Normal Distributions

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example 2

Normal Distributions

  • Also known as the Z distribution
    • Mean is 0
    • Standard deviation is 1
  • Characteristics of a Normal Distribution
    • Continuous Random Variable
    • - ∞ < x < + ∞
    • Curve is symmetrical around the mean (m).
    • Area under curve = 1
    • Mean & standard deviation uniquely determine a normal distribution.

Normal Distribution




value of Z




To find  P(a < X < b)  when  X  is distributed normally:

  1. Draw the normal curve for the problem in terms of X
  2. Translate X-values to Z-values and put Z values on your diagram
  3. Use the Standardised Normal Table

Example: Suppose X is normally distributed with mean 8 and std dev 5. Find P(X < 8.6)

Z value example






Z value table









Finding the X for a Known Probability:


  1. Draw a normal curve placing all known values on it such as mean of X and Z
  2. Shade in area of interest and find cumulative probability
  3. Find the Z value for the known probability
  4. Convert to X units using the formula:


How Large is Large Enough?

  • For most population distributions, n ≥ 30 will give a sampling distribution that is nearly normal
  • For fairly symmetric population distributions, n ≥ 5 is sufficient
  • For normal population distributions, the sampling distribution of the mean is always normally distributed



  • A point estimate is the value of a single sample statistic
  • A confidence interval provides a range of values constructed around the point estimate

Point Estimates







Confidence Level  (1-a)

  • Common confidence levels = 90%, 95% or 99%
    • Also written (1 – a) = 0.90, 0.95 or 0.99
  • A relative frequency interpretation
    • In the long run, 90%, 95% or 99% of all the confidence intervals that can be constructed (in repeated samples) will contain the unknown true parameter
  • For example, if we were to randomly select 100 samples and use the results of each sample to construct 95% confidence intervals, approximately 95 out of 100 would contain the population mean

 Confident Interval for Population when standard deviation is known












Finding the critical value Z











So, what happens if we don’t know the standard deviation of the population????

  • If the population standard deviation σ  is unknown, we can substitute the sample standard deviation, S
  • This introduces extra uncertainty, since S is variable from sample to sample
  • So we use the t distribution instead of the normal distribution

Confidence Interval Estimate:

Interval estimate



where t is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail

t distribution increasing










A random sample of n = 25 has X = 50 & S = 8.

Form a 95% confidence interval for μ:

d.f. = n – 1 = 24,  so

example of distribution





The confidence interval is  46.698 ≤ μ ≤ 53.302


Required Sample Size Example

If s = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?

example 2



So the required sample size is n = 220






Basic Math for Economists – Logs

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We have learnt indices or exponents in the algebra material. If you haven’t check then, we recommend you to do so. We will need those concepts for progressing with logs. The idea of logarithms (or simply logs) is based on indices. In fact, as you will find out very soon, the rules for logarithms are very similar to the rules for indices. Therefore, a recap of the concept of indices will be useful for us to understand how logarithms works.

Download the entire file : LOGARITHMS

Data Analysis for Economists – Part I

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Correlation r

Describing Data

Every economist needs to have the ability to collect, analyse, manipulate, understand and report data. In a daily research environment, we need to deal with randomness, variation and in order to apply our knowledge. Therefore we are going to summarize the most important and useful tools for every economist.

Key Definitions:

  • A population consists of all the members of a group about which you want to draw a conclusion. The size of the population depends on what you are interested in. (μ, σ, Ν)
  • A sample is the portion of the population selected for analysis. Collecting information on the population can be difficult and costly, therefore we sample. (x, s, n)
  • A parameter is a numerical measure that describes a characteristic of a population
  • A statistic is a numerical measure that describes a characteristic of a sample

A note on Notation

  • Greek letters (μ, σ, Ν) are used for population data
  • Roman letter (x, s, n) are used for sample data


Scatter diagrams are very common in econometrics and are used to examine possible relationships between two numerical variables.

  • In a scatter diagram one variable is measured on the vertical axis (Y) and the other variable is measured on the horizontal axis (X)
    • X = independent variable
    • Y = dependent variable

Scatter Plot










Figure 1: Plot A: Scatter Plot Relationship between Share of Food (WFOOD) and Total Expenditure (TOTEXP).

So, how do we actually describe our data?

Describing Data


We will need to know the data mean, median and mode, however, we will pretty much talk about the data Variation, Shape, Skewness, Range, Interquartile Range, Variance, Standard Deviation and Coefficient of Variation.

So, let’s start through the Central Tendency. What is the mean, median and mode?


  • Commonly called as the average
  • Calculated as the sum of values divided by the number of values
  • Affected by extreme values (outliers)
Population Mean Sample Mean
               μ           X



  • In an ordered array, the median is the ‘middle’ number, not the average, but actually the physical position.
  • The location of the median: (n + 1) /2   is not the value of the median, only the position of the median in the ranked data

Rule 1: If the number of values in the data set is odd, the median is the middle ranked  value

Rule 2: If the number of values in the data set is even, the median is the mean (average) of the two middle ranked values



  • Value that occurs most often (the most frequent). It can be more than one value.



  • Quartiles split the ranked data into 4 segments with an equal number of values per segment
  • The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger

Q1 = (n+1)/4

  • The second quartile, Q2, is the same as the median (50% are smaller, 50% are larger)

Q2 = (n+1)/2

  • Only 25% of the observations are greater than the third quartile Q3

Q3 = 3(n+1)/4



Measures of variation give information on the spread or variability of the data values.

  • RANGE:

Difference between the largest and the smallest values in a set of data

Range = Xlargest - Xsmallest



Like the median and Q1 and Q2, the IQR is a resistant summary measure. It eliminates outlier problems by using the interquartile range as high- and low-valued observations are removed from calculations:

IQR = 3rd quartile – 1st quartile


  • VARIANCE: The mean squared deviation and it shows variation about the mean.


  • Each value in the data set is used in the calculation
  • Values far from the mean are given extra weight as deviations from the mean are squared


  • Sensitive to extreme values (outliers)
  • Measures of absolute variation not relative variation


The denominator (n-1) is to adjust for the biasness of the sample statistics.




Measures relative variation i.e. shows variation relative to mean. It can be used to compare two or more sets of data measured in different units and it is always expressed as percentage (%).


Shape and Skweness

  • Describes how data are distributed
  • Measures of shape – Symmetric or skewed

Shape of a Data



Sample Covariance

  • The sample covariance measures the direction of the linear relationship between two numerical variables (direction of the association)


Sample Coefficient of the of Correlation r

  • Measures the relative strength of the linear relationship between two variables:

Correlation r

Where Sx and Sy are their Sample Variance.


Value of r Interpretation
r = -1 PERFECT negative linear relationship
-1 < r ≤ -0.7 STRONG negative linear relationship
-0.7 < r ≤ -0.3 MODERATE negative linear relationship
-0.3 < r < 0 WEAK negative linear relationship
r = 0 No relationship
0 < r < 0.3 WEAK positive linear relationship
0.3 ≤ r < 0.7 MODERATE positive linear relationship
0.7 ≤ r < 1 STRONG positive linear relationship
1 PERFECT positive linear relationship




Introduction to Economics Part II

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Now that we know about scarcity, competition, functioning markets, benevolent rule makers, homo economicus, economic rationality, equilibrium, money circulation, creative destruction, supply, and demand, elasticity and government interventions. We can go deeply covering theories of long-term economic growth and short-term economic growth.

GDP – gross domestic product

  • main measure of size of an economy
  • total value of all marketed goods and services produced in a country
  • if GDP increases from one period to the next = economic growth
  • see examples in the textbook
    • Australia in the last 150 years
    • world economy in the last 50 and next 30 years
  • Primary (~5%), secondary (~25%), tertiary (~70%)

Natural resources for the long-term economic growth:

  • nonrenewable resources
  • oil, coal, gas,
  • renewable/quasi renewable resources
  • fish stock, forestry, water
  • Accumulation = growth (not easy to increase). Countries generally have them or not
  • not necessarily good for growth
  • Dutch disease/current resources boom

Three theories of short term cycles

Keynesianism – How to avoid/get out of Keynesian recession??

  • talk optimistically
  • ‘demand management’
  • combination of the above; But hard because demand management means borrowing money & can’t keep doing this in a recession & may be offset by more saving in population.

Real business cycles

How to avoid/get out of RBC recession

  • no need
  • recessions ‘mere preludes’ to further growth & matter of choice
  • no reason to avoid them or for fear or worry about them

Creative destruction

How to avoid/get out of creative destruction recessions

  • contain the impacts in the industry
  • but part of process leading to long-term growth
  • new technologies & industries can’t emerge without demise of less productive industries and old forms of technologies


It is the increase in prices of representative bundle of goods.

