AP Stat Do Now

1
AP Stat - Do Now

• Rank order the graphs on the handout in terms of
  the strength and direction of their correlation

2
Objectives

• Chapter 7 Scatterplots, Association, and
  Correlation
• How do we quantify the strength and direction of
  a linear relationship between two quantitative
  variables?
• Explanation of the Pearson Correlation
  Coefficient (r)
• What should we always check for before reporting
  the correlation coefficient?
• Properties of the correlation coefficient

3
Correlation

• In Statistics, the word correlation is reserved
  for linear associations between two quantitative
  variables
• The correlation coefficient (r) gives us a
  numerical measurement of the strength (and
  direction) of the linear relationship between the
  explanatory and response variables.

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Correlation

• Correlation can be thought of as a measure of
  how much things co-vary. In other words, when
  one variable goes up, does the other variable
  vary by the same amount?
• The problem is What if the variables are two
  completely different things (e.g. height and
  weight)?
• If somebody increases by an inch, do we expect
  them to weigh one more pound?

5
Correlation

• Standardized (z) scores to the rescue!
• Now we can compare things regardless of their
  units, magnitudes, and variances
• Whats really cool about the Pearson Correlation
  Coefficient (r) is that it is distribution-indepen
  dent.
• Your two variables dont have to come from
  symmetric, bell-shaped distributions for it to
  give an accurate representation of the
  correlation between the two variables.

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Correlation

• Take a look at the figure 7.5 on page 121

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Correlation Conditions

• Before you use correlation, you must check
  several conditions
• Quantitative Variables Condition
• Linearity (Straight Enough) Condition
• Outlier Condition

8
Correlation Conditions (cont.)

• Quantitative Variables Condition
• Correlation applies only to quantitative
  variables.
• Dont apply correlation to categorical data
  masquerading as quantitative.
• Check that you know the variables units and what
  they measure.

9
Correlation Conditions (cont.)

• Straight Enough Condition
• You can calculate a correlation coefficient for
  any pair of variables.
• But correlation measures the strength only of the
  linear association, and will be misleading if the
  relationship is not linear.

10
Correlation Conditions (cont.)

• Outlier Condition
• Outliers can distort the correlation
  dramatically.
• An outlier can make an otherwise small
  correlation look big or hide a large correlation.
  
• It can even give an otherwise positive
  association a negative correlation coefficient
  (and vice versa).
• When you see an outlier, its often a good idea
  to report the correlations with and without the
  point.

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Correlation Properties

• The sign of a correlation coefficient gives the
  direction of the association.
• Correlation is always between -1 and 1.
• Correlation can be exactly equal to -1 or 1, but
  these values are unusual in real data because
  they mean that all the data points fall exactly
  on a single straight line.
• A correlation near zero corresponds to a weak
  linear association.

12
Correlation Properties (cont.)

• Correlation treats x and y symmetrically
• The correlation of x with y is the same as the
  correlation of y with x.
• Correlation has no units.
• Correlation is not affected by changes in the
  center or scale of either variable.
• Correlation depends only on the z-scores, and
  they are unaffected by changes in center or scale.

13
Correlation Properties (cont.)

• Correlation measures the strength of the linear
  association between the two variables.
• Variables can have a strong association but still
  have a small correlation if the association isnt
  linear.
• Correlation is sensitive to outliers. A single
  outlying value can make a small correlation large
  or make a large one small.

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What Can Go Wrong?

• Dont say correlation when you mean
  association.
• More often than not, people say correlation when
  they mean association.
• The word correlation should be reserved for
  measuring the strength and direction of the
  linear relationship between two quantitative
  variables.

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What Can Go Wrong? (cont.)

• Dont correlate categorical variables.
• Be sure to check the Quantitative Variables
  Condition.
• Be sure the association is linear.
• There may be a strong association between two
  variables that have a nonlinear association.

16
What Can Go Wrong? (cont.)

• Beware of outliers.
• Even a single outlier can dominate the
  
  correlation value.
• Make sure to check
  the Outlier Condition.

17
What should we do here?
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What Can Go Wrong? (cont.)

• Dont confuse correlation with causation.
• Not every relationship is one of cause and
  effect.

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What Can Go Wrong? (cont.)

• Watch out for lurking variables.
• A hidden variable that stands behind a
  relationship and determines it by simultaneously
  affecting the other two variables is called a
  lurking variable.
• Ice cream sales and drowning deaths

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Review Looking at Scatterplots

• Scatterplots may be the most common and most
  effective display for data.
• In a scatterplot, you can see patterns, trends,
  relationships, and even the occasional
  extraordinary value sitting apart from the
  others.
• Scatterplots are the best way to start observing
  the relationship and the ideal way to picture
  associations between two quantitative variables.

21
Looking at Scatterplots (cont.)

• When looking at scatterplots, we will look for
  direction, form, strength, and unusual features.
• Direction
• A pattern that runs from the upper left to the
  lower right is said to have a negative direction.
  
• A trend running the other way has a positive
  direction.

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Looking at Scatterplots (cont.)

• Form
• Linear / Non-linear
• Strength
• Weak/moderate/strong
• Direction
• Positive / negative
• Unusual features
• Outliers , clusters, varying scatter across the
  range of x-values

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Roles for Variables

• The explanatory or predictor variable goes on the
  x-axis, and the response variable goes on the
  y-axis.
• Think about a possible causal relationship

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Francis Galton
1822-1911
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Francis Galton

• Cousin of Darwin
• Best known for his work in anthropology and
  heredity and considered the founder of the
  science of eugenics.
• Interested in heredity and the measurement of
  humans

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Francis Galton

• Collected statistics on height, dimensions,
  strength, and other characteristics of large
  numbers of people.
• Demonstrated fundamental techniques in
  statistical measurement, notably in the
  calculation of the correlation between pairs of
  attributes.

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Karl Pearson
1857-1936
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Karl Pearson

• British mathematician and philosopher of science,
  who is best known for developing some of the
  central techniques of modern statistics, and for
  applying these techniques to the problem of
  biological inheritance .
• In the early 1900s, Pearson became interested in
  the work of Francis Galton

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Karl Pearson

• Pearson's research laid much of the foundation
  for 20th-century statistics, defining the
  meanings of correlation, regression analysis, and
  standard deviation

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Karl Pearson

• In 1911 Pearson became the Galton professor of
  eugenics at University College (London)
• He oversaw the compilation and analysis of
  information on the ways in which characteristics
  such as intelligence, criminality, poverty, and
  creativity were passed among generations.
• He hoped to apply these insights to improving the
  human race.

Source MSN Encarta
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AP Stat - Homework

• p. 132-135 7, 8, 11, 12, 14-16, 23 (due Friday)
• Quiz Friday on Ch. 7 and Standardized (z) Scores