Describing Distributions with Numbers

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Chapter 12

• Describing Distributions with Numbers

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Today Math Cookies

• Pick up one cookie a handout
• DO NOT EAT IT (yet). You may eat it later once we
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Counting rule

• Anything brown counts for a chip.
• Carefully count front and back surfaces.

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Review

• Categorical variables
• pie bar graphs
• Quantitative variables
• stemplots histograms
• Good bad graphs
• Shape
• symmetric, skewed

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Quick math overview

• ? sum

• These expressions are algebraically equivalent

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Examples
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Turning Data Into Information

• Center of the data
• mean
• median
• mode
• Spread of the data (variability)
• variance
• standard deviation
• range
• interquartile range

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Centers of Data

• Average - a single data value that represents all
  of the data
• mean (arithmetic average)
• median
• mode

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Mean ( )

• Traditional measure of center
• Sum the values and divide by the number of values

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Sample mean

• Grades 68, 79, 60, 72, 77, 76, 69, 70, 60, 79.

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Median (M)

• A resistant measure of the datas center
• Median - the center of value of ordered (ranked)
  data
• If n is odd, the median is the middle ordered
  value
• If n is even, the median is the average of the
  two middle ordered values
• Median 1/2(n1)th position in ordered set

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Median

• Example 1 data 2 4 6
• Median (M) 4
• Example 2 data 2 4 6 8
• Median 5 (avg. of 4
  and 6)
• Example 3 data 6 2 4
• Median ? 2
• (order the values 2 4 6 , so Median 4)

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Sample median

• Grades 68, 79, 60, 72, 77, 76, 69, 70, 60, 79.
• rank data
• 60, 60, 68, 69, 70, 72, 76, 77, 79, 79
• Find position
• (1/2)(n1) 11/2 51/2th position
• Locate median
• M (7072)/2 71

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• Example
• minutes waiting for the PRT (n8)
• x 5, 11, 9, 15, 33, 3, 7, 12

Median RANK DATA FIRST! 3, 5, 7, 9, 11, 12, 15,
33
Median is 1/2(n1)th position (81)/2
41/2 41/2 th position is half-way between 9 and
11. (911)/2 10 Median10
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Comparing the Mean Median

• The mean and median of data from a symmetric
  distribution should be close together. The
  actual (true) mean and median of a symmetric
  distribution are exactly the same.
• In a skewed distribution, the mean is farther out
  in the long tail than is the median the mean is
  pulled in the direction of the possible
  outlier(s).

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Mean vs. Median

• Which should we use?
• Symmetric or approx symmetric use mean
• Significantly skewed used median
• affected by outliers (extreme values)

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Outliers?

• If it is a mistake and is documented, we can
  eliminate it
• If it is not a mistake, do not eliminate it
• A statistic is robust if it is not led too far
  astray by a few outliers. Means (and standard
  deviations) are not robust.

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Mode

• Observed value that occurs with the greatest
  frequency
• Note if no mode, write none not 0
• If two modes bimodal

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Sample mode

• Grades 60, 60, 68, 69, 70, 72, 76, 77, 79, 79
• There are two modes, so this data is bimodal

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Measures of Dispersion

• spread - A general term referring to how spread
  out or variable a set of numbers is.
• Very large spread
• 0, 100, 9999, 100000
• No spread
• 12, 12, 12, 12, 12

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Spread or Variability

• If all values are the same, then they all equal
  the mean. There is no spread.
• Variability exists when some values are different
  from (above or below) the mean.
• We will discuss the following measures of spread
  range, interquartile range, variance, standard
  deviation.

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Range

• One way to measure spread is to give the smallest
  (minimum) and largest (maximum) values in the
  data set
• Range max ? min
• ( the values range from min to max )
• The range is strongly affected by outliers, and
  is rarely used

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sample range

• highest data value - lowest data value
• Grades 60, 60, 68, 69, 70, 72, 76, 77, 79, 79
• Range max-min 79-6019

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Quartiles

• Three numbers that divide the ordered data into
  four equal-sized groups.
• Q1 has 25 of the data below it.
• Q2 has 50 of the data below it. (Median)
• Q3 has 75 of the data below it.

