Data Analysis for Inequalities

Sophisticated data analysis will help you spot patterns, trends and relationships in your results. Data analysis can be qualitative and/or quantitative, and may include statistical tests. An example of a statistical test is outlined below.

Lorenz curves

The Lorenz curve is a graph showing how evenly distributed a variable is over space.
The diagonal black line represents a perfectly even distribution. The blue and red lines show uneven distributions. The further these coloured lines are from the black line, the more uneven is the distribution.

You can draw Lorenz curves based on ordinal data (see worked example 1 below) or interval data (see worked example 2 below).

Worked example 1: Lorenz curve for ordinal data

There are 32844 LSOAs in England. These have been given an IMD score, and then ranked from 0 (the most deprived) to 32844 (the least deprived). The LSOAs can be divided into five quintiles. The table shows how many LSOAs are in each of the five quintiles for Barking and Dagenham and for Hillingdon.

All LSOAs in England	All LSOAs in Barking and Dagenham	All LSOAs in Hillingdon
1st (top 20% deprived)	66	6
2nd	35	52
3rd	8	30
4th	0	31
5th (least 20% deprived)	0	33
SUM	109	152

From the raw data, it looks like there is a greater number of deprived LSOAs in Barking and Dagenham. In contrast, Hillingdon contains a more even distribution. Calculate the percentages for all three columns.

England	England	Barking and Dagenham	Barking and Dagenham	Hillingdon	Hillingdon
quintile	%	raw data	%	raw data	%
1st	20	66	60.6	6	3.9
2nd	20	35	32.1	52	34.2
3rd	20	8	7.3	30	19.7
4th	20	0	0	31	20.4
5th	20	0	0	33	21.7
SUM	100	109	100	152	100

Now calculate the cumulative percentages for all three columns.

England	England	England	Barking and Dagenham	Barking and Dagenham	Barking and Dagenham	Hillingdon	Hillingdon	Hillingdon
quintile	%	cu%	data	%	cu%	data	%	cu%
1st	20	20	66	60.6	60.6	6	3.9	3.9
2nd	20	40	35	32.1	92.7	52	34.2	38.2
3rd	20	60	8	7.3	100	30	19.7	57.9
4th	20	80	0	0	100	31	20.4	78.3
5th	20	100	0	0	100	33	21.7	100
SUM	100	100	109	100	100	152	100	100

Plot a scattergraph with axes as follows

• x-axis: cumulative percentages for England
• y-axis: cumulative percentages for a single London Borough

The black line shows a perfectly even distribution. This shows the distribution of deprivation ranks in England. The further a line is from this, the more uneven the distribution. As suspected, Barking and Dagenham has a more uneven distribution of IMD ranks than Hillingdon.

Worked example 2: Lorenz curve for interval data

Lorenz curves can also be constructed for interval data, but there are some extra steps.

Bristol City Council have divided up the city into 14 ‘Neighbourhood Areas’. For each Neighbourhood Area, the total population of each area has been counted, plus the number of people with a ‘severe limiting long-term illness’.

This information can be used to help answer the question: do certain areas of Bristol contain a greater concentration of severely ill people than other areas? Or by contrast, are severely ill people evenly distributed throughout Bristol?

Name of Neighbourhood Area in Bristol	Total population of this Neighbourhood Area	Number of severely ill in this Neighbourhood Area
Ashley	47782	3514
Avonmouth	20237	2074
Bishopston	36713	1383
Clifton	41192	1537
Dundry View	28771	3411
Filwood	38778	3553
Bedminster	22664	1864
Brislington	22107	1686
Fishponds	27575	3503
Henbury	24253	2691
Hengrove	28786	2963
Henleaze	31412	2127
Horfield	23912	2141
St Georges	24052	2123
TOTAL	418234	34570

Calculate the percentages for the ‘total population’ and ‘number of severely ill’ columns. This shows the percentage of Bristol’s population and number of severely ill people in each Neighbourhood Area. For example, Fishponds contains 6.59% of Bristol’s population and 10.13% of Bristol’s severely ill people.

