Averages, rates and risk: statistics terms explained

How averages and rates are calculated, and why the choice of median, mean, per-head or age-standardised figures changes what a number says.

Median

The middle value when all values are ordered from lowest to highest. Half are above and half below, so a few very large values cannot pull it up.

The median is the value in the middle of a list arranged in order. With an even number of values it is halfway between the two middle ones. Because it depends only on position, it is not pulled up by a small number of very high values.

Worked example

Five hypothetical salaries: £20,000, £25,000, £30,000, £35,000 and £190,000. The median is £30,000. The mean is £60,000, which is higher than four of the five salaries.

Where you will see it

The ONS Annual Survey of Hours and Earnings leads with median weekly pay for full-time employees.
DWP poverty lines are set at 60% of median household income (see relative and absolute poverty).
ONS house price statistics include median prices.

How it trips up claims

"Average pay" may mean the median, the mean, or average weekly earnings, which is a mean-based measure. These give different numbers and can move differently.
A median for full-time employees is higher than a median for all employees, because part-time pay is included in the second.

Also searched as: median average, middle value, median vs mean, median income

Official definitions: ONS: Annual Survey of Hours and Earnings, latest bulletin · ONS: Household disposable income and inequality, latest bulletin

Mean

The total of all values divided by how many there are. Often what people mean by 'average', it can be pulled up or down by a few extreme values.

The mean (or arithmetic mean) is the sum of all values divided by the number of values. It uses every value, which makes it useful for totals and budgets, but it is sensitive to extremes.

Worked example

Ten hypothetical households: nine have an income of £30,000 and one has £330,000. The total is £600,000, so the mean is £60,000. The median is £30,000. Adding one very high income doubled the mean but left the median unchanged.

When the mean is the right tool

If you want to know total spending or total tax, mean values multiplied by counts give the total. A median cannot do that.
ONS average weekly earnings is a mean-based measure, which is why it reacts to bonuses and high earners.

How it trips up claims

"The average family" built from a mean income may describe very few real families when incomes are spread unevenly.
Comparing a mean from one source with a median from another mixes measures.
When incomes at the top rise faster, the mean can rise while the median stays flat.

Also searched as: arithmetic mean, average, mean vs median, mean average

Official definitions: ONS: Average weekly earnings in Great Britain, latest bulletin · ONS: Household disposable income and inequality, latest bulletin

Per capita (per head)

A total divided by the number of people in the population, such as GDP per head or spending per person, to allow fair comparison between places or years.

Per capita means per person. A per capita figure divides a total by the population it relates to, which makes places or years of different size comparable.

Worked example

Country A has hypothetical output of £2,000 billion and 50 million people: £40,000 per head. Country B has £3,000 billion and 100 million people: £30,000 per head. B has the bigger economy, A has more output per person.

Over time, if total output grows by 1% but population grows by 1.5%, output per head falls by about 0.5%.

How it trips up claims

Total GDP and GDP per head can move in opposite directions. Check which is quoted.
The population figure matters: ONS mid-year estimates are revised, and per-head figures are revised with them.
For comparisons between countries, per-head figures are often converted with purchasing power parities rather than market exchange rates.
Per head of total population differs from per worker or per hour worked, which are productivity measures.

The ONS publishes real GDP per head as series IHXW.

Also searched as: per head, per person, GDP per capita meaning, per capita meaning

Official definitions: ONS: Real GDP per head, seasonally adjusted (IHXW) · ONS: Population estimates

Rate per 100,000

The number of events divided by the population and multiplied by 100,000, so areas of different sizes can be compared on a like-for-like basis.

A rate per 100,000 standardises a count by population size. The calculation is: number of events, divided by population, multiplied by 100,000. Crime, mortality and disease figures are often published this way; some use per 1,000 instead.

Worked example

Town A records 300 hypothetical incidents with a population of 200,000: a rate of 150 per 100,000. City B records 900 incidents with 1,000,000 residents: 90 per 100,000. B has three times the count but a lower rate.

How it trips up claims

Raw counts favour small areas in league tables; rates allow fair comparison.
Small areas have small numbers, so a few extra events can swing the rate a lot. Look for confidence intervals where they are published.
City-centre areas have many visitors who are not residents, which can raise rates per resident for crimes such as shoplifting.
A rate per 100,000 that is not adjusted for age is a crude rate; health comparisons often use an age-standardised rate.

Also searched as: per 100k, rate per 1,000, crime rate per 100,000, per 100,000 population

Official definitions: ONS: Crime in England and Wales, latest bulletin · ONS: Mortality statistics in England and Wales, quality and methodology information

Crude rate

A rate calculated by dividing events by the whole population, with no adjustment for differences in age structure between places or over time.

A crude rate is the simplest rate: total events divided by the total population, usually expressed per 100,000 or per 1,000. The crude death rate, for example, is all deaths in a year divided by the mid-year population.

Worked example

Area A has 1,000 hypothetical deaths among 100,000 people: a crude rate of 1,000 per 100,000. Area B has 600 deaths among 100,000 people: 600 per 100,000. If a third of A's residents are over 75 and only a tenth of B's are, A's higher crude rate may reflect its older population rather than worse health at any given age.

