Chapter Three

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Chapter Three

• Numerical Descriptive Measures

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Commonly used Descriptive Measures

1. Measures of Central Tendency
2. Measures of Variation
3. Measures of Position
4. Measures of Shape

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Measures of Central Tendency

• Purpose To determine the centre of the data
  values.

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Measures of Central Tendency Answer questions

• Where is the middle of my data?
• Mean, Median, Midrange
• Which data value occurs most often?
• Mode

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The Mean

• The sample mean is denoted by x-bar
• The population mean is denoted by µ (mu)
• x individual data values
• X-bar Sx / n
• µ Sx / N

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Example

• The following are accident data for a 5 month
  period
• 6, 9, 7, 23, 5

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To calculate the average number of accidents per
month

• X-bar Sx / n
• X-bar (6 9 7 23 5) 5
• X- bar 10.0

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Statistic

• What is the average persons monetary value to
  society?

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The Median

• is the centre value in a data set when the data
  are arranged from smallest to
  largest.

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What do we call this ordering process?
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By arranging the data in an Ordered Array

• 5, 6, 7, 9, 23
• With an even number of observations, the value
  that has an equal number of items to the right
  and to the left is the Median.
• Md 7

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To calculate the median with an even number of
observations,

• average the two center values of the ordered
  set.
• Example With an ordered array 5, 6, 7, 9
• Md ( 6 7 ) 2 6.5

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If there is an odd number of observations

• Md (n 1 ) 2
• where
• n of observations

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Remember

• Median describes the centrally placed location of
  a value relative to the rest of the data.

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Question

• Is the mean or median more sensitive to extreme
  values (outliers)?
• Explain.

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The mean is affected by every value.

• The median is unaffected by extreme values.

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The mean is pulled toward extreme values.

• The median does not use all data information
  available.

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Question

• When dealing with data that are likely to contain
  outliers (personal income, ages, or prices of
  houses), would the Mean or Median be preferred as
  the measure of central tendency? Why?

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Think of the Median

• as providing a more
• typical or representative
• value of the situation.

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The Mode(Mo)

• The value that occurs most frequently.

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Questions?

• Can there be more than one mode?
• Is the mode affected by extreme values?
• For continuous variables, is it possible that a
  mode does not exist? Explain?
• Is the mode always a measure of central tendency?

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Give an example of when the mode may provide more
useful information than the mean or the median.
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Example

• From a purchasers standpoint, the most common
  hat or jeans size is what you would like to know,
  not the average hat or jeans size.

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Measures of Central Tendency are useful.
MeansMediansModes
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The use of any single statistic to describe a
complete distribution fails to reveal important
facts.
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Dig Deeper!
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Measures of Variation

• Answers the question
• How spread out are my data values?

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Consider Two Scenarios

• Scenario 1
• Jack buys a car pays 1000.
• Jill buys a car pays 21,000.
• Average Price 11,000

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Scenario 2

• Bob buys a car pays 10,000.
• Mary buys a car
• pays 12,000.
• Average Price 11,000

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Based on the data, both scenarios report the same
average price.
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Whats the difference?
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Quiz

• Suppose you are a purchasing agent for a large
  manufacturing company. Your two suppliers fill
  your orders in an average of 10 days.
• The following histograms plot the delivery time
  of the two suppliers.

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Do the two suppliers have the same reliability in
terms of making deliveries on time?
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Homogeneity the degree of similarity within a
set of data values.

• The mean of a homogeneous data set is far more
  representative of the typical value than a mean
  of a heterogeneous data set.

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If all the data values in a sample are identical,
then the mean provides perfect information, the
variation is zero, and the data are perfectly
homogeneous.
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Variation

• the tendency of data values to scatter about the
  mean, x-bar.

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If all the data values in a sample are identical,
then the mean provides perfect information, the
variation is zero, and the data are perfectly
homogeneous.
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Commonly used Measures of Variation

• Range
• Variance
• Standard Deviation
• Coefficient of Variation (CV)

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The Range

• Range H L
• The value of the range is strongly influenced by
  an outlier in the sample data.

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Variance Standard Deviation

• During a five week production period, a small
  company produced 5,9,16,17, 18 computers,
  respectfully.
• The average 13 computers/wk
• Describe the variability in these five weeks of
  production.

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Variance Standard Deviation
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Formulas for Variance Standard Deviation
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Empirical Rule
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Normally Distributed Data w/ Empirical Rule
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Example Empirical Rule

• A company produces a lightweight valve that is
  specified to weigh 1365 g.
• Unfortunately, because of imperfections in the
  manufacturing process not all of the valves
  produced weigh exactly 1365 grams.
• In fact, the weights of the valves produced are
  normally distributed with a mean weight of 1365
  grams and a standard deviation of 294 grams.

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Question?

• Within what range of weights would approximately
  95 of the valve weights fall?
• 2) Approximately 16 of the weights would be more
  than what value?
• 3) Approximately 0.15 of the weights would be
  less than what value?

