AP Stat Do Now

1
AP Stat - Do Now

• Look at activity 5-2 City Temperatures

2
Objectives

• Chapter 5 Describing Distributions Numerically
• Period 9 Example regarding nuns and quartiles
  (if time)
• How to use the frequency option on the calc.
• IQR Kurtosis (if time)
• Standard deviation

NJCCCS 4.4.12.A.5
3
Using the Frequency Option on the Calc.
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Lets check out the standard deviation worksheet
5
Nuns, happiness, longevity and quartiles

• Reading from pages 3 4 of Authentic Happiness
  by Martin Seligman, Ph.D.
• Learned helplessness

6
IQR and Kurtosis
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Finding the Center The Median

• When we think of a typical value, we usually look
  for the center of the distribution.
• For a unimodal, symmetric distribution, its easy
  to find the centerits just the center of
  symmetry.

8
Finding the Center The Median (cont.)

• The median is the value with exactly half the
  data values below it and half above it.
• It is the middle data
  value (once the data
  
  values have been
  ordered) that divides
  the
  histogram into
  two equal areas.
• It has the same
  units as the data.

9
Spread Home on the Range

• Always report a measure of spread along with a
  measure of center when describing a distribution
  numerically.
• The range of the data is the difference between
  the maximum and minimum values
• Range max min
• A disadvantage of the range is that a single
  extreme value can make it very large and, thus,
  not representative of the data overall.

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Spread The Interquartile Range

• The interquartile range (IQR) lets us ignore
  extreme data values and concentrate on the middle
  of the data.
• To find the IQR, we first need to know what
  quartiles are

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Spread The Interquartile Range (cont.)

• Quartiles divide the data into four equal
  sections.
• The lower quartile is the median of the half of
  the data below the median.
• The upper quartile is the median of the half of
  the data above the median.
• The difference between the quartiles is the IQR,
  so
• IQR upper quartile lower quartile

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Spread The Interquartile Range (cont.)

• The lower and upper quartiles are the 25th and
  75th percentiles of the data, so
• The IQR contains the
  middle 50 of
  the
  values of the
  distribution,
  as shown in
  Figure 5.3
  from the text

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The Five-Number Summary

• The five-number summary of a distribution reports
  its median, quartiles, and extremes (maximum and
  minimum).
• Example The five-number summary for the ages at
  death for rock concert goers who died from being
  crushed is

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Rock Concert Deaths Making Boxplots

• A boxplot is a graphical display of the
  five-number summary.
• Boxplots are particularly useful when comparing
  groups.

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Constructing Boxplots

• Draw a single vertical axis spanning the range of
  the data. Draw short horizontal lines at the
  lower and upper quartiles and at the median. Then
  connect them with vertical lines to form a box.

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Constructing Boxplots (cont.)

• Erect fences around the main part of the data.
• The upper fence is 1.5 IQRs above the upper
  quartile.
• The lower fence is 1.5 IQRs below the lower
  quartile.
• Note the fences only help with constructing the
  boxplot and should not appear in the final
  display.

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Constructing Boxplots (cont.)

• Use the fences to grow whiskers.
• Draw lines from the ends of the box up and down
  to the most extreme data values found within the
  fences.
• If a data value falls outside one of the fences,
  we do not connect it with a whisker.

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Constructing Boxplots (cont.)

• Add the outliers by displaying any data values
  beyond the fences with special symbols.
• We often use a different symbol for far
  outliers that are farther than 3 IQRs from the
  quartiles (optional).

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Rock Concert Deaths Making Boxplots (cont.)

• Compare the histogram and boxplot for rock
  concert deaths
• How does each display represent the distribution?

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Comparing Groups With Boxplots

• The following set of boxplots compares the
  effectiveness of various coffee containers
• What does this graphical display tell you?

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Summarizing Symmetric Distributions

• Medians do a good job of identifying the center
  of skewed distributions, but it is just a
  pointer to a middle value.
• Mean takes into account every single value, so no
  one is left out of the calculation.
• Mean is also used in many of the formulas that we
  will use later in the course.
• When we have symmetric data that is free from
  outliers, the mean is a good measure of center.
• We find the mean by adding up all of the data
  values and dividing by n, the number of data
  values we have.