  • inflation rate for consumers: Consumer Price Index (CPI), derived from representative bundle of good and services consumed by Australian consumers
  • real interest rates: corrected for inflation
    • ie nominal interest rate minus inflation rate
    • Similarly, real wage increase = nominal wage increase minus

Real impacts of inflation:

  • income
    • shrinks if rate of inflation is greater than increase in nominal income
  • wealth
    • increases if nominal value of assets increases faster than rate of inflation
  • lending, saving & borrowing
    • savers and lenders lose & borrowers win if nominal interest rate is less than the rate of inflation

Fiscal Policy

  • responsibility of the government
  • instruments: G (government expenditure) & Tax, where C = Consumption; I = investment and NX = Export – Import (Net export)
GDP = C + I + G + NX
  • non-discretionary
    • Automatic stabilisers – i.e. policy measures that happen automatically. They are anti-cyclical and are to stabilise cycle
    • e. tax & welfare system (in boom times taxes increase, benefits decrease – automatic break – in downturn benefits increase, taxes decrease – automatic stimulant)
  • discretionary fiscal policy: deliberate measures (counter-cyclical policy)
  • expansionary fiscal policy to deal with recession
    • increase in government expenditure
    • deficit budget
    • often financed by government borrowing
  • contractionary fiscal policy to deal with boom
    • decrease in government expenditure
    • surplus budget
    • pay back loans, build up reserves


Monetary Policy

  • monetary policy – responsibility of the central bank
  • fiscal policy – responsibility of the government
  • instruments: interest rates, money supply
  • interest rate = price charged by a lender of money
    • the less certain it is that a loan will be paid back, the higher the interest rate (price) charged
    • RBA has lots of money at its disposal, hence can lend money to other institutions (banks) at any rate.
    • sets daily interest rate, and commercial banks align their interest rates accordingly
  • Official policy of RBA to have inflation rates between 2 and 3 percent.
    • a bit of inflation is not such a bad thing
    • goods or services becoming more expensive = more scarce à incentive for producers to produce more
    • Economic agents adapt to changing circumstances.
  • If inflation is > 3 percent, RBA increases interest rates
  • if inflation < 2 percent, RBA decreases



  • taxes affect individual behaviour
  • and produce the following effects
    • financial effect
    • organisational effect
    • general equilibrium effect
    • announcement effects
    • distortionary/non-distortionary effect
    • correction effect (ex. Alcohol)


  • if supply is more elastic than demand, incidence of tax falls more heavily on consumers
  • if demand is more elastic than supply, incidence of tax falls more heavily on producers

Remember externalities:  when action by either or both parties in the market exchange affect third parties, negatively or positively it can:

  • ie reduce or increase others’ welfare

In the particular case of a negative externality where more output is produced at a lower price than if social cost was taken into account, it can be corrected by a tax, which has the effect of increasing price, reducing output.


Domestic Market before the Trade



And… what happens if the world price is above domestic price??




And… what happens if the world price is lower than the domestic price??




Effects of tariffs on imports




Effects on imports with quota


The most important thing to understand is that both tariffs and quotas raise domestic prices; reduce the welfare of domestic consumers; increase the welfare of domestic producers; and cause deadweight losses (ie less overall welfare). However, while tariff provides revenue to government, quotas provides some surplus to foreign producers.


Introduction to Economics Part I

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Before we start to explain you what actually economics is, let’s start with some definition of many useful words related to Economics. The words I would like to bring aboard are the following:

  1. Scarcity
  2. Competition
  3. Functioning markets
  4. Benevolent rule makers
  5. Homo economicus
  6. Economic rationality
  7. Equilibrium
  8. Money circulation
  9. Creative destruction



  • basic economic problem (condition):
    • every decision-maker wants more than is possible
    • infinite needs/wants v finite resources
    • scarcity unavoidable
    • choices have to be made
    • can consider who will benefit and who will lose
  • decision-makers aim to increase their material welfare
    • have to take this into account
    • can try to maximise welfare (eg utilitarianism)
    • but how? whose?



