Transcript BCB702_Chapter_4

Measures of Location
INFERENTIAL STATISTICS & DESCRIPTIVE STATISTICS
Statistics of location
Summarise a central point
Statistics of dispersion
Summarises distribution
around central point
Measures of Location
ARITHMETIC MEAN
Sum all observation, then divide by number of observations
For a sample:
X

X 
n
For a population:
X


n
Measures of Location
18
No. of People
16
X=7.07
14
12
10
8
6
4
2
0
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5 11.5
Nightly Hours of Sleep
Measures of Location
MEDIAN
MEDIAN
Value that has equal no. of observations (n) on either side
5
For N = 15
the median is
the eighth
score = 37
Frequency
4
3
2
1
0
33
34
35
36
37
Score
38
39
40
Measures of Location
MEDIAN
Value that has equal no. of observations (n) on either side
5
For N = 16
the median is the
average of the
eighth and ninth
scores = 37.5
Frequency
4
3
2
1
0
33
34
35
36
37
Score
38
39
40
Measures of Location
MODE
• the most frequently occurring score value
• corresponds to the highest point on the frequency
distribution
5
4
Frequency
For a given sample N=16:
33 35 36 37 38 38 38 39
39 39 39 40 40 41 41 45
The mode = 39
3
2
1
0
33 34 35 36 37 38 39 40 41 42 43 44 45
Score
Measures of Location
Measures of central tendency
Summary
Advantages
Mode
quick & easy to compute
 useful for nominal data

poor sampling stability
not affected by extreme
scores

somewhat poor
sampling stability

inappropriate for
discrete data
affected by skewed
distributions


Median
sampling stability
 related to variance

Mean
Disadvantages

Measures of Location
DISPERSION
These are measures of how the
observations are distributed
around the mean
Measures of Location
Range
DISPERSION:
•
largest score minus the smallest score
16
these two
have same range (80)
but spreads look different
14
# of People
•
12
10
8
6
4
2
0
0
10
20
30
40
50
60
70
80
Scores
•
says nothing about how scores vary around the center
•
greatly affected by extreme scores (defined by them)
90 100
Measures of Location
DISPERSION:
Score
Deviation
Amy
10
-40
Theo
20
-30
Max
30
-20
Henry
40
-10
Leticia
50
0
Charlotte
60
10
Pedro
70
20
Tricia
80
30
Lulu
90
40
SUM
0
Variance
mean = 50
To see how ‘deviant’ the
distribution is relative to
another, we could sum
these scores
But this would leave us
with a big fat zero
Measures of Location
Variance
DISPERSION:
Score
Deviation
Sq. of
deviation
Amy
10
-40
1600
Theo
20
-30
900
Max
30
-20
400
Henry
40
-10
100
Leticia
50
0
0
Charlotte
60
10
100
Pedro
70
20
400
Tricia
80
30
900
Lulu
90
40
1600
0
6000
SUM
So we use squared
deviations from
the mean, which
are then summed
This is the sum
of squares (SS)
SS= ∑(X-X)2
Measures of Location
DISPERSION:
Variance
For a sample:
SS
s 
n 1
2
For a population:
SS
 
N
2
(to correct for the fact that
sample variance tends to
underestimate pop variance)
We take the “average”
squared deviation
from the mean and
call it VARIANCE
Measures of Location
Standard deviation
DISPERSION:
The standard deviation is the square root of the variance
The standard deviation measures spread in the original
units of measurement, while the variance does so in
units squared.
Variance is good for inferential stats.
Standard deviation is nice for descriptive stats.
SS
s s 
n 1
2
Measures of Location
DISPERSION
N = 28
X = 50
s2 = 140.74
s = 11.86
# of People
14
12
10
8
6
4
2
0
0
10 20 30 40 50 60 70 80 90 100
Scores
N = 28
X = 50
s2 = 555.55
s = 23.57
Measures of Location
DISPERSION
For a sample:
For a population:
Mean
Variance
Standard
Deviation
SS
s 
n 1
SS
 
N
s s
  2
2
2
2
Measures of Location
DISPERSION
The Standard Error, or Standard Error of the Mean, is an
estimate of the standard deviation of the sampling distribution
of means, based on the data from one or more random
samples e.g. 15 students each compile data sets of the
heights of 20 people
Numerically, it is equal to the square root of the quantity
obtained when s squared is divided by the size of the sample.
s = s
X
n
and

X 
n