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Mean or Median?

• Regardless of the shape of the distribution, the
  mean is the point at which a histogram of the
  data would balance

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Mean or Median? (cont.)

• In symmetric distributions, the mean and median
  are approximately the same in value, so either
  measure of center may be used.
• For skewed data, though, its better to report
  the median than the mean as a measure of center.

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Summarizing Symmetric Distributions (cont.)

• The distribution of pulse rates for 52 adults is
  generally symmetric, with a mean of 72.7 beats
  per minute (bpm) and a median of 73 bpm

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The Formula for Averaging

• The formula for the mean is given by
• The formula says that to find the mean, we add up
  the numbers and divide by n.

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What About Spread? The Standard Deviation

• A more powerful measure of spread than the IQR is
  the standard deviation, which takes into account
  how far each data value is from the mean.
• A deviation is the distance that a data value is
  from the mean.
• Since adding all deviations together would total
  zero, we square each deviation and find an
  average of sorts for the deviations.

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What About Spread? The Standard Deviation (cont.)

• The variance, notated by s2, is found by summing
  the squared deviations and (almost) averaging
  them
• The variance will play a role later in our study,
  but it is problematic as a measure of spreadit
  is measured in squared units!

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What About Spread? The Standard Deviation (cont.)

• The standard deviation, s, is just the square
  root of the variance and is measured in the same
  units as the original data.

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Thinking About Variation

• Since Statistics is about variation, spread is an
  important fundamental concept of Statistics.
• Measures of spread help us talk about what we
  dont know.
• When the data values are tightly clustered around
  the center of the distribution, the IQR and
  standard deviation will be small.
• When the data values are scattered far from the
  center, the IQR and standard deviation will be
  large.

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Shape, Center, and Spread

• When telling about a quantitative variable,
  always report the shape of its distribution,
  along with a center and a spread.
• If the shape is skewed, report the median and
  IQR.
• If the shape is symmetric, report the mean and
  standard deviation and possibly the median and
  IQR as well.

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What About Outliers?

• If there are any clear outliers and you are
  reporting the mean and standard deviation, report
  them with the outliers present and with the
  outliers removed. The differences may be quite
  revealing.
• Note The median and IQR are not likely to be
  affected by the outliers.

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What Can Go Wrong?

• Dont forget to do a reality checkdont let
  technology do your thinking for you.
• Dont forget to sort the values before finding
  the median or percentiles.
• Dont compute numerical summaries of a
  categorical variable.
• Watch out for multiple modesmultiple modes might
  indicate multiple groups in your data.

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What Can Go Wrong? (cont.)

• Be aware of slightly different methodsdifferent
  statistics packages and calculators may give you
  different answers for the same data.
• Beware of outliers.
• Make a picture (make a picture, make a picture).

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What Can Go Wrong? (cont.)

• Be careful when comparing groups that have very
  different spreads.
• Consider these side-by-side boxplots of cotinine
  levels

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Re-expressing to Equalize the Spread of Groups

• Here are the side-by-side boxplots of the
  log(cotinine) values

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What have we learned?

• We can now summarize distributions of
  quantitative variables numerically.
• The 5-number summary displays the min, Q1,
  median, Q3, and max.
• Measures of center include the mean and median.
• Measures of spread include the range, IQR, and
  standard deviation.
• We know which measures to use for symmetric
  distributions and skewed distributions.

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What have we learned? (cont.)

• We can also display distributions with boxplots.
• While histograms better show the shape of the
  distribution, boxplots reveal the center, middle
  50, and any outliers in the distribution.
• Boxplots are useful for comparing groups.

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AP Stat - Homework

• P. 73-82 3, 5, 7, 8, 9, 11, 12, 15, 16a-d,
  19-21, 24, 25, 27, 29, 35
• Worth 10 points
• You can work with one other person and hand in
  one assignment (remember that you are responsible
  for all of the content, however)
• Due Thursday
• QUIZ FRIDAY (CHAPTER 5)