  • private vices & social outcomes
  • Smith: individuals behaving ‘selfishly’ produces public benefits – through competition
    • ‘invisible hand’ – prices reflect the difficulty with which goods can be made
    • specialisation – people have unequal abilities so it is efficient to specialise
    • do have to play by the rules though
      • governments must prevent ‘negative competition’


Functioning markets

  • requirement for competition
  • competition works only in a situation where they are many sellers
  • buyers need to have access to price information of all sellers and be allowed to buy from any seller
  • neither buyers nor sellers can collude (coordinate) to defraud the others
  • buyers and sellers do not cheat by theft, reneging on contracts, murder…


Benevolent rule makers

  • who sets the rules and enforces them?
  • economists belief (often not always):
    • that the rule setter is interested in the common good!
  • who could be such rule setters?
    • independent competition boards (of benevolent experts)
    • courts with impartial benevolent judges
    • a government or the policy analysts of government
    • those who draw up constitutions
    • anyone with great independent power (a good king/dictator)
  • if none of these is actually loyal to the overall good, we talk of state capture, state failure, corruption…


Homo economicus

  • agents (people) in the real world maximise their own individual materialistic gain and they act highly rationally in doing so.
  • hence materialist explanations for, eg, cultural institutions
    • not eating cows, cannibalism, food labelling rule changes…
  • weight of observation
    • eg tax on beards, on non-believers, sales…
  • experience with socialism


Economic rationality

  • argument: non-government decision-makers are rational
  • rational expectations hypothesis:
    • expectations of households and firms as to what is going to happen in the future are going to be on average ‘right’
    • consciously predicting what is likely to happen in the future and getting it right on average
  • will only leave opportunities for advancement unused if they are unaware or surprised by them
  • note: won’t always believe a policy is inevitable or sustainable
    • policy can change behaviour
    • but not always in intended direction



  • equilibrium when it is not possible for anyone to improve on outcome by changing current behaviour
  • given rationality, economy should always be moving towards equilibrium
    • as individuals learn to recognise/take opportunities
  • support from introspection
    • would we take opportunity for improvement?
    • try to anticipate and control our environment?
    • see text examples on saving, education, indicators, traffic, traders


Money circulation

  • main roles of money: means of exchange, store of value, unit of account
  • circulation of money: from one seller to another
  • links prices to the quantity of money:
    • more money without extra production will put upward pressure on prices
    • more bank notes buying same amount of goods
    • inflation
  • importance of price stability
    • counter-example: hyperinflation


Creative destruction

  • constantly changing world opens new business opportunities and closes old ones
    • continuous search for new opportunities via continuous experimentation
    • with organisational technologies, trading partners, etc
    • needs flexibility/freedom & well-established property rights
    • needs as much useful information as possible
    • ‘creative’ in adaptation to new circumstances
      • see textbook examples
    • destructive’ for old organisations destroyed, so politically difficult


Therefore, we can now understand how some of these above concepts will connect within economics. It is known that goods and services are wanted for people/other services and neet to be met. These wants/needs are produced by using resources

  • sometimes called factors of production
  • However, resources are limited… there is only a limited amount of resources available to produce the unlimited amount of goods and services we desire.
Type of Resource Descripton Reward
Land All Natural Resources Rent
Labour The Physical and Mental Work of People Wages
Capital All human made tools and machines Interest
Enterprise/Contact All managers and organisers Profit


  • Choice implies to choose one thing and forego another

The opportunity cost principle states the cost (or value) of one good in terms of the next best alternative. The set of possible choices is called the opportunity set, the feasibility set, or the budget constraint. There are a lot of decision making and we cannot avoid the choices. They will be based on supply, demand and price. The lower the price the more quantitative demand. So, if prices changes, quantitative demand will change. However, that could be no price that will change the demand.

  • demand: buyers, consumers, customers
  • supply: sellers, suppliers, producers
  • particular market: perfectly competitive
  • price-takers
    • buyers cannot influence price, can just ‘take it or leave it’, ie buy or not buy the product
    • sellers cannot influence price, can sell at that price or not sell at all
  • assumes knowledge of:
    • all prices that are offered by some sellers
    • all the sellers (and where their shops are)
    • the exact quality/design of products offered
    • how much a good is worth to them (buyers)
    • their costs to produce/supply a good (sellers)
  • ie perfect competition
  • alternatives:
    • bargaining (buyers and sellers discuss the price),
    • market power (either the seller, eg a monopolist, or the buyer, can influence the price)
  • demand: from buyers (consumers/customers)
  • as price-takers, they will buy or not buy
  • ie are willing to pay the price, or not
  • willingness to pay depends on opportunity cost
    • buy this cannot have that (given limited budget/income)
    • have more of this will have to have less of that
  • can give this a monetary value
  • non-price factors determine overall demand for a product
    • ie all combinations of P & Qd comprising the D curve
  • budget or income
    • increase in income, increase in D (for normal goods)
  • prices of other goods: substitutes/complements
    • increase in price of substitute, increase in D
  • tastes/preferences
    • eg out of fashion, decrease in demand



