MEASURES OF

CENTRAL TENDENCY

Mean, Median, Mode

1

Numerical values used to locate the

middle of a set of data, or where the data

is clustered

The term average is often associated with

all measures of central tendency

Measures of Central Tendency

2

Arithmetic Mean

The arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are: All values are used.

It is unique.

The sum of the deviations from the mean is 0.

It is calculated by summing the values and dividing by the number of values.

3

Mean: The type of average with which you are probably most familiar.

The mean is the sum of all the values divided by the total number of values, n:

The population mean, , (lowercase mu, Greek alphabet), is the

mean of all x values for the entire population

Notes:

We usually cannot measure but would like to estimate its value

x n

x n

x x x i n = = + +

1 1 1 2 ( ) + . . .

Definition

4

Example: The following data represents the number of

accidents in each of the last 6 years at a dangerous

intersection. Find the mean number of accidents:

8, 9, 3, 5, 2, 6, 4, 5:

x = + + + + + + + =

1

8 8 9 3 5 2 6 4 5 5 25 ( ) . Solution:

In the data above, change 6 to 26:

Note: The mean can be greatly influenced by outliers

x = + + + + + + + = 1

8 8 9 3 5 2 26 4 5 7 75 ( ) . Solution:

Example

5

Population Mean

For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

6

EXAMPLE – Population Mean

7

Sample Mean

For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

8

EXAMPLE – Sample Mean

9

Properties of the Arithmetic

Mean Every set of interval-level and ratio-level data has a mean.

All the values are included in computing the mean.

A set of data has a unique mean.

The mean is affected by unusually large or small data values.

The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

10

Weighted Mean

The weighted mean of a set of

numbers X1, X2, ..., Xn, with

corresponding weights w1, w2, ...,wn, is

computed from the following formula:

11

EXAMPLE – Weighted Mean

The Carter Construction Company pays its hourly

employees $16.50, $19.00, or $25.00 per hour.

There are 26 hourly employees, 14 of which are paid

at the $16.50 rate, 10 at the $19.00 rate, and 2 at the

$25.00 rate. What is the mean hourly rate paid the

26 employees?

12

Population mean, =

Sample mean,

X = variable

N = population size

n = sample size

Ν

Χ

n

ΧX

=

Mean for Ungrouped Data

13

Mean, =

f

fX

X

Frequency (f )

fX

50

1

50

58

1

58

63

2

126

65

3

195

67

2

134

74

5

370

78

2

156

80

2

160

86

1

86

89

1

89

N = f = 20

fX= 1424

20

1424=

= 71.2

Mean for Frequency Distribution

for Ungrouped Data

14

The Mean of Grouped Data

15

Population mean, =

Sample mean,

m = class mid-point

f = class frequency

N = population size

n = sample size

Ν

Σmf

n

ΣmfX =

Mean for Grouped Data

16

Class

Interval

Class Limit

Class Mid-

point (m)

Frequency (less

than UCL) (f)

mf

50 – 54

49.5 – 54.5

52

1

52

55 – 59

54.5 – 59.5

57

1

57

60 – 64

59.5 – 64.5

62

2

124

65 – 69

64.5 – 69.5

67

5

335

70 – 74

69.5 – 74.5

72

5

360

75 – 79

74.5 – 79.5

77

2

154

80 – 85

79.5 – 84.5

82

2

164

85 – 89

84.5 – 89.5

87

2

174

N = f = 20

mf =

1420

20

1420=

=

N

mf = 71

17

We constructed a

frequency distribution

for the vehicle selling

prices. The

information is as

given. Determine the

arithmetic mean

vehicle selling price.

The Mean of Grouped Data -

Example

18

The Mean of Grouped Data -

Example

19

The Median

The Median is the midpoint of the

values after they have been ordered

from the smallest to the largest.

There are as many values above the median as

below it in the data array.

For an even set of values, the median will be the

arithmetic average of the two middle numbers.

20

Median: The value of the data that occupies the middle position when the data are ranked in order according to size

Notes:

Denoted by “x tilde”:

The population median, (uppercase mu, Greek alphabet), is

the data value in the middle position of the entire population

~x

21)~( += nxd

To find the median:

1. Rank the data

2. Determine the depth of the median:

3. Determine the value of the median

Median

21

Solution:

1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11

2. Find the depth:

3. The median is the fifth number from either end in the ranked

data:

d x(~) ( )/= + =9 1 2 5

~x =4

Example: Find the median for the set of data: {4, 8, 3, 8, 2, 9, 2, 11, 3}

Suppose the data set is {4, 8, 3, 8, 2, 9, 2, 11, 3, 15}:

1. Rank the data: 2, 2, 3, 3, 4, 8, 8, 9, 11, 15

2. Find the depth:

3. The median is halfway between the fifth and sixth observations: ~ ( )/x = + =4 8 2 6

5.52/)110()~( =+=xd

Example

22

Depth of Median =

N = Population size 2

1+N

For ungrouped data

(a) For odd N

Example: 30, 45, 48, 48, 54, 55, 60, 62, 68

N = 9

Median is at position

Median = 54

52

19=

+

Median of Ungrouped Data

23

Depth of Median =

N = Population size 2

1+N

For ungrouped data

(b) For even N

Example: 65, 65, 65, 68, 70, 74, 81, 81, 83, 86

N = 10

Median is at

5.52

110=

+

(Value at position 5 + Value at position 6) 2

Median = 722

7470=

+

Median of Ungrouped Data

24

EXAMPLES - Median

The ages for a sample of five college students are:

21, 25, 19, 20, 22

Arranging the data in ascending order gives:

19, 20, 21, 22, 25.

Thus the median is 21.

The heights of four

basketball players, in

inches, are:

76, 73, 80, 75

Arranging the data in

ascending order gives:

73, 75, 76, 80.

Thus the median is 75.5

25

Median = L + (N – s ) x c

2 f

L = LCL of median class = 69.5

N = f = total frequency = 20

s = total frequency before median class = 9

f = frequency of median class = 5

c = class size = (74.5 – 69.5) = 5

Median = 69.5 + (20 – 9) x (74.5 – 69.5) = 70.5

2 5

Median of Grouped Data

26

Class

Interval

Class Limit

Class Mid-

point (m)

Frequency (less

than UCL) (f)

cf

50 – 54

49.5 – 54.5

52

1

1

55 – 59

54.5 – 59.5

57

1

2

60 – 64

59.5 – 64.5

62

2

4

65 – 69

64.5 – 69.5

67

5

9

70 – 74

69.5 – 74.5

72

5

14

75 – 79

74.5 – 79.5

77

2

16

80 – 85

79.5 – 84.5

82

2

18

85 – 89

84.5 – 89.5

87

2

20

Median = 69.5 + (20 – 9) x (74.5 – 69.5) = 70.5

2 5

Median

class

27

Properties of the Median

There is a unique median for each data set.

It is not affected by extremely large or small

values and is therefore a valuable measure

of central tendency when such values occur.

It can be computed for ratio-level, interval-

level, and ordinal-level data.

It can be computed for an open-ended

frequency distribution if the median does not

lie in an open-ended class.

28

The Mode

The mode is the value of the observation that appears most frequently.

29

Mode: The mode is the value of x that occurs most frequently

For Ungrouped Data

(a) Distribution with one mode

Example: 52, 54, 54, 54, 57, 57, 62, 63, 63, 65

(b) Distribution with two mode or bimodal

Example: 48, 53, 62, 62, 62, 63, 65, 70, 70, 70, 75

Mode

30

Example – Mode

31

Mode = L + c

+

21

1

L = lower class limit (LCL) of modal class

1 = frequency of modal class – frequency before

2 = frequency of modal class – frequency after

c = class size

with

Mode = 69.5 + 72.8 69.5) (74.5x )1520()1020(

)1020(=

+

Mode for Grouped Data

32

Class limit

Frequency

(f)

Cumulative

Frequency

59.5 – 64.5

3

3

64.5 – 69.5

10

13

69.5 – 74.5

20

33

74.5 – 79.5

15

48

79.5 – 84.5

2

50

N = f = 50

Mode = 69.5 + 72.8 69.5) (74.5x )1520()1020(

)1020(=

+

Modal

class

Mode for Grouped Data

33

Mean, Median, Mode using Excel

Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in

Raytown, Missouri. Determine the mean and the median selling price. The mean and the median

selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the

calculations with a calculator would be tedious and prone to error.

34

Mean, Median, Mode using Excel

35

midrange=+L H

2

Midrange: The number exactly midway between a

lowest value data L and a highest value data H. It is

found by averaging the low and the high values:

Midrange

36

Example: Consider the data set {12.7, 27.1, 35.6, 44.2,

18.0}

Midrange = +

= +

= L H

2

12 7 44 2

2 28 45

. . .

When rounding off an answer, a common rule-of-thumb is to keep one more decimal place in the answer than was present in the original data

To avoid round-off buildup, round off only the final answer, not intermediate steps

Notes:

Example

37

Normal Distribution

mean

median

mode

Frequency

X

Normal Distribution

38

Frequency

X

Negatively Skewed

Frequency

X

Positively Skewed

0 +

Frequency

X

Not Normal Distribution

39

The Relative Positions of the

Mean, Median and the Mode

40

Negatively and Positively Skewed

Distribution

41

455

225==

Ν

Χ

605

300==

Ν

Χ

Group

Score

Mean

Median

A

0, 45, 50, 60, 70

50

B

40, 48, 50, 62, 100

50

Mean vs Median

42

The Geometric Mean

Useful in finding the average change of percentages, ratios, indexes, or growth rates over time.

It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other.

The geometric mean will always be less than or equal to the arithmetic mean.

The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values.

The formula for the geometric mean is written:

43

EXAMPLE – Geometric Mean

Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean.

098861151051 . ).)(.(GM ==

44

EXAMPLE 2 – Geometric Mean

The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

..).)(.)(.)(.(GM 2941808203602131 44 ===

45