Aims of these Lectures Discuss some basic statistical - - PowerPoint PPT Presentation
Aims of these Lectures Discuss some basic statistical concepts/techniques. Relate to these to Economics. Help you to help yourself These lectures are not, and are not intended to be, a substitute for the Applied Statistics for
a substitute for the Applied Statistics for Economics and Business course.
A Useful Website:
http://www.maths.murdoch.edu.au/units/statsnotes/
A useful statistics package:
MINITAB Start Networked Applications General Software Statistics and Graphing Minitab 14
Others?
Others?
faster in the past, e.g. compare 2000 with 1500
Higher s implies larger output per capita and real
wage rate in the long run.
Higher n implies lower output per capita and real
wage rate in the long run.
Countries which are far below their long run
equilibrium will grow faster than countries which are close to their steady state.
Panel data set of Macroeconomic variables 208 countries 25 macro variables 1950 to 2000 Many missing values
available at http://www.pwt.econ.upenn.edu/ Select countries/years/variables Choose CSV option Copy and Paste data into Notepad Save as, e.g., “mydata.csv” – keep “” Open in Excel (Alternatively: use Word and save as test
file.)
Tabulate, List Numerical Summary Graphical Summary
Select a suitable set of class intervals bounded by
class limits.
The class frequency is the number of data points in
each interval.
The class mark is the midpoint of the class interval. Class Boundaries may differ from Class Limits (as a
result of rounding).
The class size is the difference between the upper
and lower class boundaries.
Suppose we have a sample of n observations. The (absolute) frequency of any value is the number of times that value appears in the sample The relative frequency of a value is the proportion of the sample which has that value. The empirical frequency distribution of a random variable is the sample analogue of its probability distribution. It can be graphed by constructing a histogram.
World Distribution of Real GDP per capita 1960, 2000
rgdp1960 rgdp2000 0-999 23 15 1000-1999 27 13 2000-2999 21 7 3000-3999 13 10 4000-4999 5 8 5000-5999 3 4 6000-6999 1 4 7000-7999 6 2 8000-8999 2 2 9000-9999 2 3 10000-10999 4 2 11000-14999 3 3 15000-19999 6 20000-24999- 10 25000 + 9
Note: Class Boundaries are, e.g. 2999.5-3999.5
Smaller than 50% of the sample and larger than 50% of the
sample
Order the sample from smallest to largest, the median lies
halfway up the order.
Let n be the sample size:
if n is odd, median is at observation (n+1)/2 if n is even, average the two values at n/2 and (n/2)+1.
A useful property: the median is insensitive to (changes in)
extreme sample values.
The First Quartile, Q1, is larger than 25% of the sample values
and smaller that 75%
The Third Quartile, Q3, is larger than 75% of the sample values
and smaller that 25%
The Second Quartile, Q2, is the Median The Interquartile Range, Q3-Q1, is a robust measure of the
variability of the sample data.
Other frequently used quartiles are deciles and percentiles.
Defined as The ‘centre of gravity’ of the distribution. Sensitive to extreme values. Gives each sample value the same ‘weight’, 1/n.
=
= + + + =
n i i n
x n n x x x x
1 2 1
1 ...
44009 14877 Maximum 482 383 Minimum 14231 2893 IQR 1590 3970 Q3 4361 2305 Median 1669 1076 Q1 9088 3332 Mean RGDP2000 RGDP1960
Stem and Leaf Plots Box Plots Bar Charts Histograms
Given a set of numbers:
The leaf is the last digit considered. The leaf unit specifies which digit. The stem is the rest of the number. The first column is the count for each stem. The count where the median occurs is
enclosed in parentheses.
Stem and Leaf for World GDP, 1960
23 0 34445556677778889999999 50 1 000001111123334455666778899 (21) 2 011223333344566667799 40 3 0000122234489 27 4 12669 22 5 238 19 6 8 18 7 334778 12 8 12 10 9 26 8 10 1469 4 11 55 2 12 4 1 13 1 14 8
N = 111, Leaf Unit = 100 i.e. lowest rgdp is in range 300-400, highest is in range 14800-14900
Skewness: A measure of asymmetry of a distribution Skewness is zero for a symmetric distribution. Positive Skewness - long tail to the right, mean greater than median. Negative Skewness - long tail to the left, mean less than median. Kurtosis: a measure of thickness of the tails
Indicate symmetry and variability of the sample values. Measuring along the horizontal or vertical axis, draw a
box with edges at Q1 and Q3 so its length is the IQR.