How it trips up claims

Crude death and disease rates usually rise as a population ages, even if the risk at every age is falling.
Comparing a seaside retirement area with a university city on crude rates mostly compares their age profiles.
To compare fairly, ONS and health bodies publish an age-standardised rate alongside or instead of the crude rate.

Crude rates are still useful for planning, because they show the actual demand an area faces.

Also searched as: crude death rate, crude mortality rate, unadjusted rate

Official definitions: ONS: Mortality statistics in England and Wales, quality and methodology information · Office for Health Improvement and Disparities: Fingertips public health data

Age-standardised rate

A rate adjusted to a standard population age structure, so that places or years with older or younger populations can be compared fairly.

An age-standardised rate shows what a rate would be if each population being compared had the same age structure. The ONS uses the 2013 European Standard Population for age-standardised mortality rates.

How it works, simply

Take two hypothetical areas with the same death rates at every age: 1 per 1,000 for under-65s and 40 per 1,000 for over-65s. Area A has 10% aged over 65; Area B has 30%.

Crude rates: A is 4.9 per 1,000 and B is 12.7 per 1,000.
Applying both areas' age-specific rates to the same standard population gives identical age-standardised rates, showing that the risk at each age is the same.

How it trips up claims

A claim that one area is "unhealthier" based on crude rates may reflect age rather than health.
Age-standardised rates are for comparison; they are not the number of deaths that happened. Use counts or crude rates for actual demand.
Rates standardised to different standard populations cannot be compared with each other.

Also searched as: ASMR, age-standardised mortality rate, age-adjusted rate, age standardised

Official definitions: ONS: Mortality statistics in England and Wales, quality and methodology information · ONS: Deaths registered in England and Wales, latest bulletin

Odds, risk, relative and absolute risk

Risk is the chance an event happens out of everyone exposed; odds compare those it happens to with those it does not. Relative risk compares two groups' risks.

Risk is the proportion of people who experience an event: 5 out of 100 is a risk of 5%. Odds compare those who experience it with those who do not: 5 to 95, or about 0.053. For rare events risk and odds are close; for common events they diverge.

Absolute and relative risk: a worked example

In a hypothetical group, 2 in 1,000 people develop a condition. In a second group exposed to some factor, 3 in 1,000 do.

Absolute risk rises from 0.2% to 0.3%: an increase of 0.1 percentage points, or one extra case per 1,000 people.
Relative risk is 3 divided by 2, which is 1.5: a "50% higher risk".

How it trips up claims

A large relative risk can describe a very small absolute change. Look for both.
Odds ratios are sometimes reported as if they were relative risks; for common outcomes this overstates the difference.
Risk depends on the population and period. Check who was counted and over what time.

Official health and mortality bulletins usually state rates per population, which let you work out absolute risk.

Also searched as: relative risk, absolute risk, odds ratio, risk vs odds, risk ratio

Official definitions: ONS: Uncertainty and how we measure it · ONS: Mortality statistics in England and Wales, quality and methodology information

Deciles and quintiles

Ways of dividing a ranked population into equal groups: ten groups (deciles) or five groups (quintiles), often used for income.

When a population is ranked from lowest to highest on a measure such as income, deciles split it into ten equal-sized groups and quintiles into five. The "bottom decile" is the poorest 10%; the "top quintile" is the richest 20%. The term can also mean the cut-off point between groups, such as the income at which someone enters the top 10%.

Why it matters when checking a claim

Group or threshold? "The top 10% earn over £X" uses a threshold; "the top 10% earn an average of £Y" uses a group average. They are different numbers.
Household or person? Income distributions in HBAI rank individuals by their household's income, adjusted for household size.
Deprivation deciles. Area statistics often use deciles of the Index of Multiple Deprivation, which rank small areas, not people.
Ratios. Comparing the top and bottom decile is a common companion to the Gini coefficient.

Also searched as: deciles, quintiles, percentiles, top 10%, bottom fifth

Official definitions: DWP: Households below average income (HBAI) statistics · ONS: Effects of taxes and benefits on household income

Weighted average

An average in which some values count for more than others, according to their size or importance.

A weighted average gives each value a weight that reflects how much it should count. A simple average of three hospitals' waiting times treats a small hospital and a large one equally; a weighted average counts each in proportion to how many patients it treats.

Worked example

Hypothetical area A has 10,000 people and a rate of 2%; area B has 90,000 people and a rate of 6%. The simple average of the two rates is 4%, but the weighted average, which is the actual rate for the combined population, is 5.6%.

Why it matters when checking a claim

Averaging regional or local rates without weighting can give a national figure that does not match the real one.
Price indices such as CPI are weighted averages of price changes, with weights based on household spending. See survey weighting.

Also searched as: weighted mean, weighted average meaning, how to calculate a weighted average

Official definitions: ONS: Higher-level aggregation and weights in consumer prices · ONS: Consumer price inflation, latest bulletin