Name of Neighbourhood Area in Bristol	Total population of this Neighbourhood Area		Number of severely ill in this Neighbourhood Area
		%		%
Ashley	47782	11.42	3514	10.16
Avonmouth	20237	4.84	2074	6.00
Bishopston	36713	8.78	1383	4.00
Clifton	41192	9.85	1537	4.45
Dundry View	28771	6.88	3411	9.87
Filwood	38778	9.27	3553	10.28
Bedminster	22664	5.42	1864	5.39
Brislington	22107	5.29	1686	4.88
Fishponds	27575	6.59	3503	10.13
Henbury	24253	5.80	2691	7.78
Hengrove	28786	6.88	2963	8.57
Henleaze	31412	7.51	2127	6.15
Horfield	23912	5.72	2141	6.19
St Georges	24052	5.75	2123	6.14
TOTAL	418234	100	34570	100.00

Calculate the ratio between the two percentage columns.

\(\mathsf{ratio = \frac{\%\;severely\;ill}{\%\; population}}\)

For example, in Ashley, the ratio is \(\frac{10.16}{11.42} = 0.89\)

Name of Neighbourhood Area in Bristol	Total population of this Neighbourhood Area	Number of severely ill in this Neighbourhood Area	Ratio of % severely ill to % population
	%		%
Ashley	47782	11.42	3514	10.16	0.89
Avonmouth	20237	4.84	2074	6.00	1.24
Bishopston	36713	8.78	1383	4.00	0.46
Clifton	41192	9.85	1537	4.45	0.45
Dundry View	28771	6.88	3411	9.87	1.43
Filwood	38778	9.27	3553	10.28	1.11
Bedminster	22664	5.42	1864	5.39	1.00
Brislington	22107	5.29	1686	4.88	0.92
Fishponds	27575	6.59	3503	10.13	1.54
Henbury	24253	5.80	2691	7.78	1.34
Hengrove	28786	6.88	2963	8.57	1.25
Henleaze	31412	7.51	2127	6.15	0.82
Horfield	23912	5.72	2141	6.19	1.08
St Georges	24052	5.75	2123	6.14	1.07
TOTAL	418234	100	34570	100.00

Rank the ratio column from highest number to lowest number. You can either do this by hand or by using the Sort command in Excel.

Name of Neighbourhood Area in Bristol	Total population of this Neighbourhood Area	Number of severely ill in this Neighbourhood Area	Ratio of % severely ill to % population
	%		%		rank
Ashley	47782	11.42	3514	10.16	0.89	11
Avonmouth	20237	4.84	2074	6.00	1.24	5
Bishopston	36713	8.78	1383	4.00	0.46	13
Clifton	41192	9.85	1537	4.45	0.45	14
Dundry View	28771	6.88	3411	9.87	1.43	2
Filwood	38778	9.27	3553	10.28	1.11	6
Bedminster	22664	5.42	1864	5.39	1.00	9
Brislington	22107	5.29	1686	4.88	0.92	10
Fishponds	27575	6.59	3503	10.13	1.54	1
Henbury	24253	5.80	2691	7.78	1.34	3
Hengrove	28786	6.88	2963	8.57	1.25	4
Henleaze	31412	7.51	2127	6.15	0.82	12
Horfield	23912	5.72	2141	6.19	1.08	7
St Georges	24052	5.75	2123	6.14	1.07	8
TOTAL	418234	100	34570	100.00

Rearrange the rows in the table according to the ranks that you have just made.

Neighbourhood Area	% total population	% severely ill	ratio	rank
Fishponds	6.59	10.13	1.54	1
Dundry View	6.88	9.87	1.43	2
Henbury	5.80	7.78	1.34	3
Hengrove	6.88	8.57	1.25	4
Avonmouth	4.84	6.00	1.24	5
Filwood	9.27	10.28	1.11	6
Horfield	5.72	6.19	1.08	7
St Georges	5.75	6.14	1.07	8
Bedminster	5.42	5.39	1.00	9
Brislington	5.29	4.88	0.92	10
Ashley	11.42	10.16	0.89	11
Henleaze	7.51	6.15	0.82	12
Bishopston	8.78	4.00	0.46	13
Clifton	9.85	4.45	0.45	14

Calculate cumulative figures for the two % columns.

