Transcript Measures of Spread and Boxplots

Describing Distributions
Numerically
Measures of Variation
And
Boxplots
Boxplots
Range: highest number - lowest number
Five number summary:
Minimum
Q1
Median
Q3
Maximum
Boxplot Continued
Interquartile Range:
IQR = Q3 - Q1
*Tells us how much territory the middle half of
the data covers.
Percentile: for whole number P (where
1≤P≤99), the Pth percentile of a distribution is
a value such that P% of the data fall at or
below it and (100-P)% of the data fall at or
above it.
Histogram
Median-splits the
histogram into two
halves with equal
area
Mean-point at
which the
histogram would
balance
Measures of Variation
Deviation: how far each data value is from
the mean
Variance (s2): average (almost) of
squared deviations
Standard Deviation (s):
Thinking about Variation…
The U.S. Census Bureau reports the
median family income in its summary of
census data. Why do you suppose they
use the median instead of the mean?
What might be the disadvantages of
reporting the mean?
Thinking about Variation…
You’ve just bought a new car that claims
to get a highway fuel efficiency of 31
mpg. Of course, your mileage will vary.
If you had to guess, would you expect
the IQR of gas mileage attained by all
cars like yours be 30 mpg, 3 mpg, or 0.3
mpg? Why?
Thinking about Variation…
A company selling a new MP3 player
advertises that the player has a mean
lifetime of 5 years. If you were in
charge of quality control at the factory,
would you prefer that the standard
deviation of lifespans of the players you
produce be 2 years or 2 months? Why?
Rules about shape, center,
and spread
1. If the shape is skewed, report the median
and IQR.
2. If the shape is symmetrical, report the mean
and standard deviation. IQR is usually
larger than the standard deviation.
3. If outliers, report mean and standard
deviation with outliers present and with
outliers removed.
Summarizing a
Distribution
A man owned a 1989 Nissan
Maxima for 8 years. Being a
statistician, he recorded the
car’s fuel efficiency (in mpg)
each time he filled the tank.
He wanted to know what fuel
efficiency to expect as
“ordinary” for his car.
Knowing this, he was able to
predict when he’d need to fill
the tank again, and notice if
the fuel efficiency suddenly
got worse, which could be a
sign of trouble. What does
the data say?
When comparing boxplots
• Compare the medians, which group has the
higher center?
• Compare the IQRs; which group is more
spread out?
• Judged by the size of the IQRs, are the
medians very different?
• Check for possible outliers. Identify them if
you can.
Comparing
Boxplots
A student designed an experiment to
test the efficiency of various coffee
containers by placing hot liquid in each
of 4 different containers types 8
different times. After 30 minutes she
measured the temperature again and
recorded the difference in temperature.
What can we say about the
effectiveness of these four mugs?
*Because these are temperature
differences, smaller differences mean
that the liquid stayed hot.
Measure of Variation
Continued
Coefficient of Variation:
Chebyshev’s Theorem: