Lecture -- 1-- Start
Outline 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Science, Method & Measurement On Building An Index Correlation & Causality Probability & Statistics Samples & Surveys Experimental & Quasi-experimental Designs Conceptual Models Quantitative Models Complexity & Chaos Recapitulation - Envoi
Outline 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Science, Method & Measurement On Building An Index Correlation & Causality Probability & Statistics Samples & Surveys Experimental & Quasi-experimental Designs Conceptual Models Quantitative Models Complexity & Chaos Recapitulation - Envoi
Quantitative Techniques for Social Science Research
Lecture # 1: Science, Method, and Measurement
Ismail Serageldin Alexandria 2012
On Science
On Science
Science Is Driven by Curiosity About the Natural World
WHAT IF?
WHAT IF?
WHAT IF?
??
Defining Science “The organization of our knowledge in such a way that it commands more of the hidden potential in nature ..”
J. Bronowski
Intellectual Activities Sciences Objective
Natural Sciences Physics Chemistry Astronomy Geology Biology Etc.
Human Sciences Psychology Economics Political Science Sociology History Etc.
Subjective Curiosity Influence others
Applied Fields Technology Education Medicine Law Etc.
Humanities Esthetics Ethics Religion Philosophy Etc.
Natural Sciences (classical definitions)
• Physical Sciences: Physics, Chemistry • Life Sciences: Biology (zoology, botany) • Earth Sciences: Geology, Astronomy, Meteorology
Overlapping Domains in Science Biochemistry, Paleontology, Molecular Genetics…
Changed Outlook: Process and System Views
Example: Photosynthesis • Light: the Energy source (physics) • Photosynthesis: The food production process (chemistry) • For plants (biology) Energy…biochemical pathways…cell Biology… plant physiology…
The Nature of Scientific Knowledge • • • •
Falsifiable (Popper) Approximative Empirical Replicable
• And so much more…
Karl Popper (1902-1994)
Before scientific thinking can proceed, certain philosophical presuppositions must be made about the nature of the universe:
Philosophical Presuppositions • Objective reality exists – there really are things out there, everything is not simply a figment of the imagination. • The universe is knowable – no aspects of the universe are beyond human understanding. • The universe’s operation is regular and predictable – if events occur at random, without any warning or pattern, no amount of analysis will uncover any regularity to them.
Philosophical Presuppositions • Objective reality exists – there really are things out there, everything is not simply a figment of the imagination. • The universe is knowable – no aspects of the universe are beyond human understanding. • The universe’s operation is regular and predictable – if events occur at random, without any warning or pattern, no amount of analysis will uncover any regularity to them.
Philosophical Presuppositions • Objective reality exists – there really are things out there, everything is not simply a figment of the imagination. • The universe is knowable – no aspects of the universe are beyond human understanding. • The universe’s operation is regular and predictable – if events occur at random, without any warning or pattern, no amount of analysis will uncover any regularity to them.
Philosophical Presuppositions • Objective reality exists – there really are things out there, everything is not simply a figment of the imagination. • The universe is knowable – no aspects of the universe are beyond human understanding. • The universe’s operation is regular and predictable – if events occur at random, without any warning or pattern, no amount of analysis will uncover any regularity to them.
On The Scientific Method
The Scientific Method
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
The Method of Science OBSERVATION: Sense specific physical realities or events.
HYPOTHESIS:
PREDICTION:
nature of the phenomenon observed.
REVISED HYPOTHESIS
Forecast a future occurrence
PREDICTION
Create a statement about the general
consistent with the hypotheses.
EXPERIMENT:
Carry out a test to see if predicted
EXPERIMENT
event occurs.
If results DO match prediction, hypothesis is supported.
If results DO NOT match prediction
RECYCLE
No amount of experimentation can ever prove me right; A single experiment can prove me wrong. Albert Einstein
The Scientific Method • • • • • • • •
Conjecture Hypothesis Testing Replicability Falsifiability Cumulative evidence Explanatory power Predictive power
Math • Logical, consistent, proof is absolute within its own axiomatic rules • Math is added to, science is replaced • Math is the science of patterns • It is elegant, beautiful and concise • It demands enormous precision in thinking clearly about abstract objects
Math and Science • Enormous power for manipulating quantitative results • Hence questions of measurement are important • Quantification and qualitative analyses remain important issues
Back To The Social Sciences
Intellectual Activities Sciences Objective
Natural Sciences Physics Chemistry Astronomy Geology Biology Etc.
Human Sciences Psychology Economics Political Science Sociology History Etc.
Subjective Curiosity Influence others
Applied Fields Technology Education Medicine Law Etc.
Humanities Esthetics Ethics Religion Philosophy Etc.
Intellectual Activities Sciences Objective
Natural Sciences Physics Chemistry Astronomy Geology Biology Etc.
Human Sciences Psychology Economics Political Science Sociology History Etc.
Subjective Curiosity Influence others
Applied Fields Technology Education Medicine Law Etc.
Humanities Esthetics Ethics Religion Philosophy Etc.
Back To The Scientific Method
The Method of Science Observation Hypothesis Prediction Experiment
The Method of Science Observation Hypothesis Prediction Experiment Interpretation
But can we apply that empirical method in the social sciences?
Understanding Society To Design Social Policies
Policy Counts
Economic Growth and Poverty Reduction
Growth and Poverty Reduction • Growth is a necessary but not sufficient condition for poverty reduction • The quality of growth and the nature of the policies matters enormously
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
8.0
6.0
4.0
2.0
0.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
8.0
6.0
4.0
2.0
0.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
Costa Rica
8.0
Malaysia Taiwan
Indonesia
6.0 Singapore
Thailand Pakistan
4.0
Brazil Sri Lanka
2.0
Mexico
India Bangladesh
0.0 Jamaica
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
Costa Rica
8.0
Malaysia Taiwan
Indonesia
6.0 Singapore
Thailand Pakistan
4.0
Brazil Sri Lanka
2.0
Mexico
India Bangladesh
0.0 Jamaica
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
Costa Rica
8.0
Malaysia Taiwan
Indonesia
6.0 Singapore
Thailand Pakistan
4.0
Brazil Sri Lanka
2.0
Mexico
India Bangladesh
0.0 Jamaica
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Growth and Poverty Reduction
% annual decline in 10.0 poverty (Headcount index)
Costa Rica
8.0
Malaysia Taiwan
Indonesia
6.0 Singapore
Thailand Pakistan
4.0
Brazil Sri Lanka
2.0
Mexico
India Bangladesh
0.0 Jamaica
0.0
2.0
4.0
6.0
8.0
10.0
12.0
% annual growth in GDP/person
Policies, Inequality and Welfare
Is Inequality Built Into Economic Structure? • Is movement into knowledge based economy necessarily accompanied by inequality? • Is US economy intrinsically generating more inequality?
Poverty Observed: US and Selected European Countries, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Poverty before Government policy effects US and Selected European Countries, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Policy Effects on Poverty: Pre and Post tax and transfers, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Policy Effects on Poverty: Pre and Post tax and transfers, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Policy Effects on Poverty: Pre and Post tax and transfers, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Policy Effects on Poverty: Pre and Post tax and transfers, 1991 USA UK Sweden Netherlands Italy Ireland France Denmark Canada Belgium
0
5
10
Source: Robert Solow, “Welfare: The Cheapest Country”’in NYRB, 23 March 2000, p. 20-23
15 % POOR
20
25
30
Why Quantitative Analysis?
The Importance of Social Research
Much Economic Analysis Erases the Human Factor
The Need for Social Inputs Into Development Decisions • Social policy is more than the social consequences of economic policies • Social goals and policies complement economic ones • Economic Analysis by itself is insufficient: Social, cultural, political and ethical dimensions must be introduced
Elements Of A Social Policy - I • To maintain social cohesion • To foster equity • To reach the ultra poor and other marginalized groups • To uphold cultural identity (shared universal values and solidarity, not divisive micro-identities)
Elements Of A Social Policy - II • To promote participation (voice, choice and empowerment through access to knowledge and resources) • To facilitate social mobility (intergenerational, geographic and occupational) • To support institutional development • To enable participatory social research
Participatory Social Research • Promotes more effective understanding • Leads to sounder policy and program designs • Empowers the people participating in the research
Social Research and Decision Making • Coherent Framework • Predictive • Prescriptive
Social Research and Decision Making • Coherent Framework • Predictive • Prescriptive
Social Research and Decision Making • Coherent Framework • Predictive • Prescriptive
Crisis in the Non-economic Social Sciences • Absence of theoretical framework for the dynamics of social change • The negative impact of the postmodern currents • Confusion about quantitative and qualitative aspects pf research • The misunderstood role of models
Crisis in the Non-economic Social Sciences • Absence of theoretical framework for the dynamics of social change • The negative impact of the postmodern currents • Confusion about quantitative and qualitative aspects pf research • The misunderstood role of models
Crisis in the Non-economic Social Sciences • Absence of theoretical framework for the dynamics of social change • The negative impact of the postmodern currents • Confusion about quantitative and qualitative aspects of research • The misunderstood role of models
Crisis in the Non-economic Social Sciences • Absence of theoretical framework for the dynamics of social change • The negative impact of the postmodern currents • Confusion about quantitative and qualitative aspects of research • The misunderstood role of models
Crisis in the Non-economic Social Sciences • Absence of a theoretical framework for the dynamics of social change • The negative impact of the postmodern currents • Confusion about quantitative and qualitative aspects of research • The misunderstood role of models
Quantitative Social Analyses: Laplace “Let us apply to the political and moral sciences, the method founded on observation and mathematics that has served so well in the natural sciences.” -- Pierre Simon de Laplace (1749-1827)
Quantitative Social Analyses: Quetelet “The more advanced the sciences have become, the more they have tended to enter the domain of mathematics, which is a sort of center toward which they converge. We can judge of the perfection toward which a science has come by the facility, more or less great, with which it may be approached by calculation.” -- Quetelet (1796-1874)
Quantitative Social Analyses: Quetelet
Quetelet (1796-1874), by the way, invented the notion of the “average man.”