Neighbourhood Area	total population	severely ill
	%	cumulative %	%	cumulative %
Fishponds	6.59	6.59	10.13	10.13
Dundry View	6.88	13.47	9.87	20.00
Henbury	5.80	19.27	7.78	27.78
Hengrove	6.88	26.15	8.57	36.36
Avonmouth	4.84	30.99	6.00	42.35
Filwood	9.27	40.26	10.28	52.63
Horfield	5.72	45.98	6.19	58.83
St Georges	5.75	51.73	6.14	64.97
Bedminster	5.42	57.15	5.39	70.36
Brislington	5.29	62.44	4.88	75.24
Ashley	11.42	73.86	10.16	85.40
Henleaze	7.51	81.37	6.15	91.55
Bishopston	8.78	90.15	4.00	95.55
Clifton	9.85	100.00	4.45	100.00

Finally it is time to draw the Lorenz curve! Plot the cumulative % total population on the x-axis. Plot the cumulative % severely ill on the y-axis.

Gini coefficient

Lorenz curves are a useful visual technique for presenting your data. But it is sometimes difficult to see how one uneven distribution compares to another. The Gini coefficient is a summary statistic that will provide a precise answer.

\(\mathsf{Gini\;coefficient = \frac{area\;of\;graph\; between\;the\;diagonal\;and\;the\;curve}{area\;of\;graph \;above\;the\;diagonal}}\)

The result for the Gini coefficient ranges from 0 (completely even distribution) to 1 (completely uneven distribution).

Worked example of Gini coefficient

There are 32844 LSOAs in England. These have been given an IMD score, and then ranked from 0 (the most deprived) to 32844 (the least deprived). The LSOAs can be divided into five quintiles. The table shows how many LSOAs are in each of the five quintiles for Barking and Dagenham and for Hillingdon.

All LSOAs in England	All LSOAs in Barking and Dagenham	All LSOAs in Hillingdon
1st (top 20% deprived)	66	6
2nd	35	52
3rd	8	30
4th	0	31
5th (least 20% deprived)	0	33
SUM	109	152

Lorenz curves were plotted for the data.

To calculate the area of the graph above the diagonal, and the area of graph between the diagonal and the curve, you can count the number of squares on graph paper. Include fractions for part-squares.

There are 625 squares shown 312.5 squares are above the black diagonal line There are 61 squares between the diagonal and the red curve (for Hillingdon) There are 109 squares between the diagonal and the red curve (for Barking)

\(\mathsf{Gini\;coefficient} = \frac{109}{312.5}=0.35\) \(\mathsf{Gini\;coefficient} = \frac{61}{312.5}=0.20\)

Location Quotient

The Location Quotient is another mathematical technique for showing how unevenly distributed a variable is over space.

\(\mathsf{Location\;Quotient = \frac{\%\;in\;one \;area}{\%\;the\;whole\;population}}\)

Location Quotient (LQ) varies from 0 to infinity.

If LQ is less than 1, the variable is under-represented in a particular area. If LQ is greater than 1, the variable is over-represented in a particular area.

Worked example

Bristol City Council have divided up the city into 14 ‘Neighbourhood Areas’. For each Neighbourhood Area, the number of people in different age bands has been counted. Here are the total number of people aged 16-24 and 65-74 for each area.

Name of Neighbourhood Area in Bristol	Total population of this Neighbouhood Area	Total number of people aged 16-24
Ashley	47782	7519
Avonmouth	20237	2364
Bishopston	36713	8351
Clifton	41192	14003
Dundry View	28771	3621
Filwood	38778	4288
Bedminster	22664	2762
Brislington	22107	2294
Fishponds	27575	5535
Henbury	24253	2631
Hengrove	28786	3137
Henleaze	31412	4160
Horfield	23912	3773
St Georges	24052	2566
TOTAL	418234	67004

Calculate the percentages for the ‘total population’ and ‘number aged 16-24’ columns. This shows the percentage of Bristol’s population and number of people aged 16-24 in each Neighbourhood Area.

For example, Avonmouth contains 3.53% of all the 16-24 year olds in Bristol. Be careful not to get confused here. This does not mean that 3.53% of Avonmouth’s population is aged 16-24.

