MATHSTAT 231

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MATH/STAT 231

• Chapter 2-II
• Univariate Data Description
• Numerical Description

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Introduction

• A graph, such as a histogram, gives us a picture
  of our data which can show us the shape of the
  distribution, the number of modes and their
  locations and other various information. However,
  we also want some quantitative information about
  the distribution. The most commonly desired
  quantitative information are a value describing
  the center of the data and a value describing how
  spread-out the data is.
• 1. Measures of central tendency
• 2. Measures of Dispersion (variation)

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Measures of central tendency

• The three common measures of center are mean,
  median and mode.
• Mean
• The mean (arithmetic average) is the most
  common measure of central tendency. It simply the
  sum of the numbers divided by the number of
  numbers.

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• Example At a ski rental shop data was collected
  on the number of rentals on each of 10
  consecutive Saturdays 44, 50, 38, 96, 42, 47,
  40, 39, 46, 50. Try to find out the mean.

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• Example On his first 5 biology tests, Bob
  received the following scores  72, 86, 92, 63,
  and 77.  What test score must Bob earn on his
  sixth test so that his average (mean score) for
  all six tests will be 80?  
•      
•             72 86 92 63 77 x    
  80                            6cross
  multiply     
•                            (80)(6) 390
  x                            480 390x
•                                     x 90

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• Median
• The median is the value that occupies in the
  middle position when the data are sorted from the
  smallest to the largest. Half the data values are
  above the median and half are below the median.
• Find the median
• 1. Order the observations from the smallest to
  the largest.
• 2. Select the middle point.
• If there is an odd number (N) of values, the
  median M is the (N1)/2 th value (the middle
  value). If there is an even number of values,
  the median is the average of the two middle
  numbers is the average of the two middle numbers
  (add the two middle values and divide by 2).

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• Example Test scores for a class of 17 students
  are as follows 93, 84, 97, 98, 100, 78, 86, 82,
  85, 92, 72, 55, 91, 90, 75, 94, 83. What is the
  median score?
• Step 1 sort the data from smallest to
  largest
• 55 72 75 78 82 83 84 85 86 90 91
  92 93 94 97 98 100
• Step 2 locate the middle position
• N 17
• Middle position (171) /2 9
• The median 86 (The 9th Number. The data at
  the middle position.)

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• Example At a ski rental shop data was collected
  on the number of rentals on each of 10
  consecutive Saturdays 44, 50, 38, 96, 42, 47,
  40, 39, 46, 50. Try to find out the median.
• Step 1 sort the data from smallest to the
  largest
• 38 39 40 42 44 46 47 50 50 96
• Step 2 locate the middle position
• N 10
• The middle position is between
  5, 6
• The median (Average of the two middle numbers)
  
• 38 39 40 42 44 M 46 47 50 50 96
• (44 46 ) /2 45

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• Example The ages of the 667 people participating
  in a large workshop (to the nearest year) are
  summarized as follows
• What is the median age of the 667 people?

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• Mode
• The value that occurs most often in a dataset
  is called the mode. It is sometimes said to be
  the most typical case.
• No mode each value occurs only once.
• Example For individuals having the following
  ages -- 18, 18, 19, 20, 20, 20, 21, and 23, the
  mode age is 20.
• Example Find the mode for the following data
  5, 15, 10, 15, 5, 10, 10, 20, 25, 15. 

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• Try to find the mode based on graphs.

Data values of blood type A B B AB
O O O B AB B B B O A
O A O O O AB AB A O B
A
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Comments

• For symmetric distributions, the mean and median
  are close equal. For symmetric and unimodal
  distributions, the mean, median and mode are
  close equal.
• In skewed distributions, the mean is farther out
  in the long tail than is the median. So, if the
  distribution is skewed to right, then mean gt
  mediangt mode. If the distribution is skewed to
  left, then mean lt median lt mode.

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• The mean is sensitive to the influence of a few
  extreme observations. The median and mode are
  more resistant than the mean.
• For nominal data (such as sex or race), the mode
  is the only valid measure.
• For ordinal data (such as salary categories),
  only the mode and median can be used.

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Measuring dispersion (variation)

• Range
• Standard deviation

• Range
• Range Max Min
• Example At a ski rental shop data was collected
  on the number of rentals on each of 10
  consecutive Saturdays 44, 50, 38, 96, 42, 47,
  40, 39, 46, 50. Find the range.
• Range 96 (the largest value) 38 (the
  smallest value)

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• Standard deviation
• Assume we have a dataset with n values
  x1,x2,xn.
• the mean of the data set.
• Formula for the standard deviation
• Sample variance
• or, in more compact notation
• Sample Standard deviation
• Variance and Standard deviation
• Variance s x s s2
• s square root (Variance)

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Example Suppose we wished to find the standard
deviation of the data set consisting of the
values 3, 7, 7, and 19. Step 1 find the mean
(average) of 3, 7, 7, and 19, (3
7 7 19) / 4 9. Step 2 find the
deviation of each number from the mean, 3 - 9
- 6 7 - 9 - 2 7 - 9
- 2 19 - 9 10. Step 3 square each
of the deviations, which amplifies large
deviations and makes negative values positive,
( - 6)2 36 ( - 2)2 4 (
- 2)2 4 102 100. Step 4 find
the variance ( sum of squared deviations/(n-1)
), s2 (36 4 4 100) / (4-1)
48. Step 5 take the non-negative square
root of the variance (s2) s
sqrt(48) 6.93 So, the standard deviation of the
set is 6.93 .
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Another formula to calculate the
variance Example Find the standard deviation
of the data set consisting of the values 3, 7, 7,
and 19. Step 1 Find the mean (3 7 7
19) / 4 9. V2 4 x 92 324
Step 2 Square each of the data value, and add
all of them V1 327272192
468 Step 3 s2 (V1-V2)/(4-1) 48 The
standard deviation sqrt (s2) sqrt (48) 6.93
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• Comments
• 1. A large standard deviation indicates that
  the data points are far from the center (Mean)
  and a small standard deviation indicates that
  they are clustered closely around the center
  (Mean).
• 2. The standard deviation has the same units
  as the data points themselves.

