Some definitions

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Some definitions

• In Statistics

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A sample

• Is a subset of the population

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In statistics

• One draws conclusions about the population based
  on data collected from a sample

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Reasons

• Cost

It is less costly to collect data from a sample
then the entire population
Accuracy
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Accuracy
Data from a sample sometimes leads to more
accurate conclusions then data from the entire
population
Costs saved from using a sample can be directed
to obtaining more accurate observations on each
case in the population
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Types of Samples

• different types of samples are determined by how
  the sample is selected.

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Convenience Samples

• In a convenience sample the subjects that are
  most convenient to the researcher are selected as
  objects in the sample.
• This is not a very good procedure for inferential
  Statistical Analysis but is useful for
  exploratory preliminary work.

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Quota samples

• In quota samples subjects are chosen conveniently
  until quotas are met for different subgroups of
  the population.
• This also is useful for exploratory preliminary
  work.

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Random Samples

• Random samples of a given size are selected in
  such that all possible samples of that size have
  the same probability of being selected.

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• Convenience Samples and Quota samples are useful
  for preliminary studies. It is however difficult
  to assess the accuracy of estimates based on this
  type of sampling scheme.
• Sometimes however one has to be satisfied with a
  convenience sample and assume that it is
  equivalent to a random sampling procedure

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Some other definitions
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A population statistic (parameter)

• Any quantity computed from the values of
  variables for the entire population.

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A sample statistic

• Any quantity computed from the values of
  variables for the cases in the sample.

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• Since only cases from the sample are observed
• only sample statistics are computed
• These are used to make inferences about
  population statistics
• It is important to be able to assess the accuracy
  of these inferences

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To download lectures

• Go to the stats 244 web site
• Through PAWS or
• by going to the website of the department of
  Mathematics and Statistics -gt people -gt faculty
  -gt W.H. Laverty -gt Stats 244-. Lectures.
• Then
• select the lecture
• Right click and choose Save as

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To print lectures

1. Open the lecture using MS Powerpoint
2. Select the menu item File -gt Print

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• The following dialogue box appear

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• In the Print what box, select handouts

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• Set Slides per page to 6 or 3.

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6 slides per page will result in the least amount
of paper being printed
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3 slides per page leaves room for notes.
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Organizing and describing Data
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Techniques for continuous variables
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The Grouped frequency tableThe Histogram
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To Construct

• A Grouped frequency table
• A Histogram

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• Find the maximum and minimum of the observations.
• Choose non-overlapping intervals of equal width
  (The Class Intervals) that cover the range
  between the maximum and the minimum.
• The endpoints of the intervals are called the
  class boundaries.
• Count the number of observations in each interval
  (The cell frequency - f).
• Calculate relative frequency
• relative frequency f/N

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  Data Set 3 The following table gives data on
Verbal IQ, Math IQ, Initial Reading Acheivement
Score, and Final Reading Acheivement Score for 23
students who have recently completed a reading
improvement program   Initial Final Verbal
Math Reading Reading Student IQ IQ Acheivement
Acheivement   1 86 94 1.1 1.7 2 104 103 1.5 1.7
3 86 92 1.5 1.9 4 105 100 2.0 2.0 5 118 115 1.9
3.5 6 96 102 1.4 2.4 7 90 87 1.5 1.8 8 95 100
1.4 2.0 9 105 96 1.7 1.7 10 84 80 1.6 1.7 11 94
87 1.6 1.7 12 119 116 1.7 3.1 13 82 91 1.2 1.8
14 80 93 1.0 1.7 15 109 124 1.8 2.5 16 111 119
1.4 3.0 17 89 94 1.6 1.8 18 99 117 1.6 2.6 19 9
4 93 1.4 1.4 20 99 110 1.4 2.0 21 95 97 1.5 1.3
22 102 104 1.7 3.1 23 102 93 1.6 1.9
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In this example the upper endpoint is included in
the interval. The lower endpoint is not.
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Histogram Verbal IQ
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Histogram Math IQ
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Example

