Chapter 11 Language of Descriptive Statistics

Section 11.2 Frequency Distributions and Percentage Calculation

11.2.1 Introduction


Let X be a given property. A sample of size n resulted in the original list (sample)

x  =  ( x1 , x2 ,…, xn ) .


Info 11.2.1
 
If a is a possible property value, then

Hx (a)  =   number of    xj      within the original list   x   with    xj =a

is called the absolute frequency of the property a in the original list x=( x1 , x2 ,…, xn ).

If a1 , a2 ,… ak are the possible property values in the original list x=( x1 , x2 ,…, xn ), then we have

Hx ( a1 )+ Hx ( a2 )+…+ Hx ( ak )  =  n

or in words: each of the n values is counted by exactly one of the frequencies.
Info 11.2.2
 
The relative frequency of the property value a in the original list x=( x1 , x2 ,…, xn ) is defined by

hx (a)  =   1 n · Hx (a) .


If a1 , a2 ,… ak are the possible property values in the original list x=( x1 , x2 ,…, xn ), then we have

hx ( a1 )+ hx ( a2 )+…+ hx ( ak )  =  1 .

Relative frequencies always lie in the interval [0;1] and are often specified in percentages,e.g. hx ( a1 )=34% instead of hx ( a1 )=0.34.
Info 11.2.3
 
Collecting the absolute or relative frequencies of all occurring (or possible) property values in the original list (sample) x=( x1 , x2 ,…, xn ) in a table results in the empirical frequency distribution.

Example 11.2.4

In a data centre, the processing time (in seconds, rounded to one fractional digit) of 20 program jobs was determined. This resulted in the following original list of a sample of size n=20:
3.9 3.3 4.6 4.0 3.8
3.8 3.6 4.6 4.0 3.9
3.9 3.9 4.1 3.7 3.6
4.6 4.0 4.0 3.8 4.1

The smallest value is 3.3 s, the largest value is 4.6 s, the increment is 0.1 s. Thus, we have the empirical frequency distribution listed (in tabular form) below. To keep the table short all values less than 3.3 and greater that 4.6 are not listed.
Result a Hx (a) hx (a) Percentage
3.3 1 1 20 =0.05 5%
3.4 0 0 0%
3.5 0 0 0%
3.6 2 2 20 =0.1 10%
3.7 1 1 20 =0.05 5%
3.8 3 3 20 =0.15 15%
3.9 4 4 20 =0.2 20%
4.0 4 4 20 =0.2 20%
4.1 2 2 20 =0.1 10%
4.2 0 0 0%
4.3 0 0 0%
4.4 0 0 0%
4.5 0 0 0%
4.6 3 3 20 =0.15 15%
Sum 20 1 100%


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