Ch5: Describing Distributions Numerically Finding the Center: The Median

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Ch5 Describing Distributions Numerically
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.
• As a measure of center, the midrange (the average
  of the minimum and maximum values) is very
  sensitive to skewed distributions and outliers.
• The median is a more reasonable choice for center
  than the midrange

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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 2
  two equal areas.
• For an even number of
• data pts, average the 2
• middle ones
• median(2,4,5,6,6,7) 5.5
• It has the same
• units as the data.

Healthy Life Expectancy (HALE) Measure for all UN
Members, 2001
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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 find the
  Quartiles, which 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.
• If the data has an even of points, this
  division is straightforward. If it is odd, then
  the text tells you to count the median in both
  halves of the data.
• 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
• 5 number summary for HALEs
• max 73.6
• Q3 62.65
• median 57.7
• Q1 48.9
• min 26.5

Healthy Life Expectancy Measure for all UN
Members, 2001
6
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 66 rock concert goers who died from
  being crushed is

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Making Boxplots

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

And also some additional information, such as
other outliers
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Constructing Box-plots

• Draw a single axis spanning the range of the data
  
• you can draw box-plots vertical or horizontal,
  but this one is oriented vertically, so that is
  how the instructions are described.
• 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. (you can leave them in, if you want, but
  only as dotted lines)

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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 you look at the original data for rock
  concert deaths, this would be 29 years for the
  upper whisker. 13 is the youngest death (and 13
  gt 9.5), so thats the lower whisker end.)
• 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.)

• Now we add any outliers by displaying any data
  values beyond the fences with special symbols.
• Often, a different symbol is used for far
  outliers that are farther than 3 IQRs from the
  quartiles. (This stylistic differentiation is
  optional)
• And we erase the fences (again, optional).

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Rock Concert Deaths Making Boxplots (cont.)

• Compare the histogram and boxplot for
• Worldwide Rock Concert Deaths, 1999-2000
• 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 travel coffee mugs
• What does this graphical display tell you?
• Which coffee container would you recommend using?
• Did we really need to see all 4 histograms to
  reach this conclusion?

Temperature change for Brands of Coffee Containers
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Summarizing Symmetric Distributions

• Medians do a good job of identifying the center
  of skewed distributions.
• When we have symmetric data, 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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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

Pulse Rates of 52 Adults
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Mean or Median?
Healthy Life Expectancy Measure for all UN
Members, 2001

• 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 significantly skewed data, though, its
  better to report the median than the mean as a
  measure of center.
• Example Does the HALE data show skew? If so,
  how?

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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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First We Find the Variance

• The variance, notation of s2, is found by summing
  the squared deviations and dividing by n-1
• 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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Then We Take the Square Root

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

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Looking at Center and Spread, an Example

• As part of a Human Resources report, assume weve
  been given annual salaries (K/yr) for 9
  professors as follows.
• Describe the distribution
• What would be an appropriate measure of center
  and spread?

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Looking at Center and Spread, an Example

• Although the data is symmetric, we could still
  determine the Median and the IQR. (First sort
  the data so it is ordered)

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Looking at Center and Spread, an Example

• First Calculate the mean, then calculate the
  Standard Deviation
• We can use Excel to work out the calculations
  step-by-step

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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 describing 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 as likely to be
  affected by the outliers as the mean and SD.

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What Can Go Wrong?

• Dont forget to do a reality checkdont let
  technology do your thinking for you.
• First sort the values before finding the median
  and quartiles.
• Dont compute numerical summaries of a
  categorical variable.
• Watch out for multiple modesmultiple modes might
  indicate multiple groups in your data.
• 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 the first side-by-side boxplots of
  cotinine levels for 3 different types of subjects
• The 2nd boxplots show the same data for
  log(cotinine) values
• This example is an aside, as re-expressing data
  is not going to be tested in DS212

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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.
• 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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Examples

• Based on Pr5- A clerk entering salary data into
  a spreadsheet accidentally put in an extra 0 on
  the bosss salary, listing it as 2,000,000 /yr
  instead of 200,000 /yr. Explain how this error
  will affect these summary statistics for the
  company payroll. (Note Although the text
  doesnt say, you can assume this is a company
  with at least 5 employees and also that the boss
  earns the largest salary!)
• Measures of center Median and Mean
• Measures of Spread Range, IQR, and Standard
  Deviation

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Examples

• Pr41- The data from the CD Rom (also available
  on my website) shows 8th graders average math
  test scores for the participating 66 nations.
• Notes copy the the CD Rom, rather than typing
  all s in by hand! (Saves time and less
  mistake-prone!)
• Excel doesnt have Q1 and Q3 calculations, but
  you can get those by sorting the data (use
  Excels Sort!) adding a column for rank.
• You can use excels built-in functions AVERAGE
  and STDEV
• For more guidance, see the Problem Solution Ive
  provided. There are 2 files, one for the excel
  work and also a word document

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Examples

• Pr39- In a USA Today advertisement (7/2001)
  Net2Phone listed long distance rates to 24 of the
  250 countries they serve. (Hint use the Excel
  data set given!)

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Examples

• Pr39 (Net2Phone Example)- continued
• Display these rates
• Find the mean and median. Which is the most
  appropriate measure of center?
• Find the IQR and Standard deviation. Which is
  the more appropriate measure of spread?
• Are there any outliers? Why?
• Write a description of the rates.
• Can you conclude anything about Net2Phones rates
  to all the countries they serve?

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Examples

• Pr39 (Net2Phone Example)- answers
• A boxplot is a good way to display these rates
• What are the outliers?
• Do they have a huge effect?
• Do you feel comfortable making generalizations
  about Net2Phones service from 10 of the data,
  when the company selected that data
• (See solution set for full details!)