Basic inequality measures

An introduction to inequality measurement is available elsewhere in this guidance.

Range and gap measures

Basic range measures can be used to show to absolute or relative gap between two groups of the population.

The absolute gap is given by:

\[ AbsoluteGap = X_a - X_b \]

The relative gap is given by:

\[ RelativeGap = \frac{X_a}{X_b} \]

where:

\(X_a\) is the indicator value for group A
\(X_b\) is the indicator value for group B

If the statistic being measured is a rate, the relative gap is a rate ratio, as described in the Rates section in the Basic public health statistics chapter.

If it is a proportion, the relative gap is a relative risk, as described in the Odds ratio and relative risk section. The relative difference between two proportions is better measured using the odds ratio as explained in that section: the odds ratio is used as an inequality measure.

Confidence interval calculations are also set out in these sections.

Interactive example: Calculating the absolute and relative gap in mortality rates per 100,000 population between group A and group B

Enter mortality rates per 100,000 population for groups A and B:

viewof a1 = Inputs.number([0, 1000], {step: 1, value: 300, width: 100, label: "Group A:"})

viewof b1 = Inputs.number([0, 1000], {step: 1, value: 200, width: 100, label: "Group B:"})

input_data1 = [
  {Group: 'Group A', Mortality_rate: a1},
  {Group: 'Group B', Mortality_rate: b1}
]

Plot.plot({
  ariaLabel:"Bar chart",
  ariaDescription:"A bar chart showing mortality rate per 100000 population for Group A and Group B",
  height: 300,
  style: {fontSize: 40},
  marginLeft: 140,
  marginBottom: 100,
  x: {label: "Mortality rate (per 100,000 population)",
      labelOffset: 50},
  y: {padding: 0.2,
      label: ""},
  style: {fontSize: 16},
  marks: [
    Plot.barX(input_data1, {
      y: a => a.Group,
      x: "Mortality_rate",
      fill: "Group",
      title: a => `${a.Group}: ${a.Mortality_rate}%`
                          })
  ]
})

Figure 1: Mortality rate per 100,000 population for Groups A and B

abs_val = a1 - b1

rel_val = ((a1 / b1).toFixed(2))

md`The **absolute gap** is the difference between two values: ${a1} – ${b1} = **${abs_val} deaths per 100,000 population**.`

md`The **relative gap** is the ratio of two values: ${a1} / ${b1} = **${rel_val}**.`

Interpretation of range measures

The absolute gap is presented in the same units of the indicator values. An absolute gap of 0 indicates no inequality between the two groups.

The relative gap is presented with no units. A relative gap of 1 indicates no inequality between the two groups.

Absolute gap values can be either positive or negative, and relative gap values either greater or less than 1 depending on whether the indicator value of group A is higher or lower than group B.

Odds ratio and relative risk

The odds ratio and relative risk can be used as relative measures of inequality. These are introduced and calculations described in the Odds ratio and relative risk section.

Mean absolute difference

The mean absolute difference shows the average of the absolute differences (differences irrespective of direction: all differences treated as positive numbers) between each population subgroup and a reference group. The mean absolute difference is an absolute measure of inequality (see the inequality measurement overview).

The reference group should be one of the population subgroups. Any subgroup could be chosen as the reference group, but the same reference group should be used if the mean absolute difference measure is being used to compare areas or analyse trends over time. OHID’s approach is to use the group with the largest population as the reference group. For example, if a mean absolute difference is being calculation for an indicator broken down by ethnic group, the White British group would be selected as the reference group, if this group had the largest population compared with the other ethnic groups being considered.

The mean absolute difference (MAD) is given by:

\[ MAD = \frac {({Abs(G_1 - G_r})) + Abs((G_2 - G_r)) + ... + Abs((G_n - G_r))}{n} \] where:

\(G\) is the indicator value for each group from 1 to \(n\)
\(G_r\) is the indicator value for the reference group
\(n\) is the number of subgroups (not including the reference group)

Interactive example: Calculating the mean absolute difference in mortality rate per 100,000 population for groups A to E

Enter the mortality rate per 100,000 population for groups A to E:

viewof a = Inputs.number([0, 1000], {step: 1, width: 200, value: 163, label: "Group A (Reference group):"})
viewof b = Inputs.number([0, 1000], {step: 1, width: 200, value: 241, label: "Group B:"})
viewof c = Inputs.number([0, 1000], {step: 1, width: 200, value: 236, label: "Group C:"})
viewof d = Inputs.number([0, 1000], {step: 1, width: 200, value: 168, label: "Group D:"})
viewof e = Inputs.number([0, 1000], {step: 1, width: 200, value: 159, label: "Group E:"})

viewof a.querySelector("label").style.width = '250px'

viewof b.querySelector("label").style.width = '250px'

viewof c.querySelector("label").style.width = '250px'

viewof d.querySelector("label").style.width = '250px'

viewof e.querySelector("label").style.width = '250px'

input_data = [
  {Group: 'Group A \n(Reference group)', Mortality_rate: a, Fill: 'Ref'},
  {Group: 'Group B', Mortality_rate: b, Fill: 'Other'},
  {Group: 'Group C', Mortality_rate: c, Fill: 'Other'},
  {Group: 'Group D', Mortality_rate: d, Fill: 'Other'},
  {Group: 'Group E', Mortality_rate: e, Fill: 'Other'}
]

Plot.plot({
  ariaLabel:"Bar chart",
  ariaDescription:"A bar chart showing mortality rate per 100000 population for Groups A to E",
  height: 400,
  marginLeft: 140,
  marginBottom: 100,
  x: {label: "Mortality rate (per 100,000 population)",
      labelOffset: 50},
  y: {padding: 0.2,
      label: ""},
  style: {fontSize: 16},
  marks: [
    Plot.barX(input_data, {
      y: a => a.Group, 
      x: "Mortality_rate", 
      <!-- sort: {y: "x"}, -->
      fill: "Fill",
      title: a => `${a.Group}: ${a.Mortality_rate}%`
                          })
  ]
})

Figure 2: Mortality rate per 100,000 population for Groups A to E

B = a - b
C = a - c
D = a - d
E = a - e

meandiff = (Math.abs(B) + Math.abs(C) + Math.abs(D) + Math.abs(E))/4

md`Absolute difference between groups B and A = ${b} – ${a} = ${B} deaths per 100,000 population.`

md`Absolute difference between groups C and A = ${c} – ${a} = ${C} deaths per 100,000 population.`

md`Absolute difference between groups D and A = ${d} – ${a} = ${D} deaths per 100,000 population.`

md`Absolute difference between groups E and A = ${e} – ${a} is ${E} deaths per 100,000 population.`

md`The **mean absolute difference** is (${Math.abs(B.toFixed(1))} + ${Math.abs(C.toFixed(1))} + ${Math.abs(D.toFixed(1))} + ${Math.abs(E.toFixed(1))}) / 4 =  **${meandiff.toFixed(1)} deaths per 100,000 population**.`

It is possible to calculate confidence intervals for the mean absolute difference using a bootstrapping approach. However this has not been developed or used by OHID.

Interpretation of the mean absolute difference

The mean absolute difference value is presented in the same units as the indicator values.

The larger the value, the greater the difference between the groups: a mean absolute difference of zero would imply no difference between the groups at all: all values would be identical.

Page last updated: December 2025