• In this example we are comparing (for two drugs A
  and B) the time to metabolize the drug.
• 120 cases were given drug A.
• 120 cases were given drug B.
• Data on time to metabolize each drug is given on
  the next two slides

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Drug A
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Drug B
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Grouped frequency tables
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Histogram drug A(time to metabolize)
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Histogram drug B(time to metabolize)
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Some comments about histograms

• The width of the class intervals should be chosen
  so that the number of intervals with a frequency
  less than 5 is small.
• This means that the width of the class intervals
  can decrease as the sample size increases

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• If the width of the class intervals is too small.
  The frequency in each interval will be either 0
  or 1
• The histogram will look like this

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• If the width of the class intervals is too large.
  One class interval will contain all of the
  observations.
• The histogram will look like this

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• Ideally one wants the histogram to appear as seen
  below.
• This will be achieved by making the width of the
  class intervals as small as possible and only
  allowing a few intervals to have a frequency less
  than 5.

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• As the sample size increases the histogram will
  approach a smooth curve.
• This is the histogram of the population

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N 25
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N 100
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N 500
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N 2000
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N 8
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Comment the proportion of area under a histogram
between two points estimates the proportion of
cases in the sample (and the population) between
those two values.
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Example The following histogram displays the
birth weight (in Kgs) of n 100 births
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Find the proportion of births that have a
birthweight less than 0.34 kg.
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Proportion (11310111917)/100 0.62
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The Characteristics of a Histogram

• Central Location (average)
• Spread (Variability, Dispersion)
• Shape

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Central Location
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Spread, Dispersion, Variability
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Shape Bell Shaped (Normal)
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Shape Positively skewed
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Shape Negatively skewed
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Shape Platykurtic
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Shape Leptokurtic
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Shape Bimodal
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The Stem-Leaf Plot

• An alternative to the histogram

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• Each number in a data set can be broken into two
  parts
• A stem
• A Leaf

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• Example
• Verbal IQ 84
• 84
• Stem 10 digit 8
• Leaf Unit digit 4

Leaf
Stem
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• Example
• Verbal IQ 104
• 104
• Stem 10 digit 10
• Leaf Unit digit 4

Leaf
Stem
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To Construct a Stem- Leaf diagram

• Make a vertical list of all stems
• Then behind each stem make a horizontal list of
  each leaf

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Example

• The data on N 23 students
• Variables
• Verbal IQ
• Math IQ
• Initial Reading Achievement Score
• Final Reading Achievement Score

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  Data Set 3 The following table gives data on
Verbal IQ, Math IQ, Initial Reading Acheivement
Score, and Final Reading Acheivement Score for 23
students who have recently completed a reading
improvement program   Initial Final Verbal
Math Reading Reading Student IQ IQ Acheivement
Acheivement   1 86 94 1.1 1.7 2 104 103 1.5 1.7
3 86 92 1.5 1.9 4 105 100 2.0 2.0 5 118 115 1.9
3.5 6 96 102 1.4 2.4 7 90 87 1.5 1.8 8 95 100
1.4 2.0 9 105 96 1.7 1.7 10 84 80 1.6 1.7 11 94
87 1.6 1.7 12 119 116 1.7 3.1 13 82 91 1.2 1.8
14 80 93 1.0 1.7 15 109 124 1.8 2.5 16 111 119
1.4 3.0 17 89 94 1.6 1.8 18 99 117 1.6 2.6 19 9
4 93 1.4 1.4 20 99 110 1.4 2.0 21 95 97 1.5 1.3
22 102 104 1.7 3.1 23 102 93 1.6 1.9
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• We now construct
• a stem-Leaf diagram
• of Verbal IQ

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• A vertical list of the stems
• 8
• 9
• 10
• 11
• 12

