MEASURES of Central Tendency

Group 8BertolanoDorueloDuero

MasigonSantillana

Toledo

MEASURES OF CENTRAL TENDENCY

TYPES OF NUMBERS

• Discrete numbers

o Whole numbers; decimals have no meaning

• Continuous numbers

o Numbers derived from measurements

o May contain decimal places, depending on the calibration of the measurement tool

MEASURES OF CENTRAL TENDENCY

• Arithmetic mean

• Median

• Mode

ARITHMETIC MEAN

• Also referred to as “mean ()”

• Obtained by taking the sum of the observations, and then dividing the sum by the total number of values in the set. It is defined by the equation:

=Where: = mean, = sum of all observations, n = total no. of observations

Ex. :

Find the mean of the following test scores :5, 6, 7, 8, 9, 10.

=

=

= 7.43

ARITHMETIC MEAN

ARITHMETIC MEAN

• If given in a frequency distribution

multiply the values by its frequency

find the sum of the products

divide it by total no. of frequency

x f xf

$10 5 50

$20 3 60

$25 4 100

$30 6 180

$50 2 100

total 20 490

ARITHMETIC MEAN

• If given in a frequency distribution

Using the formula:

= 24.5

x f xf

$10 5 50

$20 3 60

$25 4 100

$30 6 180

$50 2 100

total 20 490

• There are 2 other ways to solve for the mean:

o Grouped Frequency Distribution

o Guessed Mean

ARITHMETIC MEAN

To determine the mean:

1. Find the midpoint of each interval.

2. Multiply the midpoint of each interval by its own frequency. Then determine their sum.

3. Then divide the sum by the total no. of frequencies

ARITHMETIC MEANGROUPED FREQUENCY DISTRIBUTION (GFD)

Ex: Find the mean of the following test scores: 18, 36, 16, 30, 21, 37, 15, 30, 31, 37, 25, 34, 26, 17, 37, 22, 27, 32, 35, 38, 25, 40, 19, 41, 28, 25, 23, 29, 25, 30, 26, 33, 17, 24, 19, 36, 30, 24, 26, 28, 23.

ARITHMETIC MEANGROUPED FREQUENCY DISTRIBUTION (GFD)

Class Interval Frequency (f)

39-41 2

36-38 6

33-35 3

30-32 6

27-29 5

24-26 9

21-23 4

18-20 3

15-17 4

Class Interval Midpoint (M) Frequency (f) Mf

39-41 40 2 80

36-38 37 6 222

33-35 34 3 102

30-32 31 6 186

27-29 28 5 140

24-26 25 9 225

21-23 22 4 88

18-20 19 3 57

15-17 16 4 64

TOTAL 42 1164

ARITHMETIC MEANGROUPED FREQUENCY DISTRIBUTION (GFD)

=

NOTE: the mean of a grouped observations may be an underestimate or an overestimate of the actual mean because of the use of a midpoint and frequency of each interval.

= 27.71 = = 27.67

ARITHMETIC MEANGROUPED FREQUENCY DISTRIBUTION (GFD)

By using the guessed mean (), we can determine the mean by using this formula:

= + (w)

where:

= mean = guessed mean

f = frequency n = total frequency

d = deviation w = class width

ARITHMETIC MEANGUESSED MEAN

To determine the mean (GFD Table is needed):

1. Make a guess of the mean.

2. Write the deviation for each interval by subtracting the guessed mean to the midpoint of each interval. Note that the sum of all the deviation scores must always be equal to zero.

