TABLES AND GRAPHS FOR NUMERICAL DATA

A first step in any study is to define a problem. Later, data is gathered and the need for analysis will be apparent. Appropriate statistical tools to convert data to information is the subject of this chapter.

Frequency Distribution

A frequency distribution for numerical data is a table that summarizes data by listing the classes in the left column and the number of observations in each class in the right column. However, the classes or intervals for a frequency distribution of numerical data are not as easy identifiable.

To determine the intervals of a frequency distribution for numerical data requires answers to certain questions: How many intervals should be used? How wide should each interval be? Where does the first interval begin? Where does the last interval end? There are some general rules for preparing frequency distributions that make it easier for us to answer these types of questions, to summarize data, and to communicate results.

CONSTRUCTION OF A FREQUENCY DISTRIBUTION

Rule 1: Determine k, the number of intervals (classes). k may be determined by the Sturges’s rule, where n is the total number of observations.

The number of intervals (classes) used in a frequency distribution may be decided in a somewhat arbitrary manner.

Quick Guide to Approximate Number of Intervals for a Frequency Distribution. 

Practice and experience provide the best guidelines. Larger data sets may require more class intervals. If you select too few classes, the pattern and various characteristics of the data may be hidden. If you select too many classes, you will discover that some of your intervals may contain no observations or a very small number of frequencies.

Rule 2: Intervals should be the same width, w; the width is determined by the following:
            

Both k and w should be rounded upward, possibly to the next largest integer. The interval width is often rounded to a convenient whole number to provide for easy interpretation.

Rule 3: Intervals (classes) must be inclusive and non-overlapping. Each observation must belong to one and to only one interval. Consider a frequency distribution for the ages of a particular group of people. If the frequency distribution contains the intervals “age 20 to age 30” and “age 30 to age 40” to which of these two classes would a person age 30 belong?

The boundaries, or endpoints, of each class must be defined. To avoid overlapping, age intervals could be defined as “age 20 but less than age 30,” followed by “age 30 but less than age 40” and so on. Another possibility is to define the age intervals as “20-29”, “30-39” and so forth. Boundary selection is subjective. Simply be sure to define interval boundaries that promote a clear understanding and interpretation of the data. 

Two special frequency distributions are the cumulative frequency distribution and the relative cumulative frequency distribution.

Relative, Cumulative And Relative Cumulative Frequency Distributions

A relative frequency distribution is obtained by dividing each frequency by the number of observations and multiplying the resulting proportion by 100%. A cumulative frequency distribution contains the total number of observations whose values are less than the upper limit for each interval. It is constructed by adding the frequencies of all frequency distribution intervals up to and including the present interval. A relative cumulative frequency distribution converts all cumulative frequencies to cumulative proportions or percentages.
Table-1 displays the number of days to maturity for 40 short-term investments.
Example:
Table 1: Days to maturity 40 short-term investments.

71	64	99	55	64	89	87	65	62	38
67	70	60	69	78	39	75	56	71	51
99	68	95	86	57	53	47	50	55	81
80	98	51	36	63	66	85	79	83	70

Construct a frequency distribution.

Solution:
By using formula
To get the idea as to the number of class intervals to use, we can apply Sturges’s rule to obtain

Now let us divide the range by 8 to get some idea about the class interval width. We have

It is apparent that a class interval width of 10 will be more convenient to use, as well as more meaningful to the reader.

By using Quick Guide
A frequency distribution with 6 classes is developed. The width of each class is

Table 2: Frequency distribution of the data in Table 2.

 Table 3: Cumulative Frequency and Relative Cumulative Frequency Distributions

Class interval	Frequency	Cumulative frequency	Relative Frequency
30—39	3	3	3/40
40—49	1	4	1/40
50—59	8	12	8/40
60—69	10	22	10/40
70—79	7	29	7/40
80—89	7	36	7/40
90—99	4	40	7/40
Total	40		1

Example: Jennie Bishop, marketing director for a leading mobile phone company, obtained records of minutes used by a random sample of 110 subscribers to the company’s low-end user plan (250 peak minutes per month). Table 1 is a list of minutes used by each subscriber in the sample during one particular month. What do the data indicate?

Table 4: Mobile Phone Usage (Peak Minutes)

271	236	294	252	254	263	266	222	262	278
288	262	237	247	282	224	263	267	254	271
278	263	262	288	247	252	264	263	247	225
281	279	238	252	242	248	263	255	294	268
255	272	271	291	263	242	288	252	226	263
269	227	273	281	267	263	244	249	252	256
263	252	261	245	252	294	288	245	251	269
256	264	252	232	275	284	252	263	274	252
252	256	254	269	234	285	275	263	263	246
294	252	231	265	269	235	275	288	294	263
247	252	269	261	266	269	236	276	248	298

Solution

Table 1 by itself offers little guidance to help the marketing director develop a marketing strategy. We can find some information in Table 1: The smallest amount of peak minutes used was 222 minutes, and the maximum time used was 298 minutes. However, we will need more information than this before submitting any report to senior-level executives. To better understand what the data in Table 1 indicate, we first develop a frequency distribution.

By using formula
Number of classes is determined by

By using Quick Guide
A frequency distribution with eight classes is developed. The width of each class is

Since the smallest value is 222, one choice for the first interval is “220 but less than 230”. Subsequent intervals of equal width are added to the frequency distribution, as well as the number of calls that belong to each class. The following Table is a frequency distribution for the mobile phone data in Table 1.

Table 5: Frequency and Relative Frequency Distribution for Mobile Phone Usage

Mobile phone usage (in minutes)	Tally Marks	Frequency	Percent
220 – 229		5	4.5
230 – 239		8	7.3
240 – 249		13	11.8
250 – 259		22	20.0
260 – 269		32	29.1
270 – 279		13	11.8
280 – 289		10	9.1
290 – 299		7	6.4
Total		110	100

The manager may want to know mobile usage below (or above) a certain amount of time. Table 2 is a cumulative frequency distribution and a cumulative percentage distribution.

Table 6: Cumulative Frequency and Relative Cumulative Frequency Distributions for Mobile Phone Usage.

Mobile Phone Usage (in Minutes)	Frequency	Cumulative Frequency	Cumulative %
Less than 230	5	5	4.5
Less than 240	8	13	11.8
Less than 250	13	26	23.6
Less than 260	22	48	43.6
Less than 270	32	80	72.7
Less than 280	13	93	84.5
Less than 290	10	103	93.6
Less than 300	7	110	100.0

The frequency distributions in Table 2 and Table 3 are an improvement over the original list of data in Table 1. We have at least summarized 110 observations into 8 categories and are able to tell Jennie that less than one-fourth (23.6%) of the subscribers sampled used their mobile phones within the guidelines of their plans during the month of the study. The marketing manager might suggest that an advertising campaign be initiated to promote a plan with an increase in peak minutes.