SBM Lesson 6B

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SBM Lesson 6BPutting Statistics to Work

• Measures of Variation

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Why Does Variation Matter?

• Imagine customers waiting in line at 2 different
  banks. Customers at Big Bank can enter at 3
  lines leading to a teller at each. While at Best
  Bank, customers are in one line going to 3
  tellers. The following data shows the waiting
  time at each bank (in minutes)
• Big Bank (3 lines)
• 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3
  11.0
• Best Bank (1 line)
• 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8
  7.8
• Where are there more unhappy customers?

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• Youll probably find more unhappy customers at
  Big Bank than at Best Bank, but it is not because
  the average wait is any longer. In fact the mean
  and median waiting times at both banks is 7.2
  min.
• The waiting times at Big Bank vary over a fairly
  wide range of time. This gives the illusion to
  customers of long and annoying waits in line.
• In contrast, the variation of waiting times at
  Best Bank is small, so the customers feel they
  are being treated fairly.

(See Figure 6.9 on page 371)
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What is Range?

• The Range of a data set is the difference between
  its highest and lowest data values

Range highest value (max) - lowest value (min)
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Example One

• The monthly incomes of 20 employees working for a
  certain company are as follows
• 2300, 3400, 2400, 2600, 1800, 2500, 4700, 3200,
  2400, 3100, 2000, 2600, 2700, 3300, 1900, 4200,
  3100, 1700, 2800, 2900
• Find the range of the data.

ANS 4700 -1700 3000
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Quartiles???

• The Lower Quartile (or 1st quartile) divides the
  lowest fourth of a data set from the upper
  three-fourths. It is the median of the data
  values in the lower half of a data set.
• The Middle Quartile (or 2nd quartile) is the
  overall median.
• The Upper Quartile (or 3rd quartile) divides the
  lowest three-fourths of a data set from the upper
  fourth. It is the median of the data values in
  the upper half of a data set.

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The Five-Number Summary

• The Five-Number Summary for a data set consists
  of the following 5 numbers
• Low Value
• Lower Quartile
• Median
• Upper Quartile
• High Value

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Remember Big and Best Banks?
Median
Low Value
High Value
Big Bank 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7
8.5 9.3 11.0
Best Bank 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4
7.7 7.8 7.8
Lower Quartile
Upper Quartile
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Example Two

• A set of data consists of the numbers
• 5.7, 8.3, 8.9, 8.9, 9.4, 9.7, 10.4, 10.6, 12.6,
  13.8
• Find the five-number summary for the data.
• ANS
• Low 5.7
• Lower Quartile 8.9
• Median 9.55
• Upper Quartile 10.6
• High 13.8

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A Box-and-Whisker Plot

• A Boxplot shows the five-number summary visually,
  with a rectangular box enclosing the lower and
  upper quartiles, a line marking the median, and
  whiskers extending to the low and high values.
• The next page shows a box-and-whisker plot going
  vertically of Students Heights.

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Height in inches of 100 Students
High Value (70.25 in.)
Upper Quartile (64.60 in.)
Median (62.50 in.)
Lower Quartile (60.50 in.)
Low Value (56.42 in.)
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Standard Deviation

• The single most commonly used number to describe
  variation is called the Standard Deviation.
• The Standard Deviation (S.D.) is a a measure of
  how far data values are spread around the mean of
  a data set. So, to calculate the S.D.
• Find the Mean
• Then find how much each data value deviates
  from the mean

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Remember Big and Best Banks?

• Since there mean was 7.2 minutes at each bank, we
  can find a Deviation from any data point.
• Example A waiting time of 8.5 minutes at Big
  Bank gives us a deviation of 8.5 - 7.2 1.3 min.
• Example A waiting time of 6.6 minutes at Best
  Bank gives us a deviation of 6.6 - 7.2 -0.6
  min.
• In essence, the standard deviation is a measure
  of the average of all the deviations from the
  mean. However, for technical reasons, this
  average is not a simple mean. Instead, use the
  method on the next slide. (or page 376)

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Calculating the Standard Deviation

• 1 Compute the mean of the data set. Then find
  the deviation from the mean for every data
  value.
• 2 Find the squares (2nd power) of all the
  deviations from the mean.
• 3 Add all the squares of the deviations from
  the mean.
• 4 Divide this sum by the total number of data
  values minus 1 (one).
• 5 The Standard Deviation is the square root
  of this quotient.

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The Standard Deviation Formula
Sum of (deviations from the mean?)
Total number of data values ?1
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Example Three

• The weights, in pounds, of a group of dogs are
• 34, 37, 44, 57, 64, 82
• Calculate the standard deviation.
• Hint You must use the S.D. Formula
• 1st Find the Mean (? 53.0)
• 2nd Find each Deviation, then square it
• (see next slide)
• 3rd Plug the values into the S.D. Formula
• (see next slide)

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Example Three (cont)
( Data Value - Mean ) ? _______
(34-53)? 361 (37-53)? 256 (44-53)?
81 (57-53)? 16 (64-53)? 121 (82-53)? 841
361 256 81 16 121 841
6 - 1
ANS 18.31 pounds
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Homework

• 6B s 4 - 6, 8 (skip histograms)
• These will take some time to complete, so use the
  book for a reference to complete each problem.
• We will not answer any questions over the lesson
  before we grade the assignment!