice that 29.2% of the observations are accumulated below the category of “moderate” and that 66.2% are accumulated up to and including the category “moderate.”

The median is the value of the category associated with this observation.

This middle observation falls within the category “moderate,” so the median for this distribution is “moderate.”

Cumulative Percentage Distribution:

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Table: Political Views of GSS Respondents, 1988

Political Views

Frequency (f)

Cf Percentage C%

Extremely Liberal

32 32 2.4 2.4

Liberal 175 207 12.9 15.3

Slightly Liberal 189 396 13.9 29.2

Moderate 502 898 37.0 66.2 29.2-66.2

Slightly Conservative

211 1109 15.6 81.8

Conservative 203 1312 15.0 96.8

Extremely Conservative

44 1356 3.2 100.00

Total 1356 100.00

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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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Finding the MeanCommunicable Diseases -> Tuberculosis (as of 22 March 2007) ->

Case detection rate (MDG indicator 24) -> DOTS all new case detection rate (%) -> Total

(Periodicity: Year, Applied Time Period: from 2005 to 2005)

  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

© World Health Organization, 2008. All rights reserved 22

Finding the Mean:To identify the number of new tuberculosis cases found in 2006 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 rate is (751 ÷ 11) = 68.273.

Finding the Mean

X XN

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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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The Weighted Mean

When groups differ in size, you can’t just sum their means and divide by the number of groups. Instead, you must weight each group mean by its size,

meanweighted

combinedgroupsallinnumber

groupparticularainnumber

groupparticularaofmean

X

X

XX

w

total

group

group

total

groupgroup

w

N

N

where

N

N

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Time to practice!

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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So what do you do? And then?

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)? - Home improvements/repairs 45.

What is the middlemost score (Mdn)?– 9, because (N + 1) ÷2 or (6+1)÷2=

3.5

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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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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Did you know?

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

Why is that? Well, let’s see…

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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.

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.

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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 =X∑N

where X =mean (read as X bar)

∑ = sum (expressed as the Greek letter sigma)

X = raw score in a set of scores

N = 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

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

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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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