07 measures of central tendency

8/13/2019 07 Measures of Central Tendency

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Measures ofCentralTendency


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Measures of Central Tendency

Central Tendency

any measure indicating the centerof a set of data, arranged in an

increasing or decreasing order ofmagnitude.

Most commonly used Measures ofCentral Tendency:

meanmedian

mode


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a calculated averagemost widely used

Mean

If the set of data x 1 , x 2 , …,x n , not necessarilyall distinct, represents a finite sample of sizen, the mean is

n

x

x

n

i

i

1

Mean


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Mean

Example:

A food inspector examined a random sample of7 cans of certain brand of tuna to determine

the percent of foreign impurities. Thefollowing data were recorded: 1.8, 2.1, 1.7,1.6, 0.9, 2.7 and 1.8. Compute the mean.

7

8.17.29.06.17.11.28.1 x

8.1 x


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Weighted Mean

An average in which each quantity to beaveraged is assigned a weight.

n

i i

n

iii

w

xw x

1

1

Course Grade Units

TOUR 119 1.25 4

TOUR 116 2.75 3

HRM 117 1.50 3

PHL 2 2.75 3

PDSR 1.00 3

FRENCH 2 5.00 3

ENG 3 1.25 3

IE 210 1.75 3

TOUR 111 1.00 3


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Median

a rank or position averageless widely used than mean

Median

The median of a set of observations arrangedin an increasing or decreasing order ofmagnitude is the middle value when the numberof observations is odd or the arithmetic mean of

the two middle values when the number ofobservations is even.


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Median

If odd: positionn x 21

~

32

15~ x

where: n = no. of observations

Example:

On 5 term tests in zoology a student has madegrades of 82, 93, 86, 92 and 79. Find themedian of the grades.

79 82 86 92 93

86~ x

position of the median


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Median

If even:

where: n = no. of observations

Example:The nicotine contents for a random sample of6 cigarettes of a certain brand found to be 2.3,2.7, 2.5, 2.9, 3.1 and 1.9 milligrams. Find themedian.

3.1 2.9 2.7 2.5 2.3 1.9

2

1

22~ position posi tion

nn x

2

5.27.2

2

43

2

12

6

26

~

thrd x

6.2~ x


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Mode

an inspection averagemost frequently occurring value

Mode

The mode of a set of observations is that valuewhich occurs most often or with the greatestfrequency.

Example:The number of movies attended last month bya random sample of 12 high school studentswas recorded as follows: 2, 0, 3, 1, 2, 4, 2, 5,2, 0, 1 and 4. Find the mode.

2ˆ

x


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Mode

*Remember:The mode does not always exist.

Example:

Krizzy’s scores in her Filipino quizzes are asfollows: 10, 11, 19, 20, 15 and 0. Find themode of her scores.

- No mode exist since each score occurs onlyonce.


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Mode

*Remember:For some set of data there may be several

values occurring with the greatest frequencyin which case we have more than one mode.

Example:

Jelaine’s scores in her Botany quizzes are 7, 6,3, 7, 7, 6, 4, 4 and 6. Find the mode.

- In this case, there are two modes, 6 and 7since both of them occur with the greatestfrequency. The distribution is said to be

bimodal.


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Measures of Central Tendencyfor Grouped Data

TABLE I

Score Distribution of Students in Literature Quiz

CI f CB LCB UCB

Class Mark(x)


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Mean:

N

fx x

56.1695

1573 x

where: f = frequency

x = class mark

N = total frequency

Example: Find the mean of Table I.


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TABLE I


CI f CB LCB UCB

Class Mark(x)


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Median: where: LB = lower class boundary of themedian class (locate n/2


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TABLE I


CI f CB LCB UCB

Class Mark(x)


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Mode: where: LB = lower class boundary of themodal class (highest frequency)D1 = f mo – f beforeD2 = f mo – f afterc = interval size

Example: Find the mode of Table I.

c D D D

LB x

21

1ˆ

318211921

1921

5.15ˆ

x

332

25.15

ˆ

x

7.16ˆ x

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