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Chapter 3

Measures of Central Tendency

I Mode

A. Definition: the Score or Qualitative Category that Occurs With the Greatest Frequency

1. Mode (Mo) for the following data, number of required textbooks for Fred’s four classes, is 2.

2 1 2 3

2

Table 1. Taylor Manifest Anxiety Scores_______________________________

(1) (2)

X j f_______________________________

74 173 172 071 270 7 Mo = 6969 868 5

67 2 66 1 65 1_______________________________

n = 28_______________________________

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II Mean

A. Definition: the Mean Is the Sum of Scores Divided by the Number of Scores

B. Formula

X =

X1 + X2 +L + Xn

n

1. X denotes the mean, X i denotes a score, and

n denotes the number of scores

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C. Summation Operator, (Greek capitol sigma)

X i =X1 + X2 +L + Xn

i=1

n∑

D. Mean Formula for a Frequency Distribution

X =

f j X jj=1

k∑

n=

f1X1 + f2X2 +L + fkXk

n1. k = number of class intervals

2. f j =frequencyofthejthclassinterval

3. X j =midpointofthejthclassinterval

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Table 2. Taylor Manifest Anxiety Scores

_________________(1) (2) (3)

X j f

_________________

74 17473 17372 0

071 2

14270 7

49069 8

55268 5

340 67 2

134 66 166

65 165

_________________n = 28

1,936_________________

f j X j

X =f j X j

j=1

k∑

n

=1,93628

=69.14

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III Median (Mdn)

A. Definition: the Median Divides Data Into Two Groups Having Equal Frequency

1. If n is odd and the scores are ordered, the medianis the (n + 1)/2th score from either end of the number line.

2. If n is even, the median is the midway point between the n/2th score and the n/2 + 1thscore from either end of the number line.

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B. Computational Examples

1. Determination of Mdn when n is odd

2. Determination of Mdn when n is even

Real limits of score

Mdn = 8

1 2 3 4 5 6 7 8 9 10 11 12

112 3 4 5 6 7 8 9 10 12

Mdn = 8.5

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3. Determination of Mdn when n is even (a) or odd (b), and the frequency of the middle score value is greater than 1

1 2 3 4 5 6 7 8 9 10 11 12

Mdn = 8

a.

1 2 3 4 5 6 7 8 9 10 11 12

8.257.75

8.00 8.50

b.

Mdn = 7.75

7.50

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4. Determination of Mdn when n is even and the frequency of the middle score value is greater than 1

Mdn = 7.833

1 2 3 4 5 6 7 8 9 10 11 12

7.833 8.167 8.5007.500

7.667 8 8.333

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C. Computation of Mdn for a Frequency Distribution

1. Formula when scores are cumulated from below

Mdn =Xll + in/ 2 − fb∑

fi

⎛

⎝⎜

⎞

⎠⎟

Xll = real lower limit of the class interval

containing the median

i = class interval size

n = number of scores

fb = number of scores below Xll

fi = number of scores in the class interval containing the median

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2. Formula for the Mdn when scores are cumulated

from above

Mdn =Xul −in/ 2 − fa∑

fi

⎛

⎝⎜

⎞

⎠⎟

Xul = real upper limit of class interval

containing the median

fa = number of scores above Xul

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__________________________74 1 173 1 272 0 271 2 470 7 1169 8 17 1968 5 967 2 466 1 265 1 1__________________________

n = 28__________________________

Table 3. Taylor Manifest Anxiety Scores_____________________________

X j

f j Cum f up Cum f down

Mdn =Xll + in/ 2 − fb∑

fi

⎛

⎝⎜

⎞

⎠⎟

=68.5+1

28 / 2 −98

⎛

⎝⎜⎞

⎠⎟

=68.5+ 0.625 =69.12

Mdn =Xul −in/ 2 − fa∑

fi

⎛

⎝⎜

⎞

⎠⎟

=69.5−1

28 / 2 −118

⎛

⎝⎜⎞

⎠⎟

=69.5−0.375 =69.12

(2) (3) (4) (1)

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IV Relative Merits of the Mean, Median, and Mode

V Location of the Mean, Median, and Modein a Distribution

Mean

Median

Mode

f

X

f

Mean

Median

Mode

X

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VI Mean of Two or More Means

A. Weighted Mean

XW =

n1X1 + n2X2 +L + nnXn

n1 + n2 +L + nn

VII Summation Rules

A. Sum of a Constant (c)

c =c+ c+L + c

nterms6 744 84 4

i=1

n∑ =nc

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B. Sum of a Variable (Vi)

Vi =V1 +V2 +L +Vn

i=1

n∑

C. Sum of the Product of a Constant and a Variable

cVi =c Vi

i=1

n∑

i=1

n∑

D. Distribution of Summation

Vi

2 + 2cVi + c2( )

i=1

n∑ = Vi

2

i=1

n∑ + 2c Vi + nc2

i=1

n∑