Measures of Location and variability

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Measures of Location and variability

• Chapter 2.4 Summary Measures of Location
• mean
• median
• quartiles
• Chapter 2.5 Summary Measures of variability
• range
• standard deviation (sd)
• inter quartile range (IQR)

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Measures of Location

• Chapter 2.4 Summary Measures of Location
• mean
• median
• trimmed mean

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Summary Measurements

• A parameter is a numerical summary measure of a
  population distribution. ( refers to the entire
  population )
• A statistic is a numerical quantity calculated
  from the observations in a sample. (obtained from
  information in the sample)

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Mean

• The population mean, denoted by ?, is the balance
  point of the population distribution, also called
  the center of the mass, of the population
  distribution.

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sample mean

• The sample mean is the average of the all
  observations. It gives the approximate value of
  the population mean. If a sample consists of
  observations y1, y2, , yn, then the sample mean
  is

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Example 2.4.1

• Here is the net worth of 10 residents of
  Washington state (in thousands of dollars) 100,
  1000, 250, 25, 750, 575, 2500, 3200, 670, 320.
  Compute the sample mean of the net worth.
• Solution Sample mean

The average net worth of the 10 residents is 1039
thousand dollars
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Continued

• What happens if we add Bill Gates' net worth of
  40.5 billion dollars, which is 40500000
  thousands of dollars?
• an outlier (a number that stand apart from the
  remainder of the data ).
• 3,682,763

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the net worth of residents

40500000
710
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Median

• The population median, denoted by ? , is the
  numerical value that divides the population
  distribution in half. It is also called the
  second quartile.

50
50
?
?
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Median

• The sample median, denoted by M, is the middle
  observarion if n is odd, or the average of the
  two middle observation if n is even. In either
  case, the median is located at the position
  (n1)/2 in the ordered data set.
• Example 5. 1, 2, 2, 3, 6, 7, 8
• Example 6. 8, 9, 10, 2, 6, 10

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Example 2.4.1(continued)100, 1000, 250, 25, 750,
575, 2500, 3200, 670, 320

• Steps to find median
• Step1,Order observations from smallest to
  largest.
• 25 100 250 320 670 750 1000 1575
  2500 3200
• Step 2,Count the observations, denote the total
  number as n. n10

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• Step3,Find the location of the median, which is
  in the (n1)/2 th position
• If n is odd, the median is the middle value.
• If n is even, the median is the average of the
  middle two values
• (101)/25.5 ,the median is
• (670 750)/2710

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Exercise Including Bill Gates' net worth, what
is the median of the net worth.

• 100, 1000, 250, 25, 750, 575, 2500, 3200, 670,
  320, 40500000
• Solution
• 25 100 250 320 670 750 1000 1575
  2500 3200 40500000
• n11,(111)/26
• the median 750

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Example 1

• data -1, 1
• data -2, 1,1
• data -3, -2, -1, 1, 1, 1, 1, 1, 1
• example 2
• 1, 2, 1, 2, 1, 2, 1, 2, 1, 2,
• 1, 2, 1, 2, 1, 2, 1, 2, 1, 20

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Trimmed mean

• Motivation
• A p trimmed sample mean
• Olympic game rating system
• use 1/9 trimmed mean

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Trimmed mean

• Example 3 Calculate 5 trimmed mean of the
  above example.
• 1, 2, 1, 2, 1, 2, 1, 2, 1, 2,
• 1, 2, 1, 2, 1, 2, 1, 2, 1, 20

Answer N 20 obs, 5201, then the remain data
set is 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,
2, 1, 2, 1 , Answer _____.
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Exercise

• A stem and leaf is given (n10)
• 1 078
• 2 02457
• 3 14
• Find the 10 trimmed sample mean. _____

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Quartiles

• The first quartile, denoted by ? 1 , is the
  numerical value that divides the lower half of
  the population in half. The first sample
  quartile, Q1 can estimate it.
• The third quartile, denoted by ? 3 , is the
  numerical value that divides the upper half of
  the population in half. The third sample quartile
  Q3 can estimate it.
• The first and third sample quartiles, Q1 and Q3,
  are similarly defined for samples. The median is
  the second quartile, Q2.