3. Multiply the frequency of each interval to its deviation and find the sum of all the products.

4. Compute for the mean by using the formula

= + (w)

ARITHMETIC MEANGUESSED MEAN

Using the previous GFD table:

Suppose the guessed mean is the midpoint of 27-29 or 28

ARITHMETIC MEANGUESSED MEAN

Class Interval Deviation (d) Frequency (f) fd

39-41 4 2 8

36-38 3 6 18

33-35 2 3 6

30-32 1 6 6

27-29 0 5 0

24-26 -1 9 -9

21-23 -2 4 -8

18-20 -3 3 -9

15-17 -4 4 -16

TOTAL 42 -4

Class Interval Deviation (d) Frequency (f) fd

39-41 4 2 8

36-38 3 6 18

33-35 2 3 6

30-32 1 6 6

27-29 0 5 0

24-26 -1 9 -9

21-23 -2 4 -8

18-20 -3 3 -9

15-17 -4 4 -16

TOTAL 42 -4

ARITHMETIC MEANGUESSED MEAN

= + (w) = 28 +

1. The sum of all deviation scores is zero. A deviation is the difference between an observation and the mean.

=

2. If a constant c is added to each observation in a group whose mean is , the resulting mean is + c.

3. If each score in a set of observation is multiplied by a constant, the resulting mean is the original mean multiplied by the constant.

4. The sum of the squared deviations from the mean is a minimum.

ARITHMETIC MEANPROPERTIES OF THE MEAN

MEDIAN

• “Md”

• the middle score when the observations are arranged from highest to lowest or vice versa.

*If the no. of observations (n) is odd, the Md is the middle score of the set.

{4, 5, 6, 7, 8}

Md = 6

*If n is even, then the Md is the average of the two middle scores

{4, 5, 6, 7, 8, 9}

Md = 6.5

• If data is given in an ungrouped frequency distribution:

Set up the cumulative frequency column

Find the median rank by finding the sum of the frequencies and divide it by 2.

15/2 = 7.5The observations whose cumulative frequency contains

the median rank will be the median.

MEDIAN

x f cf85 1 15

83 3 14

81 2 11

80 3 9

78 4 6

75 2 2

• If data is given in an ungrouped frequency distribution:

Set up the cumulative frequency column

Find the median rank by finding the sum of the frequencies and divide it by 2.

15/2 = 7.5The observation whose cumulative frequency

contains the median rank will be the median.

MEDIAN

x f cf85 1 15

83 3 14

81 2 11

80 3 9

78 4 6

75 2 2

If the no. of observations (n) is large, use a GFD.

To find the median:

1. Set up a GFD table.

2. Find the cumulative frequency for each interval.

3. Find the median class ( the lowest class interval whose cumulative frequency is greater than or equal to half of “n” ()).

MEDIANGROUPED FREQUENCY DISTRIBUTION

If the no. of observations (n) is large, use a GPD.

To find the median:

4. Use the formula:

Md = L + wWhere L = lower boundary of the median class

cf = cumulative frequency of the interval below the median class

fm = frequency of the median class

w = class width

MEDIANGROUPED FREQUENCY DISTRIBUTION

• Since n = 42, = 21. Thus 27-29 is the median class

• Md = L + w = 26.5 + 3 = 27.1

MEDIANGROUPED FREQUENCY DISTRIBUTION

Class Interval Frequency (f) cf

39-41 2 42

36-38 6 40

33-35 3 34

30-32 6 31

27-29 5 25

24-26 9 20

21-23 4 11

18-20 3 7

15-17 4 4

MODE

• “Mo”

• observation that occurs most frequently in a set.

• Mostly associated with nominal data.

• A set of data is Multimodal if it has more than 1 mode

o Bimodal – 2 modes

o Trimodal – 3 modes

• If all observations only occur once, then there is no mode.

To find the mode:

• Set up a GFD table

• Find the class interval which has the highest frequency

MODEGROUPED FREQUENCY DISTRIBUTION

Class Interval Frequency (f)

58-64 5

51-57 7

44-50 10

37-43 6

30-36 4

To find the mode:

• Set up a GFD table

• Find the class interval which has the highest frequency

• Find the midpoint of that interval to get the mode. Mo = = 47

MODEGROUPED FREQUENCY DISTRIBUTION

Class Interval Frequency (f)

58-64 5

51-57 7

44-50 10

37-43 6

30-36 4