Name of Neighbourhood Area in Bristol	Total population of this Neighbouhood Area	Total number of people aged 16-24 in this Neighborhood Area
	%		%
Ashley	47782	11.42	7519	11.22
Avonmouth	20237	4.84	2364	3.53
Bishopston	36713	8.78	8351	12.46
Clifton	41192	9.85	14003	20.90
Dundry View	28771	6.88	3621	5.40
Filwood	38778	9.27	4288	6.40
Bedminster	22664	5.42	2762	4.12
Brislington	22107	5.29	2294	3.42
Fishponds	27575	6.59	5535	8.26
Henbury	24253	5.80	2631	3.93
Hengrove	28786	6.88	3137	4.68
Henleaze	31412	7.51	4160	6.21
Horfield	23912	5.72	3773	5.63
St Georges	24052	5.75	2566	3.83
TOTAL	418234	100	67004	100

The Location Quotient is the ratio between the two percentage columns.
\(\mathsf{Location\;Quotient = \frac{\%\;aged\;16-24}{\%\;whole\;population}}\)

For example, in Avonmouth, the LQ is \(\frac{3.53}{4.84} = 0.73\)

Name of Neighbourhood Area in Bristol	Total population of this Neighbourhood Area	Total number of people aged 16-24 in this Neighborhood Area	Location Quotient
	%		%
Ashley	47782	11.42	7519	11.22	0.98
Avonmouth	20237	4.84	2364	3.53	0.73
Bishopston	36713	8.78	8351	12.46	1.42
Clifton	41192	9.85	14003	20.90	2.12
Dundry View	28771	6.88	3621	5.40	0.79
Filwood	38778	9.27	4288	6.40	0.69
Bedminster	22664	5.42	2762	4.12	0.76
Brislington	22107	5.29	2294	3.42	0.65
Fishponds	27575	6.59	5535	8.26	1.25
Henbury	24253	5.80	2631	3.93	0.68
Hengrove	28786	6.88	3137	4.68	0.68
Henleaze	31412	7.51	4160	6.21	0.83
Horfield	23912	5.72	3773	5.63	0.98
St Georges	24052	5.75	2566	3.83	0.67
TOTAL	418234	100	67004	100	1

The calculated figures show that people aged 16-24 are under-represented in a number of areas, such as Avonmouth, Brislington and St Georges. But people aged 16-24 are over-represented in other areas, such as Clifton, Bishopston and Fishponds. The LQ results show that the greatest concentration of young adults is in Clifton: can you find any other data to help explain this?

Index of Dissimilarility

The Index of Dissimilarility is used to compare the distribution of two variables, such as two socio-economic groups or two ethnic groups in a particular area.

\(\mathsf{Index\;of\;dissimilarity} = 1/2 ∑ |x_i/X-y_i/Y|\)

• \(x_i\)is the population of group \(x\) in small area \(i\)
• \(X\) is the total population of group \(x\) in the whole area
• \(y_i\)is the population of group \(y\) in small area \(i\)
• \(Y\) is the total population of group \(y\) in the whole area

It helps answer the question: is group X more evenly distributed in a particular place than group Y? The index ranges from 0 (complete integration) to 100 (complete segregation).

Worked example 1 of Index of Dissimilarity

Census 2011 data for wards in Sandwell (West Midlands) can be obtained from Nomis. An extract is shown below

Name of ward in Sandwell	Number of persons identifying their ethnicity as White in the ward (this is \(x_i\))	Number of persons identifying their ethnicity as Asian in the ward (this is \(y_i\))
Abbey	9078	1271
Blackheath	10808	870
Bristnall	9064	1814
Charlemont with Grove Vale	8903	1918
Cradley Heath and Old Hill	11913	1009
Friar Park	11335	619
Great Barr with Yew Tree	8300	3105
Great Bridge	10393	1626
Greets Green and Lyng	6925	3244
Hateley Heath	10295	2182
Langley	10135	1448
Newton	7879	2178
Old Warley	9388	1399
Oldbury	7648	4011
Princes End	11847	369
Rowley	10648	609
St Pauls	4252	7822
Smethwick	7128	4522
Soho and Victoria	3854	6881
Tipton Green	9262	2625
Tividale	10616	913
Wednesbury North	10331	1734
Wednesbury South	9132	2232
West Bromwich Central	6337	4857
TOTAL	215471 (this is \(X\))	59258 (this is \(Y\))

This means that Princes End contains 5.50% of people identifying as White in Sandwell. Be careful not to get confused here. This does not mean that 5.50% of the population of Princes End is White.

