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Statistics for Economist
Chap 3. The Average and the Standard Deviation
1.
2.
3.
4.
5.
6.
7.
The Center and the Spread
The Average
The mean, median and mode
The Root-Mean-Square
The Standard Deviation
The Degrees of Freedom
The Measurement Error
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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STATISTICS
1. The Center and the Spread
Center and Spread(I)
center
center
0
25
50
75
100
0
25
spread
50
75
100
spread
 A histogram can be used to summarize large amounts
of data giving the center of the histogram and the
spread around the center.
- center : mean, median
- spread : deviation, interquartile range
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1. The Center and the Spread
STATISTICS
Center and Spread(II)
 The histogram can be
summarized by the center and
the spread, but things do not
always work out so well.
 Reporting only the center
and spread of this histogram
would miss the two peaks
-8
-6
-4 -2
0
해발 (km)
2
4
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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STATISTICS
2. The Average
Average
The average of a list of numbers equals
their sum, divided by how many there
are.
EX) data : 9, 1, 2, 2, 0
sum : 9+1+2+2+0 = 14
number of data : 5
average : 14/5 = 2.8
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STATISTICS
2. The Average
Average and the variance
average
average
average
The average is important but not sufficient.
The distribution with the same average
varies if the variances are different.
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2. The Average
STATISTICS
Age-specific average heights and weights
남성
여성
70
남성
여성
180
평균 신장(cm)
평균몸무게(kg)
80
60
50
40
30
170
160
150
140
130
120
10
20
30
40
연
50
60
70
80
령
10
20
30
40
50
연
령
60
70
80
Does the average height of people decrease after
age 20, dropping about 10cm in 60years?
Does the average weight of people decrease after
age 20, dropping about 15kg in 60years?
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2. The Average
STATISTICS
Lexis diagram
100
Cross-sectional data
O Time-series data
90
80
O3
70
Longitudinal data
연령
60
50
X
40
30
O2
20
10
O1
Examine O and X along each
45 degree line.
X2
At 1978
X1
45 o
0
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
연도
O : age 70, 157cm
X : age 21, 162cm
You need to know the type of data first!
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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3. The Mean, Median and Mode
STATISTICS
Histogram for the weight
1400
1200
1000
800
600
400
200
0
25 30 35 40
45 50 55 60 65 70 75 80
85 90 95
Weight (kg)
 The left and the right of the mean are not always same.
It is the median of a histogram that has the value with half the
area to the left and half to the right.
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3. The Mean, Median and Mode
STATISTICS
Histogram for the list 1,2,2,3
50%
25%
0%
0
1
2
3
4
If the histogram is symmetric around a median,
it equals the average.
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3. The Mean, Median and Mode
STATISTICS
The average and the histogram
list 1,2,2,3
50%
0%
0
1
3
4
5
6
7
8
list 1,2,2,5
50%
0%
0
1
2
0
1
2
3
4
5
6
4
5
6
50%
7
8
list 1,2,2,7
0%
7
8
The histograms balance when supported at the average
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3. The Mean, Median and Mode
STATISTICS
The tails of a histogram
Long right hand tail
symmetric
(right-skewed)
Average is
bigger than
median
Long left hand tail
(left-skewed)
Average is
about the same
as median
Average is
smaller than
median
Half the area under the histogram lies to the left
of the median and half to the right.
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3. The Mean, Median and Mode
STATISTICS
Histogram and the mode
 mode: the most frequent value
Think of ‘fashion’. Fashion Mode!
Histogram is highest at the mode.
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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4. The Root-Mean-Square
STATISTICS
Root-mean-square
Root-Mean-Square
how to do the arithmetic, read it backwards:
(1) SQUARE all the entries, getting rid of the
signs.
(2) Take the MEAN of the squares.
(3) Take the square ROOT of the mean.
EX) For the list 0, 5, -8, 7, -3 find the RMS
0 2  52  (8) 2  7 2  (3) 2
 29.4  5.4
5
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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5. The Standard Deviation
STATISTICS
Calculating SD
 SD is a sort of average deviation
deviation from the average = each value – average
SD  RMS of deviations from the average
Ex) for the list 20, 10, 15, 15 find the SD.
average = (20+10+15+15)/4 = 15
each deviation from the average : 5, -5, 0, 0
SD:
52  (5) 2  02  02
 16.7  4.1
4 1
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5. The Standard Deviation
STATISTICS
Meaning of SD
 SD is often helpful to think of the way a list of
numbers spreads out around the average.
Roughly 68% of the entries on a list are within 1 SD of
the average. And roughly 95% are within 2 SDs of the
average
68%
mean
-1SD
mean
mean+1SD
95%
mean-2SD
mean
mean
+2SD
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5. The Standard Deviation
STATISTICS
Histogram for the heights of women
The heights of the 5,257 women in the Health and Nutrition Examination Survey
sample. The region within 1SD of the average is shaded in (i) and 2SD in (ii)
(i)
(ii)
110 120 130 140 150 160 170 180 190 110 120 130 140 150 160 170 180 190
68 – 95 rule
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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STATISTICS
6. The Degrees of Freedom
Definition of DF
 The degrees of freedom is as same as the number of actual
independent values among all the values.
 When calculating SD, we get the average by dividing the
sum of the squares of the deviations by “number of data – 1”
and then get the root squares of the average.
 Why do we use “number of data – 1” instead of “number of
data”? The sum of the deviation has to be 0, so they cannot
all vary freely. The constraint that the sum equals 0
eliminates one degree of freedom.
DF when calculating SD = number of data - 1
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INDEX
STATISTICS
1
The Center and the Spread
2
The Average
3
The mean, median and mode
4
The Root-Mean-Square
5
The Standard Deviation
6
The Degrees of freedom
7
The Measurement Error
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STATISTICS
7. The Measurement Error
Definition
 Measurement Error
No matter how carefully it was made, a measurement
could have come out a bit differently than it did. There
exists difference between the exact value and the
measurement. Such a difference is a measurement error.
Measurement error makes an individual measurement
deviate from the exact value.
Measurement error = individual measurement – exact value
Individual measurement = exact value + measurement error
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STATISTICS
7. The Measurement Error
Size of the measurement error
 The SD of a series of repeated measurements
becomes SD of the measurement error.
 The SD is an estimate of the likely size of the
measurement error in a single measurement.
Individual measurement = exact value + measurement error
SD of individual measurement = SD of measurement error
The size of the measurement error is obtained by
SD of individual measurements
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STATISTICS
7. The Measurement Error
Bias
 Bias is the systematic error.
Ex) A butcher weighs a steak with his thumb on the
scale.
Ex) A draper store uses a cloth tape measure shrunk.
Bias affects all measurements the same way, pushing
them in the same direction. When bias is present, the
long-run average of repeated measurements will
itself be either too high or too low. The error should
not cancel out.
Individual measurement = exact value + bias + chance error
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