03 central tendency and disperson

MEASURES OF CENTRALMEASURES OF CENTRAL
TENDENCYTENDENCY
06/22/16 eccphd@yahoo.com
• Mean
• Median
• Mode
 These are indices
representing the average or
typical score attained by a
group of subjects.
 This value must best
describe the group and be a
representative of all the
observations.

Mean
Scores: 89, 90, 95, 92, 91, 88, 89
_
X = ∑X/n
Mean = (89 + 90 + 95 + 92 + 91 + 88 + 89)/7
= 634/7
= 90.57
the sum of all
scores divided by
the total number
of cases
_
X

Median
Scores: 89, 90, 95, 92, 91, 88,
89
88, 89, 89, 90 , 91, 92, 95
Mdn = 90
the score located
at the middle of
the distribution
Arrange the
scores in
order

Mode
Scores frequency
89 4
90 2
95 1
92 1
91 1
88 2
Mode = 89
the score with the
highest frequency

Calculate the mean, median and modeCalculate the mean, median and mode
of the following scores:of the following scores:
10 17 16
14 31 10
32 30 27
10 23 28
35 20 12

ANSWERSANSWERS
Median = 20
10
10
10
12
14
16
17
20
23
27
28
30
31
32
35
Mode = 10
10
10
10
12
14
16
17
20
23
27
28
30
31
32
35
315
n = 15
Mean = 315/15
= 21

MEASURES OFMEASURES OF
VARIABILITYVARIABILITY
• It indicates how spread or
scatter the scores are in the
distribution (measure of
dispersion).
Ex: Range, Mean Average
Deviation, Variance, SIQR,
Standard Deviation

RANGERANGE
Group A SCORES: 88, 89, 89, 90 , 91, 92,
95
Range = 95 - 88
= 7
Group B SCORES: 75, 80, 89, 90 , 90, 92,
95
Range = 95 - 75
= 20
 the difference between the
highest and the lowest score in a
distribution•
 the most primitive way of determining
variability of scores

STANDARDSTANDARD
DEVIATIONDEVIATION
the sum of squares of the score
deviations from the distribution mean
divided by the total number of cases
less than 1
s = ∑x2
n – 1 _
where ∑x2
= ∑(X – X)2

Case 1: 4, 9, 7, 9, 10, 25
X x x2
4 -6.7 44.49 Mean = 10.67
9 -1.7 2.79
7 -3.7 13.47 s = 269.33
9 -1.7 2.79 6 – 1
10 -0.7 0.45
25 14 205.35 = 53.87
64 269.33 s = 7.34
Calculate the standard deviation of the
following scores:

X x x2
88 0.5 0.25 Mean = 87.50
90 2.5 6.25
87 -0.5 0.25
87 -0.5 0.25 s = 13.50
88 0.5 0.25 6 - 1
85 -2.5 6.25 = 2.7
525 13.50 s = 1.64
Case 2:Case 2: 88, 90, 87, 87, 88, 8588, 90, 87, 87, 88, 85

Standard Deviation
X = 0-3s +1s-1s-2s +2s +3s

Case 2 Illustration
X =87.5
82.58 89.1485.8684.22 90.78 92.42
x 87.5
s 1.64
1s 1.64
2s 3.28
3s 4.92

Standard Deviation
X-3s
68%
+1s-1s-2s +2s +3s

Standard Deviation
X-3s
98%+
+3s

Standard Deviation
X-3s
95%
-2s +2s +3s

In most cases, the
standard deviation is used
to describe the degree of
dispersion, spread or
variation of scores in a
distribution.

Calculate the standard deviation
of the following scores:
Scores
A 6 3
Scores
B 2 4
4 4 6 6
7 4 6 7
2 3 7 8
10 9 3 8
Which group is more scattered?

Calculate the standard deviation
of the following scores:
26 31 32 24
41 24 16 46
37 42 26 70
12 30 37 28
10 19 33 18

Measures of Location
• decile
•quartile
• percentile

Distribution of scores

PercentilePercentile
Formula:
xN
P =
100
where
x = the desired percentile
rank
N = number of cases

IllustrationIllustration
Determine the score equivalent to
percentile 80 in the following test
results.
45 80 45
65 23 67
67 43 66
87 56 89
65 29 78
54 88 76
52 75 72
89 67 56
76 34 55
89 51 45

ProceduresProcedures
1. Arrange the scores in an ascending
order
23 55 75
29 56 76
34 56 76
43 65 78
45 65 80
45 66 87
45 67 88
51 67 89
52 67 89
54 72 89
lowest
highest

2. Compute P80 using the formula.
P = xN/100
= (80)(30)/100
= 24 or
P80 is the 24th
score in the
distribution.
23 55 75
29 56 76
34 56 76
43 65 78
45 65 80
45 66 87
45 67 88
51 67 89
52 67 89
54 72 89P80

P80 is the 24th
score in the distribution (P80
= 78).
23 55 75
29 56 76
34 56 76
43 65 78
45 65 80
45 66 87
45 67 88
51 67 89
52 67 89
54 72 89
Thus, if your score is
78, that means you
surpassed 79% of the
class or 20% have
scores higher than
you have got.

P10 P20 P30 P40 P50 P60 P70 P80 P90
P80
80% 20%

P10 P20 P30 P40 P50 P60 P70 P80 P90
P50
50% 50%

DecileDecile
Formula:
xN
D =
10
where
rank
N = number of cases

D1 D2 D3 D4 D5 D6 D7 D8 D9
D5
50% 50%

QuartileQuartile
Formula:
xN
Q =
4
where
rank
N = number of cases

Q1 Q3
D5
p50
Q2

SummarySummary
 Percentile (divides the distribution into
100 equal parts)
 Decile (divides the distribution into 10
equal parts)
 Quartile (divides the distribution into
4 equal parts)

End of
Session

03 central tendency and disperson

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