Ward	White	White	Asian	Asian
	raw data	% of Sandwell’s population (this is \(\frac{x_i}{X}\))	raw data	% of Sandwell’s population (this is \(\frac{y_i}{Y}\))
Abbey	9078	4.21	1271	2.14
Blackheath	10808	5.02	870	1.47
Bristnall	9064	4.21	1814	3.06
Charlemont	8903	4.13	1918	3.24
Cradley Heath	11913	5.53	1009	1.70
Friar Park	11335	5.26	619	1.04
Great Barr	8300	3.85	3105	5.24
Great Bridge	10393	4.82	1626	2.74
Greets Green	6925	3.21	3244	5.47
Hateley Heath	10295	4.78	2182	3.68
Langley	10135	4.70	1448	2.44
Newton	7879	3.66	2178	3.68
Old Warley	9388	4.36	1399	2.36
Oldbury	7648	3.55	4011	6.77
Princes End	11847	5.50	369	0.62
Rowley	10648	4.94	609	1.03
St Pauls	4252	1.97	7822	13.20
Smethwick	7128	3.31	4522	7.63
Soho and Victoria	3854	1.79	6881	11.61
Tipton Green	9262	4.30	2625	4.43
Tividale	10616	4.93	913	1.54
Wednesbury N	10331	4.79	1734	2.93
Wednesbury S	9132	4.24	2232	3.77
West Bromwich C	6337	2.94	4857	8.20
SUM	215471	100.00	59258	100.00

Calculate \(\vert\;x\;-\;y\;\vert\)

This is the difference between the two columns of percentages. Remove all negative numbers.

Ward	White	White	Asian	Asian
	raw data	%	raw data	%	Differences (this is \(|x_i/X-y_i/Y|\) )
Abbey	9078	4.21	1271	2.14	2.07
Blackheath	10808	5.02	870	1.47	3.55
Bristnall	9064	4.21	1814	3.06	1.15
Charlemont	8903	4.13	1918	3.24	0.90
Cradley Heath	11913	5.53	1009	1.70	3.83
Friar Park	11335	5.26	619	1.04	4.22
Great Barr	8300	3.85	3105	5.24	1.39
Great Bridge	10393	4.82	1626	2.74	2.08
Greets Green	6925	3.21	3244	5.47	2.26
Hateley Heath	10295	4.78	2182	3.68	1.10
Langley	10135	4.70	1448	2.44	2.26
Newton	7879	3.66	2178	3.68	0.02
Old Warley	9388	4.36	1399	2.36	2.00
Oldbury	7648	3.55	4011	6.77	3.22
Princes End	11847	5.50	369	0.62	4.88
Rowley	10648	4.94	609	1.03	3.91
St Pauls	4252	1.97	7822	13.20	11.23
Smethwick	7128	3.31	4522	7.63	4.23
Soho and Victoria	3854	1.79	6881	11.61	9.82
Tipton Green	9262	4.30	2625	4.43	0.13
Tividale	10616	4.93	913	1.54	3.39
Wednesbury N	10331	4.79	1734	2.93	1.87
Wednesbury S	9132	4.24	2232	3.77	0.47
West Bromwich C	6337	2.94	4857	8.20	5.26
SUM	215471	100.00	59258	100.00	75.21

This is the sum of all the differences column.

In this example, \(\vert\frac{x_i}{X}-\frac{y_i}{Y}=75.21\vert\)

Calculate \(\mathsf{Index\;of\;dissimilarity} = 1/2 ∑ \vert \frac{x_i}{X} – \frac{y_i}{Y}\vert\)

In this example \(\mathsf{Index\;of\;dissimilarity} = 1/2 \times 75.21 = 37.61\)

This means that 37.61 of the Asian population of Sandwell would need to change residence to a different ward in order to have the same relative distribution as the White population of Sandwell.

Worked example 2 of Index of Dissimilarity

Census 2011 data for wards in Sandwell (West Midlands) can be obtained from Nomis. The Index of Dissimilarility has been calculated for ward-level data for the 7 largest ethnic groups of residents (excluding people of mixed ethnicity). A summary of the results is shown in the table.

	White British	White Other	Indian	Pakistani	Bangladeshi	Black Caribbean	Black African
White British		31.76	37.39	54.01	57.81	33.86	33.48
White Other			18.68	39.80	45.22	15.96	21.07
Indian				34.03	42.73	15.58	25.60
Pakistani					43.42	33.71	26.17
Bangladeshi						48.79	54.75
Black Caribbean							18.16

---