Transcript Median, Mode, Range - University of California, Riverside

Distribution Summaries
Measures of central tendency
Mean
 Median
 Mode

Measures of spread
Range
 Standard Deviation
 Interquartile Range (IQR)

1
Variance Formula
 X
n
 
2
i 1
X
2
i
n
Definitional Formula
2
Variance Formula


 Xi 
i 1



n
n
n
 X 
n
 
2
i 1
2
i
2
Computational Formula
3
Note


X

n
i 1
2
i


  Xi 
 i 1 
n
2
4
Other central tendencies
Median
Mode
5
Median
The median is the point in the center of
the distribution such that half of the
cases are larger and half are smaller
34 54 72 73 96 128 200
54 72 73 96 128 200
6
But, where?
When there are an even number of
observations, there is no middle
observation.
There are two middle observations
The median is between the two - half
way between the two
54 72 73 96 128 200
(73+96)/2 = 84.5
7
What happens?
If top value is raised?
54 72 73 96 128 200
54 72 73 96 128 400
Mean? Raised
Median? No change
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What happens?
If bottom value is lowered?
54 72 73 96 128 200
0 72 73 96 128 200
Mean? Lowered
Median? No change
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Mode
The mode of a distribution is the value
that has the most observations.
1 2 4 9 10 10 15 18 19 32
Mode: 10
1 2 4 9 10 15 18 19 32
No mode
10
Compare for cohabit length
Mos|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
Freq.
246
25
18
6
12
7
8
17
14
16
19
21
14
9
11
16
7
5
8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
39
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11
5
10
8
10
9
9
5
1
4
3
2
5
4
2
5
2
3
4
2
40
41
43
44
45
46
47
48
50
51
56
57
62
63
66
67
71
73
74
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1
2
1
2
1
5
2
2
2
2
1
1
4
1
2
1
2
2
1
76 |
1
79 |
1
81 |
2
97 |
2
103 |
2
-----------Tot |
626
Mode: 0
Median: 5
Mean: 11.75
11
Graphically
Mode
Median
.461661
Fraction
Mean
0
0
5 10 15 20 25 30 35 40 45 50
# Months Cohabited
103
12
Another
Fraction
.188
0
.6
.7
.8
llx
.9
1
13
Central tendency
Mean: find the “average” value
Median: find the “middle” value
Mode : find the “most common” value
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Measures of Spread
Range
Variance
Standard deviation
Interquartile range IQR
15
Quartile
Median is the half-way point -- half of
cases are above, half are below
Quartile is the quarter point
There are 3 quarter points
First quartile
 Second quartile
 Third quartile

25% below
50% below
75% below
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Consider cohabitation
# Months
|
Cohabited
|
Freq.
Percent
Cum.
------------+----------------------------------0 |
246
39.30
39.30
1 |
25
3.99
43.29
2 |
18
2.88
46.17
3 |
6
0.96
47.12
4 |
12
1.92
49.04
5 |
7
1.12
50.16
6 |
8
1.28
51.44
. . .
13 |
9
1.44
69.01
14 |
11
1.76
70.77
15 |
16
2.56
73.32
16 |
7
1.12
74.44
17 |
5
0.80
75.24
18 |
8
1.28
76.52
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Graphically
0
0
5
17
# Months Cohabited
103
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Interquartile range (IQR)
Difference between the third quartile
and the first quartile, Q3 - Q1
Compare with the range - the difference
between the maximum and the
minimum, Max - Min
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Boxplot
Graph based on median and IQR
Lower end of box: Q1
Upper end of box: Q3
Middle line: Q2 (median)
Upper whisker: 1.5*IQR above Q3
Lower whisker: 1.5*IQR below Q1
Outliers: plotted separately
20
Depression Boxplot
5
0
10
16.5
ces-d scale items s112-s123
71
IQR = 16.5 - 5 = 11.5
1.5*IQR = 17.25
Upper whisker = 16.5 + 17.25 = 33.75
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Scatterplot with boxes
Depression
ces-d scale items s112-s123
71
0
10
40
Rosenberg self-esteemscale
Esteem
22