Chapter 4 – 1

Chapter 4:Measures of Central Tendency

• What is a measure of central tendency?• Measures of Central Tendency

– Mode– Median– Mean

• Shape of the Distribution• Considerations for Choosing an Appropriate

Measure of Central Tendency

Chapter 4 – 2

• Numbers that describe what is average or typical of the distribution

• You can think of this value as where the middle of a distribution lies.

What is a measure of Central Tendency?

Chapter 4 – 3

Central Tendency

• The mode

The most frequently value.

• The median

A value in the middle of distribution, 50% vs. 50%.

• The mean

“the average”.

Chapter 4 – 4

• The category or score with the largest frequency (or percentage) in the distribution.

• The mode can be calculated for variables with levels of measurement that are: nominal, ordinal, or interval-ratio.

The Mode

Chapter 4 – 5

The Mode: An Example

• Example: Number of Votes for Candidates for Mayor. The mode, in this case, gives you the “central” response of the voters: the most popular candidate.

Candidate A – 11,769 votes The Mode:

Candidate B – 39,443 votes “Candidate C”

Candidate C – 78,331 votes

Chapter 4 – 6

Example

• Mode

Mode = 7

• Modal category

Modal category

= ’20-39’

X= 6 7 8

f= 2 4 3

age 0-19 20-39 40+

f= 40 60 55

Chapter 4 – 7

The Median

• The score that divides the distribution into two equal parts, so that half the cases are above it and half below it.

• The median is the middle score, or average of middle scores in a distribution.

Chapter 4 – 8

The Median

• Calculation

1st, place the values in an array, from lowest to highest (sorting).

2nd, find the median position by using the following formula

Md. Pos = (n+1)/2

3rd, find the value associated with the median position.

Chapter 4 – 9

Example

• X=

Md. Pos = (9+1)/2

=5

Median = 7

• X=

Md. Pos = (6+1)/2

=3.5

Median=(7+7)/2=7

p1

p2

p3

p4

p5

p6

p7

p8

p9

6 6 6 7 7 7 8 8 8

p1

p2

p3

p4

p5

p6

6 6 7 7 7 7

Chapter 4 – 10

Median Exercise #1 (N is odd)

Calculate the median for this hypothetical distribution:

Job Satisfaction Frequency

Very High 2

High 3

Moderate 5

Low 7

Very Low 4

TOTAL 21

Chapter 4 – 11

Median Exercise #2 (N is even)

Calculate the median for this hypothetical distribution:

Satisfaction with Health Frequency

Very High 5

High 7

Moderate 6

Low 7

Very Low 3

TOTAL 28

Chapter 4 – 12

Finding the Median in Grouped Data

( 1)*.5N CfMedian L w

f

L = lower limit of the interval containing the median

W = width of the interval containing the median

N = total number of respondents

Cf = cumulative frequency corresponding to the lower limit

f = number of cases in the interval containing the median

Chapter 4 – 13

Finding the Median in Grouped Data

• From this we can see that there are 50 results. To find the median add 1 to this number and halve the result.

• We are therefore looking for the 25.5th result. This lies in the fourth group, which starts at 24. 

• We must work out how far though the group we must go. Our result is 1.5 into the group (25.5-24) and the group has a frequency of 11.

x(cm)

(0,10] (10,20] (20,25] (25,30] (30,40] (40,50]

f 5 9 10 11 8 7

cf 5 14 24 35 43 50

Chapter 4 – 14

( 1)*.5

(50 1)*.5 2425 *5 25.682

11

N CfMedian L w

f

Finding the Median in Grouped Data

• We therefore must go 1.5/11 of the way through the group

• The group covers values from 25cm to 30 cm, and therefore has a 5cm width. We need to find a fraction of 5cm.

• Therefore the median is 25+(1.5/11)=25.682

Chapter 4 – 15

Example

x 1-4 5-8 9-11 12-15 16-20 21-24

f 2 5 9 10 11 6

cf 2 7 16 26 37 43

( 1)*.5N CfMedian L w

f

(43 1)*.5 1612 *4

1014.4

Chapter 4 – 16

Percentiles

• A score below which a specific percentage of the distribution falls.

• Finding percentiles in grouped data:

( 1)*.2525%

N CfL w

f

Chapter 4 – 17

The Mean

• The arithmetic average obtained by adding up all the scores and dividing by the total number of scores.

Chapter 4 – 18

The Mean• Formula for the Mean

X the mean value of the variable x

1 2 3 4 ...i nx x x x x x

Xn n

“X bar” equals the sum of all the scores, X, divided by the number of scores, N.

Chapter 4 – 19

The mean

1 1 2 2 ...i i i in n

i

f x f xf x f x f xX

N N f

•Formula for the mean w/ grouped scores

where: fi Xi = a score multiplied by its frequency

Chapter 4 – 20

Example• X=

8 8 7 7 6 6

Σx=8+8+7+7+6+6

=42

n=6

• X= f=

8 2

7 2

6 2

Σfx=2×8+2×7+6×2 =42

N=2+2+2=6

427

6X 42

76

X

Chapter 4 – 21

Mean: Grouped Scores

Chapter 4 – 22

Mean: Grouped Scores

Chapter 4 – 23

Grouped Data: the Mean & Median

Number of People Age 18 or older living in a U.S. Household in 1996 (GSS 1996)

Number of People Frequency

1 1902 3163 544 175 26 2

TOTAL 581

Calculate the median and mean for the grouped frequency below.

Chapter 4 – 24

Shape of the Distribution

• Symmetrical (mean is about equal to median)• Skewed

– Negatively (example: years of education)

mean < median– Positively (example: income)

mean > median• Bimodal (two distinct modes)• Multi-modal (more than 2 distinct modes)

Chapter 4 – 25

Skewness

• When a curve is skewed, we indicate the direction of skewness with reference to the longer tail.

• Skewed to the right or positively skewed

• Skewed to the left or negatively skewed

Chapter 4 – 26

Chapter 4 – 27

Chapter 4 – 28

Chapter 4 – 29

Chapter 4 – 30

Chapter 4 – 31

Distribution Shape

Chapter 4 – 32

Considerations for Choosing a Measure of Central Tendency

• For a nominal variable, the mode is the only measure that can be used.

• For ordinal variables, the mode and the median may be used. The median provides more information (taking into account the ranking of categories.)

• For interval-ratio variables, the mode, median, and mean may all be calculated. The mean provides the most information about the distribution, but the median is preferred if the distribution is skewed.

Chapter 4 – 33

Central Tendency