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Quartiles

• Q3 Upper quartile median of upper half
• (include median if
  n is odd)
• Q1 Lower quartile median of lower half
• (include median if
  n is odd)
• Q2median

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Example 1
Data (sorted!) 35 37 45 46 49 56 57 57 59
61 62 64 68 71 72 76 80 89 94
Calculate Max, Min, n, Mean, Median, Q1 and
Q3

• Max 94, Min 35, n19, Mean 62, Median
  61
• Upper half
• 35 37 45 46 49 56 57 57 59 61 62 64 68 71 72
  76 80 89 94
• Q3 (7172)/2 71.5
• Lower half
• 35 37 45 46 49 56 57 57 59 61 62 64 68 71 72 76
  80 89 94
• Q1 (49 56)/2 52.5

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Example 2

• Researchers have investigated lead absorption in
  children of parents who worked in a factory where
  lead is used to make batteries. A stem and leaf
  is given (n10)
• 4 07
• 5
• 6 14
• 7 1349
• 8
• 9 2
• 10 3
• Compute the following quantities
• The sample mean , 10 trimmed mean,
• sample median M, first quantile Q1 and third
  quantile Q3.

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Chapter 2.5 Summary Measure of Variability

• range,
• standard deviation (sd)
• inter quartile range (IQR) (Q spread)

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One open question

• The following two data sets are scores of student
  A and student B in some tests.
• A60, 60, 80, 80, 80, 90, 90
• B30, 50, 80, 80, 80, 100, 120
• Can the location measures tell the difference
  between them ?

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A60, 60, 80, 80, 80, 90, 90 B30, 50, 80,
80, 80, 100, 120
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• Range H-L
• Q-spread is the distance between the first and
  third sample quartile, Q3 Q1.
• The corresponding q-spread is similarly
  defined using the population quartiles in place
  of the sample quartiles. (This measure of
  variability is resistant to the influence of
  outliers)
• Standard deviation is the most widely used.

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• The sample variance, denoted by s2, is the
  average squared distance of all measurements from
  the sample mean.
• A small question why do we square distance?
• The expression in the numerator is referred to as
  a Sum of squares

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Standard deviation
Standard deviation is the positive square root of
the variance.
The population standard deviation is denoted by
?, the sample standard deviation is denoted by
s.
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Example

• Data set is given as follows
• 3 4 10 7 6
• mean median
• variance
• standard deviation

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Interpreting the standard deviation s

• If we have two samples, a larger value of s
  in one sample reflects greater variation of the
  observations from the mean than the other sample.

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• While, if we have one sample, once we know
  standard deviation, we can tell the percent of
  the data that is with in a specified number of
  standard deviation. E.g., what percent of the
  distribution is within one standard deviation of
  the mean? The answer depends on the shape of the
  distribution.

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Variability- The standard deviation

• Standard deviation has also meaning when used
  with only one sample. The number of measurements
  that fall within 1, 2 and 3 standard deviations
  of the mean are calculated by the following two
  rules
• -Chebyshevs rule
• -Empirical rule
• Chebyshevs rule applies to any set of data.
• The empirical rule applied only to bell shaped
  symmetrical distributions of data.

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• Empirical rule

-Approximately 68 of the measurements fall
within 1 std of the mean. -Approximately 95 of
the measurements fall within 2 std of the
mean. -Essentially all the measurements will fall
within 3 std of the mean.
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Chebyshev's rule

• Chebyshev's rule (regardless of the shape of the
  distribution)
• (1) At least 3/4 of the measurements will fall
  within two standard deviation of the mean.
  
• (2) At least 8/9 of the measurements will fall
  within three standard deviation of the mean.

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Example

• The recorded temperature on the 24 launches
  previous to the Challenger accident are given
  here in a stem and leaf plot. Calculate the mean
  and the standard deviation and use them to give
  an interpretation of the amount of variability in
  the data using either the empirical rule or
  Chebyshevs rule (page 111).
• 5 378
• 6 3677789
• 7 000023556689
• 8 01

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Answer

• Mean70
• Sd7.2
• 17/2470.868
• 23/2495.895

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z-score

• In the above example, we observed that 31 degrees
  is unusually low. When 31 is included in the data
  set, mean68.44, stDev10.53. How low is it? To
  evaluate a single score, we calculate its
  z-score
• The z-score corresponding to a particular
  observation x is given by
• z(observation-mean)/standard deviation

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z-score

• Negative z-score indicates that the observation
  is below the mean. It is generally assumed that
  any observation with a z-score greater than 3 in
  absolute value is an outlier

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Exercise

• We have a data set of ages of 10 students in one
  university.
• 22 21 27 32 19 20 22 23 18 25
• Draw the stem-and-leaf plot and histogram
• Compute the sample mean and 10 trimmed mean
• Compute the range and Q-spread .