Quantitative Social Analyses: Boorstin “Today, the Cassandras of social science speak the language of numbers”. -- D.J. Boorstin (1914-2004) Source: Daniel J. Boorstin, Cleopatra , (op.cit., p142)
Vehement Reactions • Dehumanizing the humanities • Denies individualism • Treats people like products or machines • Economics is not the whole story • Etc. etc.
Serageldin On Reductionist Views
Serageldin on Reductionist Views • Three buckets of water and a handful of minerals held together by chemical reactions… • A society is more than the sum of its economic and financial transactions…
+
≠
Serageldin on Reductionist Views • Three buckets of water and a handful of minerals held together by chemical reactions… • A society is more than the sum of its economic and financial transactions…
≠
Conclusions • We need more, not less, sophisticated approaches… • Clever word games are not helpful to either explain social realities or to help formulate polices and interventions that improve the wellbeing of people
But we need quantitative analysis to understand, and to measure and to devise appropriate Social policies
Quantitative Analysis complements Qualitative analysis and frequently undergirds it.
On Measurement
Measurement counts
Measurement Is Important • We treasure what we measure • Prescription and dosage depend upon accurate estimation of magnitudes • Establishing trends is as – or more -important than snapshots of magnitudes • Monitoring of progress over time
Measurement Is Important • We treasure what we measure • Prescription and dosage depend upon accurate estimation of magnitudes • Establishing trends is as – or more -important than snapshots of magnitudes • Monitoring of progress over time
Measurement Is Important • We treasure what we measure • Prescription and dosage depend upon accurate estimation of magnitudes • Establishing trends is as – or more -important than snapshots of magnitudes • Monitoring of progress over time
Measurement Is Important • We treasure what we measure • Prescription and dosage depend upon accurate estimation of magnitudes • Establishing trends is as – or more -important than snapshots of magnitudes • Monitoring of progress over time
Measurement Is Important • We treasure what we measure • Prescription and dosage depend upon accurate estimation of magnitudes • Establishing trends is as – or more -important than snapshots of magnitudes • Monitoring of progress over time
On Measurement • Accuracy & Precision • Resolution & Randomness • Types of Scales
143/xxx
On Measurement • Accuracy & Precision • Resolution & Randomness • Types of Scales
144/xxx
Accuracy in Measurement • Using the right tool • The quality of the tool is important • How carefully we measure with it is also important • Let’s use a ruler to measure the length of a piece of wood…
Accuracy in Measurement • Using the right tool • The quality of the tool is important
Accuracy in Measurement • Using the right tool • The quality of the tool is important • How carefully we measure with it is also important
Accuracy in Measurement • Using the right tool • The quality of the tool is important • How carefully we measure with it is also important • Let’s use a ruler to measure the length of a piece of wood…
Accuracy & Precision • Accuracy: how close the measured value is to reality (i.e. what it ought to be) -- So if the ruler is defective and two rulers yield different results that is an error of accuracy • Precision: is a measure of the reproducibility of the measurement,
Accuracy & Precision • Precision: is a measure of the reproducibility of the measurement, our confidence that uncertainty of measurement has been reduced to a minimum. • Sometimes the problem is instrumental precision (level of resolution) or the randomness of the event being measured.
On Measurement • Accuracy & Precision • Resolution & Randomness • Types of Scales
Resolution vs. Randomness • We do not re-measure the piece of wood 100 times and take the average. • Assuming the wood was measured carefully, the error here is due to the resolution of the ruler, not the randomness of the event being measured.
