Technical Briefing 3Commonly used public health statistics andtheir confidence intervals

PurposeThis is the third in a series of technical briefingsproduced by the Association of Public HealthObservatories (APHO), designed to support publichealth practitioners and analysts and to promote theuse of public health intelligence in decision making.

In this briefing we describe the calculation of themore common types of statistic used within publichealth intelligence, specifically: rates, proportions,means and age-standardised rates and ratios. Wealso describe the use of confidence intervals, i.e.what they are and why they are used, and presentthe APHO-preferred methods for calculating theintervals for the types of statistic described. Furthermaterial, including detailed bibliography, methodsand tools, is available from http://www.apho.org.uk

Contents

Key public health measures 2

Counts, proportions, 2percentages and rates

Means 2

Age-standardised ratios, 2rates and proportions

Confidence intervals 4

Use and presentation 5

Calculating the statistics 6and their confidence intervals

Proportions 6

Rates, counts and means 7

Age-standardised ratios 8and rates

References 10

Glossary 11

AuthorDaniel Eayres(National Centre for Health OutcomesDevelopment)

Acknowledgements

Anne CunninghamPaul Fryers Jake AbbasJulian FlowersDiane Stockton

Published March 2008 - Reprinted April 2010

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Key public health measuresCounts, proportions, percentages and rates

The most basic measure used in public health is the count.This may be a count of events such as deaths oradmissions to hospital, or a count of people with aparticular attribute such as people who smoke. This countitself is essential information for planning the healthservices for prevention and/or treatment. However, toproperly investigate the distribution of disease and riskfactors and to make comparisons between differentpopulations, the denominator population or population-years at risk in which the count was observed must also betaken into consideration. The simplest way of doing this isto divide the numerator count by the denominatorpopulation to give a proportion, percentage or rate.

At this point it is worth distinguishing between these terms.The term ‘rate’ in particular tends to be rather looselyapplied to describe many public health statistics, some ofwhich would be better described as proportions. Thedistinction is not important to the calculation of the statisticitself but is necessary to best determine the confidenceinterval.

Proportions are statistics where the denominator is thecount of a ‘closed’ population, and the numerator is thecount of members of this population that have a specifiedcharacteristic. Common examples in public health occurwhere a population is evaluated at a finite time point for acertain characteristic, e.g. survey prevalence of smoking orobesity, day cases as a proportion of elective admissions,discharge to usual place of residence, or stillbirths as aproportion of all live and still births.

For other statistics the population is an ‘open’ oneevaluated over a period of time. Individuals may enter orleave the population during this period (through ageing,migration, birth, death, loss to follow-up etc), eachcontributing different periods at risk. The denominator is thesum of the population-periods at risk experienced by theindividuals (or an approximation to this, such as a mid-yearpopulation estimate). The numerator is usually a count ofevents that occurred in the population over the period. Weshall refer to such statistics as ‘rates’ and examples includemortality rates and cancer incidence rates.

Both proportions and rates are frequently multiplied by ascaling factor for presentation purposes, e.g. per 100,000.When this factor is 100 the statistic is usually described asa percentage. In most, but not all, cases percentages areproportions.

Means

Some public health indicators are reported as the mean ofthe individual values observed in a sample or population.For example, the mean number of decayed, missing orfilled teeth in children, or the mean length of stay inhospital.

Age-standardised ratios, rates andproportions

Disease and mortality rates may vary widely by age. Suchvariation complicates any comparisons made between twopopulations that have different age structures. Forexample, consider two areas A and B with equal-sizedpopulations and identical crude all-age death rates. At firstglance they appear to have a similar mortality experience.Suppose, however, that area A has a younger agestructure than area B. Given that mortality rates increasewith age, one would expect the older population in area Bto experience more deaths. The fact that the two haveidentical rates means that the younger population in areaA must have a relatively worse mortality experience.