Shift of the Demand Curve


  • Law of demand holds: if Pá Qdâ
    • but sometimes a small change in P results in a large change in Qd (elastic)
    • sometimes hardly any change in Qd (inelastic)
  • elasticity explains this:
    • can calculate price elasticity (%∆Qd / %∆P)
    • & use answer as a measure of how elastic the product is (>1 elastic, < 1 inelastic)
  • important information for seller in terms of total revenue & for government for effect of tax
  • Can also calculate cross elasticity of demand
    • how much demand for product responds to a change in price of a substitute or complement
    • %∆Qd good B / %∆ P good A
  • Income elasticity of demand
    • how much Qd changes in response to a change in income
    • %∆Qd / %∆Y
  • supply: from suppliers, sellers, producers
  • production process: combining inputs
    • resources: land, labour, capital, enterprise
  • to produce outputs
    • final consumptions goods, intermediate goods, services
  • represented by production function
    • in the short-run, initially output increasing at increasing rate (increasing returns)
    • but after a certain point diminishing returns set in
    • output still increasing but at a decreasing rate i
  • total cost = sum of costs to produce a certain quantity
    • fixed costs (don’t change, independent of quantity)
    • + variable costs (do change as output increases)
  • average cost = total cost/quantity
  • marginal cost = change in total cost when output increased by one unit
  • cost function: minimum cost to produce certain Q
  • Profit = TR – TC so if TR>TC, profit made to maximize profit, firms need to compare MR (marginal revenue) and marginal cost (MC). It is important thing in terms of total revenue. For example, if you want to increase the revenue go up, you can put the price up, but you can take the risk to decrease the demand. Thus, they need to no weather the good is elastic or inelastic (ex: medical equipment). If it is elastic, we put the price down, increasing the demand and consequently the total revenue.
  • in perfect competition MR = P
  • rule is to keep producing as long as MC < P
  • ie produce Q where P = MC
    • if MC < P, increase output/production to make more profit
    • if MC > P, decrease output/production, to make more profit
    • when P (MR) = MC, profit maximized, make the most profit

Can we improve on a situation such that nobody is worse off and at least one person is better off?

  • if not, situation is Pareto efficient.
  • Consumer Surplus: benefits consumers derive from market interaction. consumers who are willing to pay $80 can have this good for $40 so they benefit (by $?) and those who are willing to pay $60 can have this good for $40.


Producer surplus:  benefits producers derive from market interaction. Producers who are willing to sell at $25 can sell this good for $40 so they benefit (by $?) and those who are willing to sell at $30 can sell this good for $40.




  • Efficiency means that society maximizes the sum of all surpluses
    • ie sum of consumer surplus and producer surplus
  • An efficient allocation is one where the sum of producer and consumers surplus is maximised




Now producers gain from the higher price but consumers lose and there is still an overall loss as less quantity is being demanded & supplied also inefficient allocation:




But when we have a Government Intervention. If the government believes/proves that a market is not working efficiently, it can intervene setting prices (cap) or quantities or even create a rate of return regulation: the authorities calculate the physical capital needed to run the business and sets prices such that the profit yields a fair return. The Government can also set a tax.


The higher the tax rate, the larger the deadweight loss, holding all else constant. The more elastic (horizontal) the Demand curve, the higher the deadweight loss. Policy conclusion: It is better to tax goods where demand is inelastic.

Hello world!

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economia verde

Welcome to Dutra’s Webpage.

This website was created to help people understand economy concepts and link then with current issues around the world.

The idea is to teach slow, step by step some key economic theories, and discuss issues using and comparing theories that big economists are debating right now. The website use statistics tools and heaps of econometrics applications.

Karen is a economist graduated from Queensland University of Technology. She is currently undertaking her Master Degree at University of Queensland. She is passionate about economic resources and their value. She is a central-left wing economist with a green mentality. She believes the economy can keep growing even if we become greener and greener.  She also holds a degree in Science. Her aim is to put herself in a position where she can help spread the world what people can be taught about economy, provide possible solutions for current issues throughout dialogue and promote the exchanging of ideas.

Feel free to add contributions and encourage the debate. Let’s start talking about economy!