Download Measures of Central Tendency

Measures of Central Tendency
Measures of Central Tendency
X
Mean (Arithmetic Average)
Mean Computed from raw scores.
N
X
X
1
N
Mean (Arithmetic Average): Raw Score Formula
Score
Coach
John Buchanan 55
Holt Cornell 46
Robert Turner 50
Charlie King 51
Jim McCright 48
John Coate 50
Ed Passmore 30
David Moser 53
Jack Patton 57
Jane Benedict 62
Coach
William Foster
David O'Steen
Bill Collmer
John Kopplin
Tom Tuley
Mike Bratcher
John Achor
Joseph Karels
Lewis Flynn
Brent Stephens
N=
Sum=
Score
39
47
52
54
48
46
68
44
49
52
30
1493
Coach
Frank Young
Dave Abbott
Danny Cooper
Ron Clarke
Charles Gerbing
Winston Edwall
Dean Wyman
Rinehart Slife
Roger Frish
Michael Barker
Mean=
Score
45
33
50
51
54
59
49
42
56
53
49.77
Measures of Central Tendency
Mean (Arithmetic Average)
X
Mean Computed from grouped scores.
(A frequency Distribution)
k
X
 fX
1
N
mp
= 3408.5 =
33
103.29
IQ
ƒ
MP
ƒMP
150-159
140-149
130-139
120-129
110-119
100-109
90-99
80-89
70-79
60-69
50-59
1
0
2
3
6
8
5
5
1
1
1
154.5
144.5
134.5
124.5
114.5
104.5
94.5
84.5
74.5
64.5
54.5
154.5
0.0
269.0
373.5
687.0
836.0
472.5
422.5
74.5
64.5
54.5
N=
33
Sum= 3408.5
Mean= 103.29
Measures of Central Tendency
Mean (Arithmetic Average)
X
Mean Computed from Guessed Average
k
X

X GA  i
(  fx ' )
1
N
4
 104.5  (10)  104.5  (10)(.121)  10329
.
33
IQ
ƒ
x'
ƒx'
150-159
140-149
130-139
120-129
110-119
100-109
90-99
80-89
70-79
60-69
50-59
1
0
2
3
6
8
5
5
1
1
1
5
4
3
2
1
0
-1
-2
-3
-4
-5
5
0
6
6
6
0
-5
-10
-3
-4
-5
33
Sum=
-4
N=
Measures of Central Tendency
The Mean (Average)
• Advantages
– Most stable measure
– Can perform algebraic
operations
– Basis for advanced
statistics
– Gives us the Center of
Gravity of a dist.
– Value depends on
every score in dist.
• Disadvantages
– Weights extreme
scores more than other
measures of Central
Tendency.
To Summarize:
• Calculations for both grouped and ungrouped data.
• Use raw scores when possible
• Use grouped formula to calculate from freq..... dist. or
graph.
Measures of Central Tendency
• Guessed Average & Arbitrary Origin methods are
seldom used today, but we need to be aware of them.
•Raw score methods have the advantage of being
more precise, because they use the exact value of
every score in the distribution.
• Next we will discuss - The Median
The Median
Measures of Central Tendency
The Median
Mdn
The median is the point on the scale of
measurement above which, and below which
50% of the scores are located.
Measures of Central Tendency
The Median
Mdn
18 18 19 20 23
For raw scores the median
is just the score in the
middle.
If no score fall in the
middle, we just
interpolate.
Measures of Central Tendency
The Median
Mdn
• Median Computed from Group Data n
 
Mdn  X l  i 2
fi


f


b



IQ
150-159
140-149
130-139
120-129
110-119
100-109
90-99
80-89
70-79
60-69
50-59
f
Cum U
1
0
2
3
6
8
5
5
1
1
1
21
13
8
3
2
1
Cum D
1
1
3
6
12
20
Measures of Central Tendency
The Median
Mdn
• Median Computed from Group Data n
 
Mdn  X l  i 2
fi


f


b



33
 13
99.5  10( 2
)  10388
.
8
IQ
150-159
140-149
130-139
120-129
110-119
100-109
90-99
80-89
70-79
60-69
50-59
f
Cum U
1
0
2
3
6
8
5
5
1
1
1
21
13
8
3
2
1
Cum D
1
1
3
6
12
20
Measures of Central Tendency
The Median ...
• Advantages
– Easy to calculate.
– Not influenced by
extreme scores, so it
can be used when we
have extreme values.
– The median divides the
distribution into two
equal groups.
• Disadvantages
– It is less stable than the
mean.
– The median will not
permit all algebraic
operations (addition &
subtraction) because
we usually have
ordinal scales
The Mode
Measures of Central Tendency
• The Mode
– Ungrouped Data
• The mode is the score which occurs most frequently.
– Grouped Data
• The mode is the midpoint of the class interval containing the
largest number of cases.
– Estimating the Mode from the Mean and Median
• Mo = 3(Mdn)-2(Mean)
• For use in skewed distributions.
• Also distributions which are bimodal.
Measures of Central Tendency
The Mode ...
• Advantages
– Easy to calculate.
– It is most appropriate
for discreet Data.
– It gives the most
typical case.
• Disadvantages
– It is the least stable measure of
central tendency.
– Different size class intervals
yield different results
– If two non-adjacent class
intervals have the same
frequency, the distribution is
bimodal and the mode is
meaningless.
• May suggest two
distributions.
– Can’t perform arithmetic or
algebraic expressions with Mo.
Measures of Central Tendency
Next:
Measures of Variability