The most comprehensive way of comparing the diseaseexperience of two populations is to present and comparetheir age-specific rates. However, when the number ofpopulations being compared increases, the volume ofdata that needs to be considered quickly becomesunmanageable. What is needed is a single, easilyinterpreted, summary figure for each population that isadjusted to take into account its age structure. Suchsummary figures are calculated using age standardisationmethods. It may also be desirable to standardise for othervariables, such as sex or level of deprivation, that may alsopotentially confound any comparisons.

The two most common methods of age standardisationare:

l Indirect: The age-specific rates of a chosen standardpopulation (usually the relevant national orregional population) are applied to the agestructure of the subject population to give anexpected number of events. The observednumber of events is then compared to thatexpected and is usually expressed as a ratio(observed/expected). A common example is thestandardised mortality ratio (SMR).

l Direct: The age-specific rates of the subject populationare applied to the age structure of the standardpopulation. This gives the overall rate that wouldhave occurred in the subject population if it hadthe standard age-profile.

Table 1 shows the advantages and disadvantages of eachmethod.

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Indirect

Advantages

l The indirect method requires only the total number of observed events in the subject population and may therefore be used in some situations where the direct method cannot.

l Indirect standardisation is more stable as it minimisesthe variance, giving a smaller standard error andnarrower confidence intervals than the direct method.It is therefore more appropriate when dealing with thestatistical significance of small populations.[1]

Disadvantages

l Indirect standardisation does not necessarily preserveconsistency between the populations beingcompared, and in extreme situations may givemisleading results. See Figure 1.

l Indirectly standardised ratios for areas A and B maybe compared to the standard but should only bedirectly compared to each other if the age structuresof areas A and B are similar to the standard, or if theratio of their age-specific mortality rates to that of thestandard is consistent across the age groups.[2]

l Indirect standardisation can be used for comparisonsover time, but only by freezing the standard rates at afixed point in time. This is often felt to be less thanintuitive. The validity of the comparisons is subject tothe same conditions as described above forcomparison between areas.

Direct

Disadvantages

l The direct method requires that the observed eventsin the subject population are available broken downby age. If this information is not available the directlystandardised rate cannot be calculated.

l For small subject populations the age-specific ratesof the subject population are based on smallnumbers and consequently are unstable. Smallchanges in the number of deaths in a particular ageband may result in large changes in the directlystandardised rate.

Advantages

l The direct method preserves consistency between thepopulations being compared, i.e. if each age-specificrate in area A is greater than each of thecorresponding age-specific rates in area B, then thedirectly standardised rate for area A will always behigher than that of area B irrespective of the standardpopulation used.[1;3] Consequently it is the preferredmethod for comparing a number of differentpopulations against each other.

l Directly standardised rates can readily be comparedover time provided the same standard population isused.

Table 1. Advantages and disadvantages of indirect and direct age standardisation.

Figure 1. Example of misleading indirectly age-standardised ratios.

Figure 1 illustrates the circumstances in which indirect standardisation can lead to loss of consistency and producemisleading results.

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In Figure 1, even though Population A has double the rateof Population B in every age-group, the SMR forPopulation A is lower than that for Population B. This arisesbecause the population distribution is very different, andthe ratios of the local rates to the reference rates are veryuneven. Although it is not unusual for different populationsto have very different age-distributions, it is unusual for theratios to be so different - in general rates will tend to varywith age in a consistent way across different populations.

In practice, the two methods generally give comparableresults. Moreover, it has been demonstrated by Breslowand Day[4] that when the two do differ it is not necessarilytrue that the direct method is the more ‘correct’. Where theratio between the age-specific rates of areas A and Bvaries by age group the choice of standard population/rates becomes crucially important for both indirect anddirect methods. This choice is often as, or more, importantthan the choice of method.[5;6]

The European Standard Population (ESP) is often used fordirect standardisation. This is a hypothetical populationstructure which does not change and is the same for bothgenders. The advantage of using such a hypotheticalpopulation is one of greater comparability, for example,between different countries, across time periods, andbetween genders. However, the ESP dates from the 1970s,and compared to current UK age structures it puts muchgreater weight on the younger age groups. This can havesome undesirable effects as ideally the standardpopulation should be similar to the populations beingstandardised. For the same reasons it is usuallyinappropriate to use the ESP for indicators where thedenominator is a particular subgroup of the population, forexample those with a certain condition or occupation,patients admitted to hospital, etc. The age structure ofthese subgroups may differ substantially from that of theESP. In summary, the ESP must be used whencomparisons are to be made with other rates which havebeen standardised to the ESP, but for self-containedcomparative analyses, a more suitable (e.g. average)population may be preferred.