The width is up to you. Draw a line across the box at the median value Draw lines - whiskers - from the box to the sample
maximum and sample minimum values (excluding
Observations lying more than 1.5*IQR from the edges of
the box are ‘outliers’ and are represented by asterisks.
Boxplot of World Income Distribution, 1960
16000 14000 12000 10000 8000 6000 4000 2000
The Evolution of World Income Distribution, 1960- 2000
rgdp2000 rgdp1990 rgdp1980 rgdp1970 rgdp1960 50000 40000 30000 20000 10000
Bar Chart of World GDP per capita, 1960
1960 5 10 15 20 25 30
9 9 2
9 9 9 4
9 9 9 6
9 9 9 8
9 9 9 1
9 9 9 1 2
2 9 9 9 1 4
4 9 9 9 1 6
6 9 9 9 1 8
8 9 9 9 2
9 9 9
2
2 9 9 9 2 4
4 9 9 9 2 6
6 9 9 9 2 8 +
Bar Chart of World GDP per capita, 2000
2000 2 4 6 8 10 12 14 16
9 9 2
9 9 9 4
9 9 9 6
9 9 9 8
9 9 9 1
9 9 9 1 2
2 9 9 9 1 4
4 9 9 9 1 6
6 9 9 9 1 8
8 9 9 9 2
9 9 9
2
2 9 9 9 2 4
4 9 9 9 2 6
6 9 9 9 2 8 +
Given a sample of size n,
sample value falls into one of the classes.
class frequency.
(a) the base of each bar is the class width (b) the height is the frequency for that class
Note: Sometimes useful to have unequal bin widths.
Histogram of World GDP per capita, 1960
rgdp1960 F requency
14000 12000 10000 8000 6000 4000 2000 40 30 20 10
Histogram of World GDP per capita, 2000
rgdp2000 Frequency
40000 30000 20000 10000 35 30 25 20 15 10 5
Note: Badly chosen class intervals.
Do Poorer Countries Grow Faster?
rgdp1960 avgrowt h
16000 14000 12000 10000 8000 6000 4000 2000 0.06 0.05 0.04 0.03 0.02 0.01 0.00
Scatterplot of Growth, 1960-2000, vs Initial RGDP
Choose Y as the dependent variable and X
as the independent variable.
What a and b best represent, the data?
Could join any two points but line may be a
long way from others.
Any line drawn through the data generates a
set of residuals, some positive some negative.
The distance of a point from the line can be
measured by the squared residual.
The Least Squares criterion: ‘minimise the
sum of the squared residuals’.
n i i n i i i n i i n i i i
= = = = 1 2 1 1 2 1
0.06 b =
a =
0.28 5.50 Means 144.42 9.26 0.00 0.00 2.84 55.05 Sums 9.13 0.52
0.11 2.48 10 0.00 0.01
0.13 5.44 9 9.12 0.53
0.11 2.48 8 4.76 0.27 0.13 2.18 0.41 7.69 7 11.80 0.65
0.10 2.07 6 0.01
0.18
0.47 5.41 5 0.00 0.00 0.07
0.36 5.49 4 0.58 0.11
0.13 4.74 3 4.05 0.42
0.08 3.49 2 104.97 6.77 0.66 10.25 0.94 15.75 1 xx xy y x Y X Obs
rgdp1960 avgrowt h
15000 12500 10000 7500 5000 0.04 0.03 0.02 0.01 0.00
A verage Growth vs RGDP 1960: 30 Richest Countries
rgdp1960 avgrowt h
1200 1100 1000 900 800 700 600 500 400 300 0.05 0.04 0.03 0.02 0.01 0.00
Average Growth vs RGDP1960: 30 Poorest Countries