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• Practice use of the standard deviation
• If you repeat a measurement several times on
  the same object over the period of measurement,
  you may get a series of readings that differ from
  each other. The cause may be small differences in
  how you use the instrument each time. The
  differences could also be due to random changes
  in the instrument, and they could be due to small
  changes in the object you are measuring. Whatever
  the cause, you would be inclined to take the
  average of the readings as the best value you can
  quote or use. You can get an idea of the
  variability from standard deviation of the
  readings. The bigger the standard deviation
  (variance) is, the less precise the readings are,
  and vice versa.

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Measures of position

• How do compare two data values from different
  groups.
• 1. Five students have taken different forms of
  the spelling test. The difficulty level of
  different tests could be different. How to
  compare their performance?
• 2. A training program only accepts top 25
  students according to a standard test. what grade
  does a student need to make to be in the top 25?

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• Measures of position are used to determine the
  relative position of a specified data value
  within a group of data values.
• Percentile
• Quantile
• z-score

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Percentiles and quartiles

• Percentiles divide the data values into 100 equal
  groups. pth percentile of a distribution is the
  value such that p percent of data values fall at
  or below it.
• The median is the 50th percentile.
• The first quartile Q1 (lower quartile)
  is the 25th percentile, the third quartile Q3
  (upper quartile) is the 75th percentile. The
  median is the second quartile. Quartiles divide
  the distribution into four equal groups,
  separated by Q1, Q2, and Q3.

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• To calculate the quartiles
• 1. Arrange the data values in increasing
  order and locate the median.
• 2. Use the median to divide the ordered data
  values into two halves. Do not include the median
  into the halves.
• 3. The lower quartile (Q1) is the median of
  the lower half of the data. The upper quartile
  (Q3) is the median of the upper half of the data.

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• Example. Suppose a group of 10 students have the
  following heights (in inches) 60,72, 64, 67, 70,
  68, 71, 68, 73, 59. Find Q1 ,Q3 and IQR.
• 1 Sort the data from smallest to largest
• 59, 60, 64, 67, 68, 68, 70, 71, 72,
  73.
• 2 Divide the observations into lower
  half and upper half by median 59, 60, 64, 67,
  68 M 68, 70, 71, 72, 73.
• 3. The first (second) half of the data is
  considered in calculating the first (third)
  quartile, and Q1 (Q3) is the median of this part
  of data
• First half 59, 60, 64, 67, 68
• Second half 68, 70, 71, 72, 73
• IQR Q3-Q1 71-64 7

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• Related Measures
• Midquartile
• midquartile (Q1Q3)/2
• The Interquartile Range
• IQR Q3-Q1
• The IQR is essentially the range of the middle
  50 of the data.

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Standardized values or z-scores

• If x is an observation (data value) from a
  population (a group of data)that has mean µ and
  standard deviation s, the z-score (standardized
  value) of x is
• The z score for an data value, indicates how far
  and in what direction, that item deviates from
  the mean, expressed in units of the standard
  deviation.

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• Example
• 1. Find the z-score corresponding to a raw
  score of 132 from a group of data with mean 100
  and standard deviation 15.
• 2. The lengths of an adult South American rain
  forest beetle species are distributed with mean
  5.6cm and standard deviation 0.32cm. What is the
  z-score for a beetle of length 5.1 cm?
• 3.A z-score of 1.7 was found from an
  observation coming from a population with mean 14
  and standard deviation 3. Find the value of the
  observation .

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• The z-score transformation is especially useful
  when seeking to compare the relative standings of
  items from distributions with different means
  and/or different standard deviations.
• Five students have taken different forms of the
  spelling test. The scores of different forms are
  distributed with different mean and standard
  deviation. How to compare their performance?

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The five-number summary and Boxplot

• Five number summary
• Minimum, Q1, Median, Q3, Maximum.
• These five numbers offer a reasonable and
  complete description of distribution.
• Boxplot (Box and whiskers display)
• Visual version of five number summary. The
  five number summary is easier to understand when
  it is displayed in a graph.

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• Here are the four steps you follow to draw a
  boxplot
• 1. Draw a box from the 25th (Q1) to the
  75th (Q3) percentile.
• 2. Split the box with a line at the median.
• 3. Draw a thin lines from the 75th
  percentile up to the maximum value. Draw another
  thin line from the 25th percentile down to the
  minimum value.

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Information obtained from a Boxplot
__________
_________
___________
Symmetry versus Skewness
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• Example The following boxplot is of the birth
  weights (in ounces) of a sample of 160 infants
  born in a local hospital.

1. Describe the shape of the distribution? 2.
Find out the five number summary. 3. About 40
infants of the birth weights were below ____? 4.
IQR ?