Unit 3 Summary Statistics (Descriptive Statistics) FPP Chapter 4

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Unit 3Summary Statistics(Descriptive
Statistics)FPP Chapter 4

• For one variable -
• - Center of distribution
• "central value", "typical value"
• - Spread of distribution
• How variable are the values in a set of data?
• - Measure how many / what proportion of
  observations are above / below a given value.

W.01
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Summary Statistics
Purposes compact reporting easy
comparison Important considerations
interpretable stable

• We will discuss
• how the statistics are defined
• when each is (in)appropriate
• how to interpret them
• how to compute them
• "guesstimation" techniques

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Example Hospital Charges

• Total charge (in dollars) of the hospital stay
  for 29 normal deliveries of babies

Charges 1,905 2,324 2,048 2,888 2,907
2,840 2,607 2,823 2,310 2,953 2,138
3,418 4,903 3,729 3,709 5,063 3,932
3,392 3,287 3,819 4,248 2,640 2,921
2,785 2,804 2,955 2,219 2,184 2,681
14,898
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Definitions
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10
8
freq.
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4
2
1500 2500 3500 4500 5500
Hospital Charges (in Dollars)
mode most frequently occurring value
_______________ median "middle value"
__________________ mean sum /
measurements in the data set
__________/___________ _________
another way to compute the mean
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Locating These SummaryStatistics on a Histogram
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10
8
freq.
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4
2
1500 2500 3500 4500 5500
Hospital Charges (in Dollars)

• mode
• median
• mean
• comparing mean median
• For skewed histograms, the mean could be
  deceiving.

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Event Day Abnormal Returns

• (ref. "Marketing Science", Fall 1987, vol 6, no
  4, pages 320-335, "Does It Pay to Change Your
  Company's Name?")
• -1.84 -0.31 0.02 0.30 0.53 1.09
• -1.38 -0.24 0.06 0.34 0.55
  1.12
• -1.00 -0.24 0.09 0.36 0.58
  1.23
• -0.59 -0.20 0.10 0.39 0.78
  1.43
• -0.57 -0.16 0.13 0.40 0.81
  1.50
• -0.56 -0.06 0.21 0.41 0.96
  1.60
• -0.51 -0.05 0.23 0.43 0.98
  1.64
• -0.44 -0.02 0.24 0.45 0.99
  1.79
• -0.39 -0.02 0.25 0.48 1.00
• -0.33 -0.01 0.29 0.50 1.03

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• mode most frequently occurring value ______
• median "middle value" __________
• mean "average"
• (sum of values in list)/( values
  in list)
• _____ / _____ _____
• p th percentile the value with p
  percent of the list less than (or equal to it)
  and 100-p percent greater than it
• 10 th percentile _____
• 25 th percentile _____

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Histogram for Abnormal Returns
0.4
20
0.3
15
0.2
10
5
0.1
-2.0 -0.5 1.0 2.5 4.0
RETURNS
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Does This Statistic Make Sense?

• Some summary statistics make sense only for
  certain types of data.
• mean
• median
• mode

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Water Watch
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• Aug 1-22 the average consumption was 223.7
  million gallons per day.
• Aug 1-25 the average consumption was 224.4
  million gallons per day.
• Q1 Was the average consumption higher Aug 1-22
  or Aug 23-25?
• Q2 What was the total amount of water consumed
  Aug 23-25?
• Q3 What was the average daily consumption Aug
  23-25?

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Baseball Batting Averages

• Suppose
• batting average ( hits / at bats) x 1000
• Before the game starts, a player has batting
  average 250.
• - first at bat, strikes out
• - new batting average 200
• Q1 How many times has this batter been up?
• Another player starts the game with batting
  average 500. After his first at bat, his new
  batting average is 524.
• Q2 Did he get a hit?
• Q3 How many times has this batter been up?