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QuartilesUniform Histogram
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Obtaining the Quartiles

• Order the data.
• For Q2, just find the median.
• For Q1, look at the lower half of the data
  values, those to the left of the median find
  the median of this lower half.
• For Q3, look at the upper half of the data
  values, those to the right of the median find
  the median of this upper half.

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Interquartile Range (IQR)

• Used to measure dispersion (spread) with the
  median
• Sample IQR Q3-Q1

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• minutes waiting for the PRT (n8)
• 3, 5, 7, 9, 11, 12, 15, 33
• Recall Median is half-way between 9 and 11
• M10
• Q1 position is half-way between 5 and 7
• Q1 6
• Q3 is half-way between 12 and 15
• Q3 131/2
• IQR Q3-Q1 13.5-6 7.5

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The five-number summary boxplots
M Q1 Q3 Min
Max

• 5 summary
• Min
• Q1
• M
• Q3
• Max

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Boxplot(from Five-Number Summary)

• Central box spans Q1 and Q3.
• A line in the box marks the median M.
• Lines extend from the box out to the minimum and
  maximum.

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106 13.53
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PRT example 5 summary and boxplot
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29 30 31 32 33
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OUTLIER BOX PLOTS Whats the fastest you have
ever driven a car? ____ mph.
Males (87 Students)
110 95 120 55 150
Females (102 Students)
89 80 95 30 130

• Outliers greater than 1.5(IQR) below Q1 or above
  Q3

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Standard deviation?
The length of human pregnancies has a mean, or
average, of 266 days for the entire population of
women. It also has a standard deviation of 16
days. What do you think is meant by the term
standard deviation?
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Variance and Standard Deviation

• When variability exists, each data value has an
  associated deviation from the mean
• What is a typical deviation from the mean?
  (standard deviation)
• Small values of this typical deviation indicate
  small spread in the data
• Large values of this typical deviation indicate
  large spread in the data

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Variance

• Find the mean
• Find the deviation of each value from the mean
• Square the deviations
• Sum the squared deviations
• Divide the sum by n-1
• (gives typical squared deviation from mean)

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Variance Formula
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Standard Deviation Formulatypical deviation from
the mean
standard deviation square root of the
variance
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• Let's say I have two classes, class A and class
  B, and I want to know how my students are doing.
  I take a random sample of 10 grades from each
  class. Suppose the average in both classes turns
  out to be 71. We might infer that class A and
  class B are very similar, right?

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• class A 68, 79, 60, 72, 77, 76, 69, 70, 60, 79.
• class B 99, 99, 98, 96, 97, 97, 30, 35, 20, 39.
• Something is going terribly wrong in class B!
  Some students are doing exceptionally well and
  some are failing. Using only the sample mean, I
  would think the classes are performing about the
  same.

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Calculating s

• class A 68, 79, 60, 72, 77, 76, 69, 70, 60, 79.

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sample SD for class A
Sample std dev is approx 7
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• class B 99, 99, 98, 96, 97, 97, 30, 35, 20, 39
• s 35
• (5 times as much as class A!)

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Choosing a Summary

• Outliers affect the values of the mean and
  standard deviation.
• The five-number summary should be used to
  describe center and spread for skewed
  distributions, or when outliers are present.
• Use the mean and standard deviation for
  reasonably symmetric distributions that are free
  of outliers.

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Distn of calories in popular candy bars
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Todays concepts

• Numerical Summaries
• Center (mean, median)
• Spread (variance, std. dev., range, IQR)
• Five-number summary Boxplots
• Choosing mean versus median
• Choosing standard deviation versus five-number
  summary