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Answers

• 1) 1365 /- 2s 777 to 1953
• 2) 1365 1 s 1659
• 3) 1365 - 3 s 483

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Example 2 Standard Deviation the Empirical Rule

• A recent report states that for California the
  average statewide price of a gallon of regular
  gasoline is 1.52.
• Suppose regular gas prices vary across the state
  with a standard deviation of 0.08 are normally
  distributed.

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With x-bar 1.52 s 0.08

1. Nearly all gas prices (97.7) should fall between
   what prices?
2. Approximately 16 of the gas prices should be
   less than what price?
3. Approximately 2.5 of the gas prices should be
   more than what price?

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Answers

1. µ /- 3s 1.28 and 1.76
2. 1.44 (Since 68 of the prices lie w/in 1s of the
   mean, 32 lie outside this range 16 in each
   tail.
3. 1.68 (Since 95 of the price lie w/in 2 s of the
   mean, 5 lie outside this range 2.5 in each
   tail.

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Coefficient of Variation

• Compares the variation between
• two data sets with different means
• and different standard deviations
• and measures the variation in
• relative terms.

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Coefficient of Variation (CV) formula

• CV s / µ (100)

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CV Example 1

• Spot, the dog, weighs 65 pounds. Spots weight
  fluctuates 5 pounds depending on Spots exercise
  level.
• Sea Biscuit, the horse, weighs 1200 pounds. Sea
  Biscuits weight fluctuates 125 pounds depending
  on the number of rides Sea Biscuit goes on.

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Question?

• Relatively speaking, which animals weight, Spot
  or Sea Biscuits, varies the most?

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Coefficient of Variation vs. Standard Deviation

• Some financial investors use the coefficient of
  variation or the standard deviation or both as
  measures of risk.

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What does the Coefficient of Variation tell us
about the risk of a stock that the standard
deviation does not?
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Relative to the amount invested in a stock, the
coefficient of variation reveals the risk of a
stock in terms of the size of standard deviation
relative to the size of the mean (in percentage).
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CV Example 2

• SUPPOSE
• Five weeks of average prices for stock A are
• 57, 68, 64, 71, and 62.
• While five weeks of average prices for stock B
  are
• 12, 17, 8, 15, and 13.

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QUESTION

• Relative to the amount of money invested in the
  stock, which stock, A or B, is riskier?

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Stock A vs. Stock B in terms of Risk

• Stock B
• µ 13
• s 3.03
• CV s/ µ (100) 23.3

• Stock A
• µ 64.40
• s 4.84
• CV s/ µ (100) 7.5

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Measures of Position

• Indicate how a particular value fits in with all
  the other data values.
• Commonly used measures of position are
• Percentiles
• Quartiles
• Z-scores

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TO FIND THE LOCATION OF THE Pth PERCENTILE

• Determine n P /100 and use one of the following
  two location rules
• Location rule 1. If n P /100 is NOT a counting
  number, round up, and the Pth percentile will be
  the value in this position of the ordered data.
• Location rule 2. If n P /100 is a counting
  number, the Pth percentile is the average of the
  number in this location (of the ordered data) and
  the number in the next largest location.

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Use the two rules of percentiles and the
following data to determine both the 85th and the
50th percentile for starting salary.
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Step 1 Arrange the data in ascending order
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Step 2

• Use the formula for percentiles
• n P /100
• Identify the 85th percentile given 12
  observations
• i n (p /100) 12 (85/100) 10.2

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Because i is not an integer, round up. The
position of the 85th percentile is the next
integer greater than 10.2, the 11th position.
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• From the data, the 85th percentile is the value
  in the 11th position, or 3130.

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To calculate the 50th percentile, apply step 2

• n P /100
• i 12 (50/100) 6
• Because i is an integer, the 50th percentile is
  the average of the sixth and seventh values
• (2890 2920) /2 2905.

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Quartiles

• Quartiles are merely particular percentiles that
  divide the data into quarters
• Q1 1st quartile 25th percentile (P25)
• Q2 2nd quartile 50th percentile (P50)
• Q3 3rd quartile 75th percentile (P75)
• Quartiles are used as benchmarks, much like the
  use of A,B,C,D, and F on exam grades.

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Z- Scores

• A z-score determines the relative position of any
  particular data value x, and is expressed in
  terms of the number of standard deviations above
  or below the mean.

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Measures of Shape

• Measures of shape address skewness and kurtosis.

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Skewness

• Symmetric data the sample mean sample median
• Right-skewed (positive) mean gt median
• Left-skewed (negative) mean lt median

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Closing Example

• The number of defects in 10 rolls of carpets
  are
• 3, 2, 6, 0, 1, 3, 2, 1, 0, 4
• What are the 75th percentile and the 50th
  percentile?
• What are the mean, standard deviation, and
  coefficient of variation?

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