Graphing Grouped Data

Grouped (quantitative) data can be displayed by using a histogram or a polygon. A pie chart can also be drawn to display the percentage distribution for quantitative data. The procedure to construct a pie chart is similar to the one for quantitative data explained earlier on.

Histograms

A histogram is constructed in such a way that the area of the bar must represent class frequency.  The procedure is to work out.

1. The Class Width
   
2. The frequency density by dividing frequency by class width
   
3. Produce a histogram where the frequency density is on the vertical axis and the variable on the horizontal axis.

Example 2.5: Draw a histogram for the following data on price-earnings ratio for 25 companies.

Price-Earning Ratio	f
10 – 14	4
15 – 19	6
20 – 24	8
25 – 29	4
30 – 34	3

Solution

First we need to work out the class boundaries class width and the frequency densities. To construct a histogram we use the class boundaries so that the bars are adjacent to each other.

Table 2.8: Frequency Densities for Price-Earnings Ratio

Price-Earnings Ratio	Class Boundaries	f	Class Width	Frequency Density
10 – 14	9.5 – 14.5	4	5	4/5 = 0.8
15 – 19	14.5 – 19.5	6	5	6/5 = 1.2
20 – 24	19.5 – 24.5	8	5	8/5 = 1.6
25 – 29	24.5 – 29.5	4	5	4/5 = 0.8
30 – 34	29.5 – 34.5	3	5	3/5 = 0.6

Figure 2.3: A Histogram for Price-Earnings Ratio

Polygons

A frequency polygon is drawn by putting the frequency density against the class midpoints.  It gives the same information as the histogram, but without bars.

Example 2.6: Construct a frequency polygon for the data on price-earnings ratio.

Solution:  The class midpoints are 12,17,22,27 and 32.

Fig 2.4: A Frequency Polygon for Price-Earnings Ratio.

Shapes of Histograms

• The most common shapes of a histogram are
• Symmetric
• Uniform or rectangular

CUMULATIVE FREQUENCY DISTRIBUTIONS

A cumulative frequency distribution gives the total number of observations that fall below the upper boundary of each class.

Example 2.7: Using the frequency distribution of Table 2.7 prepare a cumulative frequency distribution table for the price earnings ratio.

Solution

Table 2.9: Cumulative Frequency Distribution of Price-Earnings Ratios of 25 companies

Class Limits	Class Boundaries	Cumulative Frequency
10 – 14	9.5 – 14.5	4
10 – 19	9.5 – 19.5	4 + 6 = 10
10 – 24	9.5 – 24.5	4 + 6 + 8 = 18
10 – 29	9.5 – 29.5	4 + 6  + 8 + 4 = 22
10 – 34	9.5 – 34.5	4 + 6 + 8 + 4 + 3 = 25

Note: This is a less than cumulative frequency distribution table where each class has the same lower limit but a different upper limit.

We can also work the cumulative relative frequency and cumulative percentages of follows:

Cumulative Revative Frequency =	cumulative frequency total observation in the data set
Cumulative Percentage =	(cumulative relative frequemcy) x 100

OGIVES

An ogive is a curve drawn for the cumulative frequency distribution by joining with as smooth a curve as possible the dots marked above the upper boundaries of classes at height equal to the cumulative frequencies of respective classes.

Example 2.8: The figure below gives an illustration of an ogive for the price-earnings ratio frequency distribution.

Figure 2.5 Ogive for Table 2.9:

LORENZ CURVES

Lorenz curves show how the total value of the measurements of some economic variable is shared out among the subjects or items involved. The curve is obtained by plotting cumulative percentage frequency class totals and compared with a line of equal distribution. The procedure is illustrated in the following example.

Example 2.9: 

Table 2.10 is the distribution that gives the personal wealth of a certain cross section of the population of country X for a particular year. Draw a Lorenz Curve to illustrate this data and use it to estimate:

(a) The percentage of total personal wealth that the least wealthy 30% of persons have their disposal

(b) The percentage of the most wealthy who command one half of all personal wealth.

