DescribingData:$ Two$Variables SAT:(μ(=(1500,(σ=325(9/14/15 2 Statistics:UnlockingthePowerofData (...

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Statistics: Unlocking the Power of Data Lock5

STAT 250 Dr. Kari Lock Morgan

Describing Data: Two Variables

SECTIONS 2.4, 2.5 •  One quantitative variable (2.4) •  One quantitative and one categorical (2.4) •  Two quantitative (2.5)


z-‐score

Which is better, an ACT score of 28 or a combined SAT score of 2100? �  ACT: μ = 21, σ = 5 �  SAT: μ = 1500, σ = 325 �  Assume ACT and SAT scores have approximately bell-‐shaped distributions

a)  ACT score of 28 b)  SAT score of 2100 c)  I don’t know


Honeybee Waggle Dances

https://www.youtube.com/watch?v=-‐7ijI-‐g4jHg Statistics: Unlocking the Power of Data Lock5

Honeybee Waggle Dance � Honeybee scouts investigate new home or food source options; the scouts communicate the information to the hive with a “waggle dance”

� Scientists took bees to an island with only two possible options for nesting: one of very high quality and one of low quality. They recorded ¡ Quality of nesting site ¡ Distance to nesting site ¡ Number of waggle dance circuits performed ¡ Duration of waggle dance

Seeley, T., Honeybee Democracy, Princeton University Press, Princeton, NJ, 2010, p. 128


Questions of the Day How many circuits of the waggle dance

do honey bees do?

How is this related to quality of a nesting site?

How is duration of the dance related to distance to a nesting site?


Other Measures of Location

Maximum = largest data value

Minimum = smallest data value

Quartiles: Q1 = median of the values below m. Q3 = median of the values above m.

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Five Number Summary �  Five Number Summary:

Min Max Q1 Q3 m

25% 25% 25% 25%

Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics


Five Number Summary

The distribution of number of circuits is

a)  Symmetric b)  Right-‐skewed c)  Left-‐skewed d)  Impossible to tell


Percentile

The Pth percentile is the value which is greater than P% of the data

� We already used z-‐scores to determine whether an SAT score of 2100 or an ACT score of 28 is better

� We could also have used percentiles: ¡ ACT score of 28: 91st percentile ¡ SAT score of 2100: 97th percentile


Five Number Summary �  Five Number Summary:

Min Max Q1 Q3 m

0th percentile

100th percentile

50th percentile

75th percentile

25th percentile

25% 25% 25% 25%


Measures of Spread

� Range = Max – Min

� Interquartile Range (IQR) = Q3 – Q1 �  Is the range resistant to outliers? a)  Yes b)  No

�  Is the IQR resistant to outliers? a)  Yes b)  No


Comparing Statistics � Measures of Center:

¡ Mean (not resistant) ¡ Median (resistant)

� Measures of Spread: ¡ Standard deviation (not resistant) ¡  IQR (resistant) ¡ Range (not resistant)

� Most often, we use the mean and the standard deviation, because they are calculated based on all the data values, so use all the available information

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Boxplot

Median Q1

Q3

Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier

Minitab: Graph -> Boxplot -> One Y -> Simple

Middle 50% of data

Outlier

Outlier

*For boxplots, outliers are dejined as any point more than 1.5 IQRs beyond the quartiles (although you don’t have to know that)


Boxplot

This boxplot shows a distribution that is

a)  Symmetric b)  Left-‐skewed c)  Right-‐skewed


One Quantitative and One Categorical

� How is number of waggle circuits related to the quality of the nesting site?

� Two variables ¡ One quantitative (number of circuits) ¡ One categorical (quality – low or high)

� Can do anything for one quantitative variable, broken down by categorical groups


Side-‐by-‐Side Boxplots

Minitab: Graph -> Boxplot -> One Y -> With Groups


Stacked Dotplots

Minitab: Graph -> Dotplot -> One Y -> With Groups


Overlaid Histograms

Minitab: Graph -> Histogram -> With Groups

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Quantitative Statistics by a Categorical Variable

� Any of the statistics we use for a quantitative variable can be looked at separately for each level of a categorical variable

Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics -> By variables


Difference in Means � Often, when comparing a quantitative variable across two categories, we compute the difference in means

!!xH − xL = 90.5−30=60.5Honeybees perform 60.5 circuits more, on average, for the high quality site as opposed to the low quality site.


Association?

Does there appear to be an association between number of waggle circuits and quality of potential nesting site?

a)  Yes b)  No


Summary: One Quantitative and One Categorical

�  Summary Statistics ¡  Any summary statistics for quantitative variables, broken down by groups

¡  Difference in means

�  Visualization ¡  Side-‐by-‐side graphs


Two Quantitative Variables

�  How is duration of the dance related to distance to a nesting site?

�  Two quantitative variables

�  Summary Statistics: correlation �  Visualization: scatterplot


Scatterplot

A scatterplot is the graph of the relationship between two quantitative variables.

Minitab: Graph -> Scatterplot -> Simple

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Direction of Association �  A positive association means that values of one variable tend to be higher when values of the other variable are higher �  A negative association means that values of one variable tend to be lower when values of the other variable are higher �  Two variables are not associated if knowing the value of one variable does not give you any information about the value of the other variable


Correlation

The correlation is a measure of the strength and direction of linear association

between two quantitative variables

•  Sample correlation: r •  Population correlation: ρ (“rho”)

Minitab: Stat -> Basic Statistics -> Correlation

r = 0.994 for duration of dance and distance to site


Correlation 1.  -‐1 ≤ r ≤ 1 2.  The sign indicates the direction of association

1.  positive association: r > 0 2.  negative association: r < 0 3.  no linear association: r ≠ 0

3.  The closer r is to ±1, the stronger the linear association 4.  r has no units and does not depend on the units of measurement 5.  The correlation between X and Y is the same as the correlation between Y and X


Correlation Guessing Game http://www.istics.net/Correlations/

Enter PennState for the group ID. Highest scorer in the class by the Birst exam gets one extra credit point on Exam 1!


3.0 3.5 4.0 4.5 5.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Malevolence Rating of Uniform

z-sc

ore

for P

enal

ty Y

ards

Correlation

r = 0.43

NFL Teams


Correlation Cautions 1.  Correlation can be heavily affected by outliers. Always plot your data!

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Testosterone Levels and Time What is the correlation between testosterone levels and hour of the day? a)  Positive b)  Negative c)  About 0

Are testosterone level and hour of the day associated? a)  Yes b)  No



2.  r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data!


TVs and Life Expectancy

0 200 400 600 800 1000

4050

6070

80

TV and Life Expectancy

TVs per 1000 People

Life

Exp

ecta

ncy

Angola

Australia

Cambodia

Canada

ChinaEgypt

France

Haiti

Iraq

Japan

Madagascar

Mexico

Morocco

Pakistan

Russia

South Africa

Sri Lanka

Uganda

United KingdomUnited States

Vietnam

Yemen

r = 0.74



2.  r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data!

3.  Correlation does not imply causation!


Summary: Two Quantitative Variables

�  Summary Statistics: correlation �  Visualization: scatterplot


To Do � Read Sections 2.4 and 2.5

� Do HW 2.2, 2.3, 2.4, 2.5 (due Friday, 9/18)

DescribingData:$ Two$Variables SAT:(μ(=(1500,(σ=325(9/14/15 2 Statistics:UnlockingthePowerofData (...

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