We now list the leafs behind stem
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8
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10
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0
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10
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11
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2
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0
10
9
11
1
8
9
9
9
9
4
9
9
9
5
10
2
10
2

• 8
• 9
• 10
• 11
• 12

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8
6
10
4
8
6
10
5
11
8
9
6
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0
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10
5
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4
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11
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10
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1
8
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5
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2
10
2

• 8
• 9
• 10
• 11
• 12

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• 8 6 6 4 2 0 9
• 9 6 0 5 4 9 4 9 5
• 10 4 5 5 9 2 2
• 11 8 9 1
• 12

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The leafs may be arranged in order

• 8 0 2 4 6 6 9
• 9 0 4 4 5 5 6 9 9
• 10 2 2 4 5 5 9
• 11 1 8 9
• 12

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The stem-leaf diagram is equivalent to a histogram

• 8 0 2 4 6 6 9
• 9 0 4 4 5 5 6 9 9
• 10 2 2 4 5 5 9
• 11 1 8 9
• 12

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The stem-leaf diagram is equivalent to a histogram

• 8 0 2 4 6 6 9
• 9 0 4 4 5 5 6 9 9
• 10 2 2 4 5 5 9
• 11 1 8 9
• 12

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Rotating the stem-leaf diagram we have
80
90
100
110
120
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The two part stem leaf diagram

• Sometimes you want to break the stems into two
  parts
• for leafs 0,1,2,3,4
• for leafs 5,6,7,8,9

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Stem-leaf diagram for Initial Reading Acheivement

• 01234444455556666677789
• 0
• This diagram as it stands does not
• give an accurate picture of the
• distribution

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• We try breaking the stems into
• two parts
• 1. 012344444
• 1. 55556666677789
• 2. 0
• 2.

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The five-part stem-leaf diagram

• If the two part stem-leaf diagram is not adequate
  you can break the stems into five parts
• for leafs 0,1
• t for leafs 2,3
• f for leafs 4, 5
• s for leafs 6,7
• for leafs 8,9

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• We try breaking the stems into
• five parts
• 1. 01
• 1.t 23
• 1.f 444445555
• 1.s 66666777
• 1. 89
• 2. 0

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• Stem leaf Diagrams
• Verbal IQ, Math IQ, Initial RA, Final RA

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Some Conclusions

• Math IQ, Verbal IQ seem to have approximately the
  same distribution
• bell shaped centered about 100
• Final RA seems to be larger than initial RA and
  more spread out
• Improvement in RA
• Amount of improvement quite variable

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Numerical Measures

• Measures of Central Tendency (Location)
• Measures of Non Central Location
• Measure of Variability (Dispersion, Spread)
• Measures of Shape

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Measures of Central Tendency (Location)

• Mean
• Median
• Mode

Central Location
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Measures of Non-central Location
Non - Central Location

• Quartiles, Mid-Hinges
• Percentiles

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Measure of Variability (Dispersion, Spread)

• Variance, standard deviation
• Range
• Inter-Quartile Range

Variability
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Measures of Shape

• Skewness
• Kurtosis

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Measures of Central Location (Mean)

• Summation Notation
• Let x1, x2, x3, xn denote a set of n numbers.
• Then the symbol
• denotes the sum of these n numbers
• x1 x2 x3 xn

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• Example
• Let x1, x2, x3, x4, x5 denote a set of 5 denote
  the set of numbers in the following table.

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• Then the symbol
• denotes the sum of these 5 numbers
• x1 x2 x3 x4 x5
• 10 15 21 7 13
• 66

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• Meaning of parts of summation notation

Final value for i
each term of the sum
Quantity changing in each term of the sum
Starting value for i
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• Example
• Again let x1, x2, x3, x4, x5 denote a set of 5
  denote the set of numbers in the following table.

