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 2 k > n, where n is the number of observations.

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.

Class Midpoint Each class has a lower limit and an upper limit. Class midpoint, M i, is the arithmetic mean of the two limits. M i = (lower limit + upper limit) / 2

Sample Mean The sample mean of grouped data is: where, f i is the frequency of the i th class, and M i is the midpoint of the i th class.

Sample Median The sample median of grouped data is: Med = L + ( n 1 / n 2 ) i where, L is the lower limit of the median class, n 1 is the number of data values in the median class that lie below the median position, n 2 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:

Sample Standard Deviation Sample Standard Deviation is: s = SQRT( Variance )