AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

1
AAEC 4302ADVANCED STATISTICAL METHODS IN
AGRICULTURAL RESEARCH

• Part I A Quick Review of Familiar Topics
• Chapter 4 Frequency Distributions

2
Introduction

• Chapter 4 examines grouping and classification
  techniques for organizing and summarizing data on
  a given variable
• There are a few but important differences in the
  techniques applied to discrete vs. continuous
  variables

3
Discrete Data Variables

• Suppose that n observations are available on a
  discrete variable X that only takes m different
  values
• The n observations can be divided into m
  different groups within which all of the
  observations have exactly the same value

4
Discrete Data Variables

• These m different groups can be indexed by k, so
  that k 1,,m
• Example Family size variable on Table 2.2 (p.
  18)
• X ranges from 2 to 10, m 9 (k 1,,9), the
  100 observations can be arranged into nine
  groups, one for each family size, value observed
  in the sample

5
Discrete Data Variables

• Table 4.1 (p. 63) is an example of a Frequency
  Table for the variable family size

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Discrete Data Variables

• Xk (column 2) denotes the value of the variable
  X (family size) taken by all of the observations
  grouped into category k (for example X3 4
  indicates that the third category groups all
  observations with a value -family size- of 4)

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Discrete Data Variables

• nk (column 3) denotes the absolute frequency
  which is the number of observations that fall on
  that category (for example n3 34 denotes the
  of families in the sample with 4 family members)

8
Discrete Data Variables

• Notice that nk n (the sample size)
• fk nk/n (column 4) is the relative frequency or
  the proportion of the total observations that
  fall on that category (for example f3 0.34
  denotes the proportion of families in the sample
  with 4 family members)

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Discrete Data Variables

• Notice that fk 1
• Can you prove it?

10
Discrete Data Variables

• Figure 4.1 (p.64) is a conventional graph of the
  relative frequency distribution of the discrete
  variable family size (X) it is a graph of the
  value of fk for each value of Xk

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Discrete Data Variables

• Family size (Xk) is in the horizontal axis, and
  the corresponding relative frequency (fk) on the
  vertical axis

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Discrete Data Variables

• This is a summary graphical representation of
  the sample data where the only information lost
  is the sample size n
• How do you read the graph?

13
Discrete Data Variables

• Since the relative frequency distribution
  implicitly orders the values of X, the median
  (Xmed) is easily found by examining the
  cumulative frequencies

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Discrete Data Variables
For example, in Table 4.1 10 of the obs. are
in grp 1, where X1 2 16 of the obs. are in grp
2, where X2 3 34 of the obs. are in grp 3,
where X3 4 Obviously the middle observation
(50th) is in group 3, and its value (family size)
equal to 4
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Discrete Data Variables

• Clearly the mode or most frequently occurring
  family size is also 4 in this case

16
Discrete Data Variables

• The mean can also be calculated from a frequency
  table instead of the raw data

17
Discrete Data Variables

• The standard deviation (SX) can also be
  calculated from a frequency table

18
Discrete Data Variables

• Columns 5 and 6 shows the steps for calculating
  X and SX from a frequency table

19
Continuous Data Variables

• The formerly discussed approach does not make
  sense with continuous variables, since there will
  likely be one group for each observation
• Instead m class intervals are created, so that
  each observation can be placed into only one of
  them
• The 3 principles to follow when creating these
  intervals are
• The number of classes (m) should be between 5
  15
• The range (width) of each interval should be the
  same
• The mean point of each interval should be a
  convenient number

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Continuous Data Variables

• Example Let X be family income from Table 2.2
• X ranges from 0.75 to 32.08 thousand dollars
• Lets set up m 9 intervals starting from 0 with
  an interval range of 4.0 (4,000 dollars)
• Observations that lie right on the boundary
  between two classes should be divided between
  lower and higher classes

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Continuous Data Variables

• Table 4.2 is the so constructed frequency table
  for family income
• Column 1 includes the class interval index k and
  the boundary values of X that define the class

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Continuous Data Variables

• In the second column is the class mark (Xk) which
  is defined as the mid-point of the class
  interval
• In the third column is the absolute frequency
  (nk) which, as before, is the number of
  observations in the sample whose value falls in
  the kth class (i.e. within the boundaries of the
  kth class interval)

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Continuous Data Variables

• In the fourth column is the relative frequency
  (fk) which, as before, is the proportion of
  observations in the sample whose value falls in
  the kth class (i.e. within the boundaries of the
  kth class interval)
• Also as before, nk n, fk nk/n, fk 1

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Continuous Data Variables

• Figure 4.2, is a graph of the values of the class
  marks (Xk) in the horizontal axis coupled with
  the corresponding relative frequency (fk) in the
  vertical axis, which represents the relative
  frequency distribution of family income (X)
• This graph is known as a histogram or bar chart
  where each box represents each class and the
  height of the box gives the relative frequency
  (fk) of the corresponding class

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Continuous Data Variables

• Figure 4.3 presents the most common shapes taken
  by histograms or bar charts
• Unimodal There is only one peak
• Bimodal There are two peaks
• Unimodal Skewed to the Right It has a longer
  tail in that direction (length of tail signifies
  direction of skewness).

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Continuous Data Variables

• The mean and standard deviation of a continuous
  variable calculated form a frequency table (using
  the formulas given in the case of discrete
  variables) are only approximations.

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Continuous Data Variables

• There is a correspondence between relative
  frequencies and areas under histogram
• The ratio of the area of the kth bar to the total
  area of the histogram
• wfk/ S wfk wfk / w S fk wfk / w fk

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Proportions

• Question What proportion of the observations
  have X values between Xa and Xb ?
• Prop (Xa X Xb ) ?
• Proportion of observations that lie in the
  one-standard-deviation interval 10.120 5.755 (X
  Income, Xa 4.365 and Xb 15.875
• Proportion of observations having incomes less
  then or equal to 15 thousand dollars
• Prop (0 X 15 ) ?
• Uniform distribution assumption X values of the
  observations in any class interval are spread
  smoothly throughout it.

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Proportions

• Determine proportion in question by calculating
  the sum of the relative frequencies of the class
  intervals and parts of class intervals that make
  up the interval from Xa to Xb
• Prop (4.365 X 15.875) (8-4.365)/4.0f2
• f2 (15.875-12)/4.0f4
• (3.635/4.0)(0.35) (0.35)
  (3.875/4.0)(0.16)
• 0.318 0.35 0.155 0.823

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Proportions

• Graphical method for determining proportions
• Figure 4.4a, where proportion is given by the
  ratio of the shaded area to the total area in the
  histogram
• Another application of proportion calculations is
  determining the median
• Prop(XXmed) 0.5
• Xmed 8 (0.10/0.35) (4.0) 9.14