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Measures ofLocation Spreadof a Data Set

• LOCATION
• mean
• median
• mode
• SPREAD
• standard deviation (SD)
• range
• variance

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Range

• RANGE
• (largest measurement) - (smallest measurement)
• example

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Deviation from Average

• definition
• deviation from average data value - average
• note
• A deviation can be zero.
• 1 2 5 7 10 data
  value

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Standard Deviationof a list of numbers

• definition
• standard deviation SD
• rms size of the deviations from average

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rms (root mean square) size of a list of
numbers

• root-mean-square (rms) operation
• 1 2 5 7 10 data
  value
• deviation

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Standard Deviation Try another list of numbers.

• Find the standard deviation (rms size of the
  deviations from average) for this list of
  numbers.
• 2, - 6, 12, 4, 6
• I. Find the average of this list of numbers.
• II. Find the deviation of each value from
  this average.
• III. Find the rms size of the list of
  deviations.
• -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
  6 7 8 9 10 11 12 data

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Standard Deviation

• The STANDARD DEVIATION (SD) OF A DATA SET
  measures how far away numbers are from their
  average.
• Most entries on the list will be somewhere
  around one SD away from the average. Very few
  will be more than two or three SDs away.

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Interpreting theStandard Deviation

• Roughly 68 of the entries on a list (roughly
  2/3 of the entries) are within one SD of the
  average.
• The other 32 (approximately 1/3) are further
  away.
• Roughly 95 (19 out of 20) are within two SDs
  of the average.
• The other 5 are further away.
• The 2/3 rule is true for most data sets.
• The 95 rule is true for many data sets, but not
  all.

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Delivery Times Example
TIME IN DAYS 27 68 79 91 107 43 71 80 91 1
08 43 71 81 93 108 44 71 83 94 116 47 73 84 94
120 49 73 84 94 120 50 74 84 97 122 54 75 86 97
123 58 76 88 103 127 65 77 88 106 128

• Class Limits Tallies Frequency
• 25-34 1 35-44 3 45-54 4 55-64 1
  65-74 8 75-84
  10 85-94 9
  95-104 3 105-114 4
  115-124 5 125-134 2

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Delivery Times Continued
Days Elapsed Between Order Date and Delivery Date
for 50 Orders
.20
rel. freq.
.16
.12
.08
.04
25 45 65 85 105 125 days
Elapsed Time to Delivery

• average (mean)
• median
• SD

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Delivery Times - 3
The 2/3 Rule says that Roughly 2/3 or 68 of
the entries on a list are within one SD of the
average. 108.0 days

• Actually, in this data set, 34 out of 50
  deliveries took between 59.4 and 108.0 days.
• 34/50 0.68 68

The 95 Rule says that Roughly 95 of the
entries on a list are within two SDs of the
average. 108.0 days
Actually, 49 out of 50 deliveries took between
35.1 and 132.3 days. 49/50
0.98 98
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Guesstimating the SDMiddle 2/3 Rule

• 1. Locate the middle 2/3 of the data.
• 2. The range of the middle 2/3 of the data is
  approximately 2 SD's.
• So, 1/2 of this range is approximately 1 SD.

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Variance
The variance of a list of numbers is the SD
squared. That is, the SD is the square root of
the variance.
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z-score

• The z-score says how many SD's above () or below
  (-) the average a value is.
• The sample z-score for a measurement is
• z
• The population z-score for a measurement is
• z
• example

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Interpreting z-scores

• Interpretation of z-Scores for "Mound-Shaped"
  Distributions of Data
• 1. Approximately 68 of the measurements will
  have a z-score between -1 and 1.
• 2. Approximately 95 of the measurements will
  have a z-score between -2 and 2.
• 3. All or almost all of the measurements will
  have a z-score between -3 and 3.

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Wonderlic Scores
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USC had average team score 20.3. What is their
z-score? Is this value extreme among NCAA
Division I teams? How about Michigan State whose
average team score is 16.6? Find their z-score
and interpret it. How about Stanford whose
average team score is 28.2? Find their z-score
and interpret it.
.