Frequency Distribution

1
Frequency Distribution

• A Frequency Distribution organizes data into
  classes, or categories, with a count of the
  number of observations that fall into each class.

2
Characteristics of Classes

• Class Limits
• smallest and largest observed values that can
  belong to a class
• Boundaries
• actual values that separate successive classes
• Intervals
• the distance spanned by the boundaries of a class
• Class Midpoint
• the arithmetic mean of its class boundaries

3
Steps for Constructing a Frequency Distribution

• Array the data values in order by size from
  lowest to highest (or vice versa)
• Compute the range
• Divide the range into a convenient number of
  class intervals of equal size
• Count the number of observations in each class to
  determine the total frequency and
• Display the class intervals with their
  frequencies.

4
How to Select a Class Interval?Some Rules of
Thumb!

• Select a class interval that allows from 6 to 15
  classes. Too many classes can destroy the
  summary effect of the grouping too few classes
  can produce oversimplification of the data and
  result in inaccuracies from subsequent
  calculations.

• The number of classes, k, should be the smallest
  integer such that 2k gt n, where n is the number
  of observations.

5
The Two Firm Rules in Grouping Data

• The All-Inclusive Rule classes must be
  All-Inclusive. All-inclusive classes are classes
  that together contain all the data.

• The Mutually-Exclusive Rule classes must be
  mutually exclusive. Classes must be arranged
  such that every piece of data can be placed in
  only one class.

6
Class Midpoint

• Each class has a lower limit and an upper limit.
  Class midpoint, Mi, is the arithmetic mean of the
  two limits.
• Mi (lower limit upper limit) / 2

7
Sample Mean

• The sample mean of grouped data is
• where, fi is the frequency of the ith class, and
  Mi is the midpoint of the ith class.

8
Sample Median

• The sample median of grouped data is
• Med L ( n1 / n2 ) i
• where, L is the lower limit of the median class,
  n1 is the number of data values in the median
  class that lie below the median position, n2 is
  the number of observations in the median class,
  and i is class interval.

9
Sample Mode

• Sample Mode is the midpoint of the class having
  the greatest frequency.

10
Sample Variance

• Sample Variance is

11
Sample Standard Deviation

• Sample Standard Deviation is
• s SQRT( Variance )