Life expectancy

Life expectancy is another common public health statisticthat allows comparisons (of mortality experience) betweenareas with different population structures. It is based on alife table approach rather than direct or indirectstandardisation. The methodology is beyond the scope ofthis briefing and the reader is referred to other PHO andONS resources.[7-9]

Confidence intervalsA confidence interval is a range of values that is used toquantify the imprecision in the estimate of a particularvalue. Specifically it quantifies the imprecision that resultsfrom random variation in the estimation of the value; itdoes not include imprecision resulting from systematicerror (bias).

In many studies the source of this random variation issampling. Even in the best designed studies there will be

random differences between the particular sample groupselected and the overall target population of inference. Any measurement taken from the sample group thereforeprovides an imprecise estimate of the true populationvalue.

In public health many indicators are based on what can beconsidered to be complete data sets and not samples,e.g. mortality rates based on death registers. In theseinstances the imprecision arises not as a result ofsampling variation but of ‘natural’ variation. The indicator isconsidered to be the outcome of a stochastic process, i.e.one which can be influenced by the random occurrencesthat are inherent in the world around us. In such instancesthe value actually observed is only one of the set thatcould occur under the same circumstances. Generally inpublic health, it is the underlying circumstances or processthat is of interest and the actual value observed gives onlyan imprecise estimate of this ‘underlying risk’.

The width of the confidence interval depends on threethings:

1) The sample size from which the estimate is derived (orpopulation size if derived from a complete data set).Larger samples give more precise estimates withsmaller confidence intervals.

2) The degree of variability in the phenomenon beingmeasured. Fortunately, observed phenomena often areknown, or assumed, to follow certain probabilitydistributions, such as the Poisson or Binomial. Thisallows us to express the amount of variabilitymathematically, and build it into the confidence intervalformulae.

3) The required level of confidence - this is an arbitraryvalue set by the analyst giving the desired probabilitythat the interval includes the true value. In medicineand public health the conventional practice is to use95% confidence but it is not uncommon to seealternatives. Within the APHO community 99.8%confidence intervals are increasingly being usedalongside 95% intervals to reflect the control limitsused in Statistical Process Control approaches.[10]

Increasing the level of confidence results in widerlimits.

For a given level of confidence, the wider the confidenceinterval, the greater the uncertainty in the estimate.

Figure 2 demonstrates confidence intervals for smokingprevalence generated by 20 random samples of 100persons. For sample one the smoking prevalenceobserved is 16.9% and the associated 95% confidenceinterval is 10.8% to 25.4%. This means that we are 95%confident that the range 10.8% to 25.4% includes the truepopulation prevalence. On average, 95 out of every 100 ofthese intervals will include the population value. This canbe seen in the figure where 19 of the 20 intervals (ie. 95%)include the 25% population prevalence, the exceptionbeing sample 16. In many public health applications we donot have the luxury of repeated samples, but theconfidence intervals are calculated in such a way as toensure that if we did, then 95% of them would contain thetrue value.

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When should confidence intervals be used?

Confidence intervals should be presented alongside thepoint estimate wherever an inference is being made from asample to a population or from a set of observations to theunderlying process (or ‘risk’) that generated them.

Presentation of confidence intervals

When presenting the confidence interval the level ofconfidence should be clearly stated and the lower andupper limits of the interval clearly labelled or presented asa range. The use of the ± sign or a dash to indicate therange should be avoided.

For example:

Area Statistic 95% Confidence Interval

Lower Limit Upper Limit

Area 1 95.2 93.5 96.9

Area 1 95.2 93.5 to 96.9

Area 1 95.2 (93.5, 96.9)

For Area 1 the statistic was 95.2 (95% confidence interval:93.5 to 96.9).

