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Lorenz Curve This is a form of cumulative frequency graph, which shows how evenly a variable is distributed, e.g. how evenly wealth is distributed over a population.
To construct a Lorenz Curve
- If not already so, change all values to % form (value divided by total multiplied by
- Find the cumulative %, the total to the end of each interval
- Draw a 100 by 100 graph and plot the cumulative % against each other
- Draw in the line of equal Distribution, i.e. a line from (0,0) to (100,100). The further the curve is from this line, the more unequal the distribution.
Example Shops Turnover 1996 Turnover 2002
1996 Cum. %
2002 Cum. % Top 2% 50% 54% 50% 54% Next 3% 18% 20% 68% 74% Next 5% 13% 12% 81% 86% Next 10% 5% 6% 86% 92% Next 30% 6% 5% 92% 97% Lowest 50% 8% 3% 100% 100%
Plot the points (0,0), (50,54), (68, 74), (81,86), (86,92), (92,97), (100,100)
EXAMPLE 2
The following table shows the personal wealth of a certain section of the population of the U.K. for a particular year. Draw a Lorenz curve to illustrate the data.
Personal Number of Persons Total Personal Wealth (€) (00,000) Wealth (€000 mil) (a) (b)
0-2000 19 2. 2000-5000 26 7. 5000-10000 74 55. 10000-15000 41 49. 15000-20000 16 25. 20000-25000 8 16. 25000-50000 5 15 ≥ 50000 1 6.
First calculate the % for column (a) and column (b)
Personal Number of Persons Total Personal Col (a) Col (b) Wealth (€) (00,000) Wealth (€000 mil) as % as % (a) (b) 0-2000 19 2.4 10.0 1. 2000-5000 26 7.8 13.7 4. 5000-10000 74 55.5 38.9 31. 10000-15000 41 49.2 21.6 27. 15000-20000 16 25.7 8.4 14. 20000-25000 8 16.8 4.2 9. 25000-50000 5 15 2.6 8. ≥ 50000 1 6.3 0.5 3.
190 178.
Then Find the cumulative % Personal Number of Persons Total Personal Col (a) Col (b) Cum % Cum % Wealth (€) (00,000) Wealth (€000 mil) as % as % (a) (b) (a) (b) 0-2000 19 2.4 10.0 1.3 10.0 1. 2000-5000 26 7.8 13.7 4.4 23.7 5. 5000-10000 74 55.5 38.9 31.1 62.6 36. 10000-15000 41 49.2 21.6 27.5 84.2 64. 15000-20000 16 25.7 8.4 14.4 92.6 78. 20000-25000 8 16.8 4.2 9.4 96.8 88. 25000-50000 5 15 2.6 8.4 99.5 96. ≥ 50000 1 6.3 0.5 3.5 100.0 100.
190 178.
On a 100X100 graph the cumulative pairs:
Z Chart
A Z chart is a form of time series that is extremely useful for presenting business data over a year. It shows a. The monthly figures for each of the 12 months b. The cumulative totals from the beginning of the year to the end of each month c. The total for the previous 12 months to the end of each month, the annual moving totals. For this you need the figures for the year before the year for which you are drawing the chart.
Example: The sales figures for a company for the years 2004 and 2005 are 2004 Sales 2005 Sales January 7 8 February 7 8 March 8 8 April 7 9 May 9 8 June 8 8 July 8 7 August 7 8 September 6 9 October 7 6 November 8 9 December 8 9 Total 90 97
We first find the cumulative monthly sales (Col CMS) and the annual moving totals (Col AMT)
2004 Sales 2005 Sales CMS AMT January 7 8 8 91 February 7 8 16 92 March 8 8 24 92 April 7 9 33 94 May 9 8 41 93 June 8 8 49 93 July 8 7 56 92 August 7 8 64 93 September 6 9 73 96 October 7 6 79 95 November 8 9 88 96 December 8 9 97 97 Total 90 97
We now put in the 2005 Sale, the CMS and the AMT figures on the same graph. The 12 months are on the X axis and the figures on the Y axis
Z Chart
88 8 8 9 8 8 7 8 9 6 9 9
16
24
33
41
49
56
64
73
79
88
97 91 92 92 94 93 93 92 93
96 95 96 97
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10 11 12 Months 2005
Sales
Sales 2005 CMS AMT