Resolution of the tool • So, instead of a ruler use higher resolution instrument s like precision Vernier calipers:
But that is very different from dealing with random events
Random events
Random events/outcomes require a probabilistic treatment
Social Science studies of events/outcomes usually require a statistical probabilistic treatment
Here multiple measurements and probabilistic techniques are used
We will get back to Probabilities Later
On Measurement • Accuracy & Precision • Resolution & Randomness • Types of Scales
What kind of scale do we use in Measurement?
Four Kinds (Types) of Scales • Nominal • Ordinal • Interval • Ratio
Types of Scales: I. Nominal Scales Numbers are used to name, identify or classify.
Level
The only permissible arithmetical procedures are counting and statistical techniques based on counting. Limitations
Example 163
Social Science Examples of Nominal Scales • Marital Status: Married, Unmarried • Nationality: Chinese, American, European, Egyptian • Religion: Muslim, Christian, Jewish, Buddhist, … • Ethnic or tribal group • Race: Black, white • P/F Evaluation: Pass/Fail
Types of Scales: II. Ordinal Scales
Numbers indicate rank or order.
Level
Ranking methods and other statistical techniques based on interpretations of “greater than” or “less than” are permissible. Limitations
Example
165
Social Science Examples of Ordinal Scales • Grading of interpersonal skills • Evaluating managerial skills • You can say greater than, but you cannot really quantify the amount or degree objectively.
Types of Scales: III. Interval Scales The intervals or Addition and subtraction distances between and statistical each number and techniques based on the next are equal, these two operations are but it is not known permissible. how far any of Multiplication and dithem is from zero. vision are not permissible. Level
Limitations
Example 167
Natural Science Examples of Interval Scales • Temperature: – Two days: 20 and 40 degrees Celsius – Difference between them is 20
• Cannot say twice as hot because zero could be: – Celsius scale – Fahrenheit scale – Kelvin scale
Social Science Examples of Interval Scales • Grading school exams: • Say two students took a test: results score was 20 and 40 points (difference is 20 points) but should not say twice as much. • BUT…teacher could have added a few easy questions that would have obtained each student 10 more points • Results would have been 30 and 50.
Social Science Examples of Interval Scales • Height or weight of people: – Say two persons 1.6 m and 1.80 m – Or two persons weigh 60kg and 80 kg
• Note: – No one is really 0.0 height or weight
Types of Scales: IV. Ratio Scales
Each number can be thought of as a distance measured from zero
Level
There are no limitations. All arithmetical operations and all statistical techniques are permissible Limitations
Example 171
Social Science Examples of Ratio Scales • Income and expenditure • Years of schooling • Number of respondents selecting something • Number of respondents who have a particular cardinal quality (e.g. married, unmarried) • Etc.
Social Science Examples of Ratio Scales • Income and expenditure • Years of schooling • Number of respondents selecting something • Number of respondents who have a particular cardinal quality (e.g. married, unmarried) • Etc.
It is important that not all relationships or all mathematical operations can be applied to all scales.
Now let’s move on to some descriptors of groups
What Is An Average?
We say: I examined 20 students and the average score was x
What is the meaning of the word “average”?
Average? • Mean: usually add up the values for all the observations and divide them by the number of observations • Median: the number at which half the observations are smaller and the other half are bigger • Mode: the number that appears most frequently in the distribution of observations.
Average? • Mean: usually add up the values for all the observations and divide them by the number of observations • Median: the number at which half the observations are smaller and the other half are bigger • Mode: the number that appears most frequently in the distribution of observations.
Lets take 20 observations • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 20 • What is the Mean ? • The Median ? • The Mode?
Lets Find the Mean • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 20 • Mean = Total / number of observations • Total = 1+2+3+…. +10+10+20 = 120 • Mean = 120 / 20 = 6
Formula for the Mean
n
=
⋯
183
Lets Find the Median • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 20 • Median = 5
Lets Find the Mode • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 20 • Mode = 4
So for these Observations • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 20 • Mean = 6 • Median = 5 • Mode = 4
Lets Change one Observation • 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 8, 10, 10, 1000 • Mean = 1100 / 20 = 55 • Median = 5 • Mode = 4
Introducing: The Normal Curve (The Bell Curve, The Gaussian Distribution)
Mean, Median, Mode
Average? • Mean: usually add up the values for all the observations and divide them by the number of observations • Median: the number at which half the observations are smaller and the other half are bigger • Mode: the number that appears most frequently in the distribution of observations.
Mixing between these is one of the most common fallacies in reporting social statistics
We will come back to the Normal Curve (The Bell Curve, The Gaussian Distribution) many times in this course
But that will be for later…
SO…
Let’s make sure that we keep our heads above water!
I want you all to be swimming, not drowning!
Then we will all learn to fly…
Thank You