In graphs or charts, the confidence limits are usuallyrepresented as bars or whiskers extending above andbelow the value of the estimate. Figure 2 is a typicalexample.

Using confidence intervals for makingcomparisons

The confidence interval may be used to compare anestimate against a target or benchmark value in a similarway to performing an exact statistical test. In such a testwe proceed on the assumption that the true value is equalto the target or benchmark value. The appropriateprobability distribution is then used to determine theprobability (P-value) of observing a value as extreme, ineither direction (a two-sided test), as the actual observedestimate. If the P-value is below a given significancethreshold, α, then the original assumption is rejected andthe difference is said to be statistically significant.

If the target or benchmark value is outside the confidenceinterval it can be inferred that the difference between theestimate and the target is statistically significant at thecorresponding significance level. For instance, if the targetvalue is outside the 95% confidence interval then thedifference is significant at the α = 5% level (P < 0.05).

Figure 2. Smoking prevalence: sample estimates and 95% confidence intervals for 20 random samples of 100persons from a population with a smoking prevalence of 25%.

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Comparisons are often made between two or moredifferent estimates, for example between differentorganisations or time periods (see Figure 3). Sometimes insuch cases statistical testing is undertaken by comparingthe confidence intervals of the estimates to see if theyoverlap - with non-overlapping confidence intervals beingconsidered as statistically significantly different. This isquick and easy to do, but not necessarily conclusive.Whilst it is safe to assume that non-overlappingconfidence intervals indicate a statistically significantdifference, it is not always the case that overlappingconfidence intervals do not. A more exact approach is tocalculate the ratio of the two estimates, or the differencebetween them, and construct a test or confidence intervalbased on that statistic. Such methods are not covered inthis briefing, but can be found in a standard textbook.

Calculating the statistics and theirconfidence intervalsThe following section presents the calculation of thestatistics and their confidence intervals in detail. Theformulae become rapidly more complicated, but please donot let that deter you, as APHO will be turning them intouser-friendly spreadsheet tools available fromhttp://www.apho.org.uk

Proportions

The proportion p is given by: p =_ Formula 1

where:O is the observed number of individuals in thesample/population having the specified characteristic;n is the total number of individuals in thesample/population.

Figure 3. Using confidence intervals for making comparisons.

Target Line

3Correct

3Correct

?Not necessarily

On

Significantly

differentNot significantly

differentSignificantly

different

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Confidence intervals for proportions are determined usingthe Binomial distribution. A Normal approximation methodis often presented in statistical textbooks but does notperform well when the numerator and/or denominator issmall. The preferred APHO method is the Wilson Scoremethod [11] which has been evaluated and recommendedby Newcombe and Altman.[12;13] It can be used with anydata values and, unlike some methods, it does not fail togive an interval when the numerator count, and thereforethe proportion, is zero.[14]

The 100(1_α)% confidence interval limits for the proportionp are given by:

Formula 2a

Formula 2b

where:q = (1_p) is the proportion without the specifiedcharacteristic;z is the 100(1_ α/2)th percentile value from the StandardNormal distribution. For example for a 95% confidenceinterval, α = 0.05, and z = 1.96 (i.e. the 97.5th percentilevalue from the Standard Normal distribution).

Rates

The rate of events r is given by: r =_ Formula 3

where:O is number of observed events;n is the population-years at risk.

Provided the rate is low and the denominator at risk islarge, the variability in the observed count O is describedby the Poisson distribution. This can be used to give aconfidence interval for O and hence r. The preferredmethod is Byar’s approximation as it is computationallysimple and gives very accurate approximations to theexact Poisson probabilities even for small counts.[15]

However, tables of the exact probabilities are widelyavailable and should be used for the lower limits ofobserved counts under 5.

Using Byar’s method, the 100(1_ α)% confidence intervallimits for the rate r are given by:

Formula 4a

Formula 4b

where:O is the observed count;z is the 100(1_ α/2)th percentile value from the StandardNormal distribution.

Where the rate r is not low, the appropriateness of thePoisson distribution is more debatable but it is stillgenerally used.[16] In extreme circumstances, for examplelooking at high risk groups in small populations, the use ofthe Poisson distribution may lead to limits beyond sensiblebounds.

Counts

The limits of the observed count should be determinedusing Formulae 2a and 2b, or 4a and 4b, depending onwhether the count is based on a Binomially distributedproportion or a Poisson rate respectively. For example, fora smoking prevalence indicator the observed count isbased on a proportion and its limits should be calculatedfrom the Binomial distribution; for a mortality or cancerincidence count Poisson limits should be used. In bothcases the limits for the proportion or rate should bemultiplied by n to convert them into limits for the observedcount. (In the case of Formula 4a and Formula 4b, thissimply means using the elements referred to as ‘Olower’ and‘Oupper’.)

Means

The sample mean x is given by: x =

Formula 5

Where:xi is the observed value for the ith individual;n is the total number of individuals in thesample/population.

On

_ _

When the size of the sample is large (say >50), the samplemean is Normally distributed and its standard error can beestimated by s/√n, where s is the observed standarddeviation of the individual values in the sample:

The traditional Normal approximation method can be used togive the 100(1_ α)% confidence interval limits:

Formula 6a

Formula 6b

where z is the 100(1_ α/2)th percentile value from theStandard Normal Distribution.

When samples are small, s/√n will vary substantiallybetween samples and cannot be considered a reliableapproximation to the standard error of the mean. However, ifthe population distribution of the individual values itself isNormal, with mean μ, then the distribution of (x-μ)/(s/√n)follows the Student’s t-distribution for (n_1) degrees offreedom. The confidence intervals are then found bysubstituting the appropriate percentile point values of theStandard Normal distribution with those from the t-distribution with the appropriate degrees of freedom. Thesevalues are found from tables or the built-in functions found inmost spreadsheet and statistical software.

When using such tools the reader should ensure that theyare using the correct set of probabilities (two-tailed) and thatthey are evaluating the distribution against the appropriatevalue, i.e. the confidence level or probability level orpercentile point, as this requirement may vary between tools.

Formula 7a

Formula 7b

where t is the 100(1_ α/2)th percentile value from the t-distribution with (n_1) degrees of freedom. For example fora 95% confidence interval for a mean of 10 observations, α= 0.05 and t = 2.26 (i.e. the 97.5th percentile value fromthe t-distribution with 9 degrees of freedom).

For small samples where the individual values are notNormally distributed the data can be adjusted so that theybecome so by applying a suitable transformation.

Indirectly age-standardised ratios and rates

The statistic most commonly presented for the indirectmethod of age standardisation is the standardised ratio -this is the ratio of the observed number of events relativeto the number of events that would be expected ifstandard age-specific rates were applied to the particularobserved population’s age structure. A common exampleis the standardised mortality ratio (SMR).

The indirectly standardised ratio (ISR) is given by:

Formula 8

where:Oi is the observed number of events in the subjectpopulation in age group i;Ei is the expected number of events in the subjectpopulation in age group i given the standard rates;ni is the number of individuals in the subject population inage group i;λi is the crude age-specific rate in the standard populationin age group i.

For presentation purposes, the ratio is usually multiplied by100. By definition, the standard population will have a ratioof 100. Ratios above 100 indicate that the number ofevents observed was greater than that expected from thestandard rates, and ratios below 100 that it was lower. Insome instances the ratio is multiplied by the overall cruderate of the standard population and presented as anindirectly age-standardised rate.

For the purposes of calculating the confidence interval ofthe ratio, the expected count is considered to be precise.The imprecision in the ratio is therefore dependent only onthe imprecision of the observed count.

The 100(1_α)% confidence interval limits of the ratio aregiven by:

Formula 9a

Formula 9b

where:Olower and Oupper are as defined in Formula 4a and Formula4b respectively;E is the expected count.The confidence limits as calculated above should then bemultiplied by any scaling factor that has been used inpresenting the ratio itself.

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Directly age-standardised rates

Directly age-standardised rates express an indicator interms of the overall rate that would occur in a standardpopulation age-structure if it experienced the age-specificrates of the observed population.

The direct rate (DSR) is given by:

Formula 10

where:wi is the number, or proportion, of individuals in thestandard population in age group i;Oi is the observed count in the population in age group i;ni is the observed denominator population-period-at-risk inage group i.

The directly standardised rate is a weighted sum of theindependent age-specific rates. Therefore, its variance is aweighted sum of the variances of each of those age-specific rates. The preferred method for calculating theconfidence interval is one described by Dobson.[17] In thismethod the exact interval is found for the crude count andthen weighted and scaled to give the interval for thedirectly standardised rate. The weight used is the ratio ofthe standard error of the DSR to the standard error of thecrude count.For rates that assume the Poisson distribution, theconfidence limits for the DSR are given by:

Formula 11a

Formula 11b

where O is the total number of observed events in thepopulation;and the limits Olower and Oupper of the observed count aredetermined using Byar’s approximation (Formulae 4a and4b) or exact look-up tables.

The variance of the DSR and the variance of the observedcount are estimated by:

Formula 11c

where:wi is the number, or proportion, of individuals in thestandard population in age group i;Oi is the observed count in the subject population in agegroup i;ni is the denominator population-period-at-risk in thesubject population in age group i.

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References

1. Breslow NE, Day NE. Statistical methods in cancerresearch, volume II: The design and analysis of cohortstudies. Lyon: International Agency for Research onCancer, World Health Organization; 1987: 72-3.

2. Tsai SP, Wen CP. A review of methodological issues ofthe standardised mortality ratio (SMR) in occupationalcohort studies. Int J Epidemiol 1986;15(1):8-21.

3. Inskip H et al. Methods for age-adjustment of rates.Stat Med 1983;2:455-66.

4. Breslow NE, Day NE. Statistical methods in cancerresearch, volume II: The design and analysis of cohortstudies. Lyon: International Agency for Research onCancer, World Health Organization; 1987: 67.

5. Goldman DA, Brender JD. Are standardised mortalityratios valid for public health data analysis? Stat Med2000;19:1081-8.

6. Higham JH et al. Direct standardisation has noadvantage over indirect standardisation. Letter. BMJ2002; 7 Aug. Available at

7.

8.

9.

10.

11.

12. Newcombe RG. Two-sided confidence intervals for thesingle proportion: comparison of seven methods. StatMed 1998;17:857-72.

13. Newcombe RG, Altman DG. Proportions and theirdifferences. In Altman DG et al. (eds). Statistics withconfidence (2nd edn). London: BMJ Books; 2000: 46-8.

14. Agresti A, Coull BA. Approximate is better than ‘exact’for interval estimation of binomial proportions. Am Stat1998;52:119-26.

15. Breslow NE, Day NE. Statistical methods in cancerresearch, volume II: The design and analysis of cohortstudies. Lyon: International Agency for Research onCancer, World Health Organization; 1987: 69.

16. Brillinger DR. The natural variability of vital rates andassociated statistics. Biometrics 1986;42:693-734.

17. Dobson A et al. Confidence intervals for weightedsums of Poisson parameters. Stat Med 1991;10:457-62.

Other contributory material

http://bmj.com/cgi/eletters/325/7357/177#24481(accessed 18 February 2008).

Flowers J. Life expectancy. INphoRM (3). Cambridge:ERPHO; 2003. Available athttp://www.erpho.org.uk/Download/Public/9479/1/3rd%20proof%20INPHORM%20Issue%203.pdf (accessed18 February 2008).

Williams E et al. Technical report: calculating lifeexpectancy in small areas. Oxford: SEPHO; 2005.Available athttp://www.sepho.org.uk/Download/Public/9847/1/Life%20Expectancy%20Nov%2005.pdf (accessed 18February 2008).

Toson B, Baker A. Life expectancy at birth:methodological options for small populations(National Statistics Methodological Series No 33).London: ONS; 2003. Available athttp://www.statistics.gov.uk/downloads/theme_other/GSSMethodology_No_33.pdf (accessed 18 February2008).

Flowers J. Technical Briefing 2: Statistical processcontrol methods in public health intelligence. York:APHO; 2007. Available athttp://www.apho.org.uk/resource/item.aspx?RID=39445 (accessed 18 February 2008).

Wilson EB. Probable inference, the law of succession,and statistical inference. J Am Stat Assoc1927;22:209-12.

18. Bland M. An introduction to medical statistics (2ndedn). Oxford: Oxford University Press; 2000.

19. Esteve J et al. Statistical methods in cancer research,volume IV: Descriptive epidemiology. Lyon:International Agency for Research on Cancer, WorldHealth Organization; 1994.

20. Merrick D. Confidence intervals for standardised ratesand ratios. Durham: NHS Executive (Northern &Yorkshire Region); 1997.

21. Washington State Department of Health. Guidelinesfor using confidence intervals for public healthassessment. Washington: Washington StateDepartment of Health; 2002.

22. Eayres D. Annex 3 - Explanations of statisticalmethods used in the Compendium. In Lakhani A,Olearnik H, Eayres D (eds). Compendium of clinicaland health indicators. London: The Information Centrefor Health and Social Care/National Centre for HealthOutcomes Development; 2007.

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Glossary

Binomial distribution: the probability distribution thatapplies to discrete data with only two possibilities - e.g.success or failure, alive or dead. It is used to generateconfidence intervals for proportions. Its shape isdependent on two parameters: the number of trials, orobservations; and the proportion of successes.

Normal distribution: familiar bell-shaped curve, which isa good representation of the probability distribution ofmany naturally-occurring variables. Its shape is dependenton two parameters: the mean and the standard deviation.

Parameter: a quantity that defines certain characteristicsof a system, function or distribution.

Poisson distribution: probability distribution representingthe number of events (e.g. accidents) in a fixed space oftime. Its mean and variance are equal, so its shape isdetermined by a single parameter.

Probability distribution: a function that gives theprobability of a specific value, or value within a giveninterval, being observed at random given the underlyingparameters.

Standard deviation/Standard error: statistical measureof the amount of variation around the mean of adistribution. It is defined as the square root of the variance.By convention, the term ‘standard deviation’ is used onlywhen the distribution is describing variation in individualvalues within a population or sample. The standarddeviation of a summary measure, such as the samplemean, is known as the ‘standard error’.

Standard Normal distribution: a Normal distribution withmean μ = 0 and standard deviation σ = 1. Any Normallydistributed variable x, with mean μ and standard deviationσ, can be transformed into a ‘Z-score’ which follows theStandard Normal distribution, by using the relationship z = (x- μ)/σ. This relationship means that there is no needto publish tables for any Normal distribution other than theStandard Normal distribution.

Standard population: a reference population age-structure used for direct age standardisation. It may be areal population, e.g. England mid-year estimates 2006, ora hypothetical one, e.g. European Standard Population.

Standard rates: a set of reference age-specific ratesused for indirect age standardisation. In nearly allinstances these are real rates rather than hypotheticalones and are taken from the reference or benchmarkpopulation against which the populations beingstandardised are to be compared.

Statistical significance/Statistical test: to test a givenhypothesis, we initially proceed on the assumption that it istrue. If, on this basis, an observation as extreme as thatobtained would have been very unlikely to arise by chance,we then ‘reject’ the hypothesis. The cut-off point or‘significance level’ below which we consider the probabilityto be too small is arbitrary, but is traditionally set at 0.05(5%). Results which lead to the rejection of the hypothesisare said to be ‘statistically significant’.

Transformation: the application of a mathematicalfunction to a set of data values, e.g. the logarithm, squareroot or reciprocal. The purpose is to ‘transform’ the dataso that the distribution of values more closely resembles aNormal distribution.

Variance: statistical measure of the amount of variationaround the mean of a distribution. The greater the variancethe more widely distributed are the observed values.

Z-score: a value that follows a Standard Normaldistribution.

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