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• Then the symbol
• denotes the sum of these 3 numbers
• 153 213 73
• 3375 9261 343
• 12979

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Mean

• Let x1, x2, x3, xn denote a set of n numbers.
• Then the mean of the n numbers is defined as

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• Example
• Again let x1, x2, x3, x4, x5 denote a set of 5
  denote the set of numbers in the following table.

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• Then the mean of the 5 numbers is

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Interpretation of the Mean

• Let x1, x2, x3, xn denote a set of n numbers.
• Then the mean, , is the centre of gravity of
  those the n numbers.
• That is if we drew a horizontal line and placed a
  weight of one at each value of xi , then the
  balancing point of that system of mass is at the
  point .

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xn
x1
x2
x3
x4
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In the Example
21
10
7
15
13
20
10
0
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The mean, , is also approximately the center
of gravity of a histogram
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The Median

• Let x1, x2, x3, xn denote a set of n numbers.
• Then the median of the n numbers is defined as
  the number that splits the numbers into two equal
  parts.
• To evaluate the median we arrange the numbers in
  increasing order.

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• If the number of observations is odd there will
  be one observation in the middle.
• This number is the median.
• If the number of observations is even there will
  be two middle observations.
• The median is the average of these two
  observations

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• Example
• Again let x1, x2, x3, x3 , x4, x5 denote a set of
  5 denote the set of numbers in the following
  table.

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• The numbers arranged in order are
• 7 10 13 15 21

Unique Middle observation the median
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• Example 2
• Let x1, x2, x3 , x4, x5 , x6 denote the 6 denote
  numbers
• 23 41 12 19 64 8
• Arranged in increasing order these observations
  would be
• 8 12 19 23 41 64

Two Middle observations
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• Median
• average of two middle observations

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Example

• The data on N 23 students
• Variables
• Verbal IQ
• Math IQ
• Initial Reading Achievement Score
• Final Reading Achievement Score

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  Data Set 3 The following table gives data on
Verbal IQ, Math IQ, Initial Reading Acheivement
Score, and Final Reading Acheivement Score for 23
students who have recently completed a reading
improvement program   Initial Final Verbal
Math Reading Reading Student IQ IQ Acheivement
Acheivement   1 86 94 1.1 1.7 2 104 103 1.5 1.7
3 86 92 1.5 1.9 4 105 100 2.0 2.0 5 118 115 1.9
3.5 6 96 102 1.4 2.4 7 90 87 1.5 1.8 8 95 100
1.4 2.0 9 105 96 1.7 1.7 10 84 80 1.6 1.7 11 94
87 1.6 1.7 12 119 116 1.7 3.1 13 82 91 1.2 1.8
14 80 93 1.0 1.7 15 109 124 1.8 2.5 16 111 119
1.4 3.0 17 89 94 1.6 1.8 18 99 117 1.6 2.6 19 9
4 93 1.4 1.4 20 99 110 1.4 2.0 21 95 97 1.5 1.3
22 102 104 1.7 3.1 23 102 93 1.6 1.9
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• Computing the Median
• Stem leaf Diagrams

Median middle observation 12th observation
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Summary
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Some Comments

• The mean is the centre of gravity of a set of
  observations. The balancing point.
• The median splits the obsevations equally in two
  parts of approximately 50

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• The median splits the area under a histogram in
  two parts of 50
• The mean is the balancing point of a histogram

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50
median
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• For symmetric distributions the mean and the
  median will be approximately the same value

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50
Median
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• For Positively skewed distributions the mean
  exceeds the median
• For Negatively skewed distributions the median
  exceeds the mean

50
50
median
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• An outlier is a wild observation in the data
• Outliers occur because
• of errors (typographical and computational)
• Extreme cases in the population

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• The mean is altered to a significant degree by
  the presence of outliers
• Outliers have little effect on the value of the
  median
• This is a reason for using the median in place of
  the mean as a measure of central location
• Alternatively the mean is the best measure of
  central location when the data is Normally
  distributed (Bell-shaped)