Table 2.10: Personal Wealth

Personal Wealth	Number of Persons (Hundred thousands) (f)	Total Personal Wealth (v)
0 – 2000	19	2.4
2000 – 5000	26	7.8
5000 – 10 000	74	55.5
10 000 – 15 000	41	49.2
15 000 – 20 000	16	25.7
20 000 – 25 000	8	16.8
25 000 – 50 000	5	15.0
50 000 and over	1	6.3
Total	190	178.7

Solution

• Step 1:  Calculate the cumulative percentage frequency for each class.
• Step 2: Calculate cumulative percentage class totals for each class – in this case these are already given.  If they are not given then use the formula.
• Step 3: Plot cumulative percentage frequency (y – axis) against cumulative percentage class totals (x-axis) as a set of points.
• Step 4:  Join the points, giving the Lorenz curve.

These are illustrated in the table below where f = frequency F = cumulative frequency,
v = values of class totals and  V = cumulative v. (Fig 2.6 shows the required Lorenz Curve)

f	F	F%	v	V	V%
19	19	10	2.4	2.4	1
26	45	24	7.8	10.2	6
74	119	63	55.5	65.7	37
41	160	84	49.2	114.9	64
16	176	93	25.7	140.6	79
8	184	97	16.8	157.4	88
5	189	99	15.0	172.4	96
1	190	100	6.3	178.7	100

Figure 2.6: A Lorenz Curve of Distribution of Personal Wealth.

1. From the graph, when F% = 30, V% = 10.  That is, the least wealthy 30% of the persons have only 10% of the total personal wealth at their disposal.
2. From the graph, when V% = 50, F% = 73.  This means that 50% of all personal wealth is at the disposal of the lower 73%, or the most wealthy 27%.

Note: If the Lorenz curve is fairly close to the line of equal distribution then it means that property values are shared equally between the properties.

There are some other graphs that you need to know how to construct.  These include:

• Component, percentage and multiple bar charts
• Multiple pie charts
• Strata charts
• Z – charts
• Gantt charts
• Semi logarithmic graphs
• Pictograms

EXERCISES

1. Fifteen workers of a company were asked if they thought the salaries of CEOs of Malawian companies were too high. The responses of the workers are listed below. (H, N, and D indicate that a worker thinks the salaries of CEOs are too high, not too high, or that the worker has no opinion/does not know, respectively.)

H	H	D	N	H
H	N	H	D	H
N	H	H	N	D

a. Prepare a frequency distribution table.

b. Calculate the relative frequencies and percentages for all categories.

c. What percentage of the workers in this sample think the salaries of CEOs are too high?
d. Draw a pie chart for the percentage distribution.

2. The data in Table 2.11 give the market value (in billion dollars) of 30 companies as of March 5, 1993.

a. Construct a frequency distribution table using the less than method to write classes

b. Calculate the relative frequencies and percentages for all classes.

c. What percentage of the companies in this sample had a market value of $30 billion or higher on March 5, 1993?
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d. From the frequency distribution of part a, can you tell whether the data are symmetric or skewed? If skewed, are they to the left or right?

Table 2.11:  Market value in billion dollars of 30 international companies as of March 5, 1993

Company	Market Value (billion dollars)	Company	Market Value (billion dollars)
American Express	12.2	IBM	31.5
Ameritech	20.0	J. C. Penney	9.7
Bank America	17.9	Johnson & Johnson	26.5
Boeing	11.7	McDonald’s	18.7
Caterpillar	5.9	Microsoft	23.1
CBS	2.8	Motorola	16.2
Chase Manhattan	5.1	Nike	5.3
Chrysler	13.6	Pepsico	33.1
Cigna	4.3	Pfizer	19.1
Citicorp	9.6	Procter & Gamble	36.5
Dow Jones	3.1	Sears, Roebuck	18.3
Ford Motor	24.1	Travellers	3.9
General Motors	27.6	Walt Disney	24.3
Gillette	13.1	Wells Fargo	5.6
H. J. Heinz	11.2	Xerox	7.8

Source: Mann 1995

3. The data in Table 2.12 estimate the total value of the repayments made by the members of each of the eight classes of the distribution

a. Use these data values to draw up a table of cumulative percentage number of home owners against cumulative percentage repayments.

b. Construct a Lorenz curve.

c. Use the Lorenz curve to estimate what percentage of the given sample pay the first 20% of total repayments.

Table 2.12:

Repayment (MWK000)	Number of Home Owners
Under 0.4	8
0.4 and under 0.8	60
0.8 and under 1.2	100
1.2 and under 1.6	108
1.6 and under 2.0	72
2.0 and under 2.4	32
2.4 and under 2.8	12
2.8 and under 3.2	8