Transcript Frequency Distribution - Eastern Illinois University

Frequency Distribution
A Frequency Distribution organizes
data into classes, or categories, with a
count of the number of observations
that fall into each class.
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
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.
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 > n, where n is
the number of
observations.
The Two Firm Rules in Grouping
Data:
• The All-Inclusive
• The Mutually-Exclusive
Rule: classes must be
Rule: classes must be
All-Inclusive. Allmutually exclusive.
inclusive classes are
Classes must be arranged
classes that together
such that every piece of
contain all the data.
data can be placed in only
one class.
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
Sample Mean
X

f i Mi
n
The sample mean of grouped data is:
X 

f i Mi
n
where, fi is the frequency of the ith class, and
Mi is the midpoint of the ith class.
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.
Sample Mode
Sample Mode is the midpoint of the
class having the greatest frequency.
Sample Variance
Sample Variance is:
k
s 
2

i 1
f i ( Mi  X )
n 1
2
Sample Standard Deviation
Sample Standard Deviation is:
s = SQRT( Variance )