Statistics.pdf.pdf for Research Physiotherapy and Occupational Therapy

Statistics
GR Miyambu
School of Science & Technology
Department Statistical Sciences

Introduction
• Statistical analysis features in the majority of papers
published in health care journals
• Most health care practitioners will need a basic
understanding of statistical principles, but not necessarily
full details of statistical techniques
• Medical statistics contribute to good research by improving
the design of studies as well as suggesting the optimum
analysis of the results
• Medical statisticians (Biostatisticians) should be consulted
early in the planning of a study
• They can contribute in a variety of ways at all stages and not
just at the final analysis of the data once the data have been
collected

Statistics is about common sense and good
design
• A well-designed study, poorly analysed, can be rescued by a
reanalysis but a poorly designed study is beyond the
redemption of even sophisticated statistical manipulation.
• Many experimenters consult the medical statistician only at
the end of the study when the data have been collected.
• They believe that the job of the statistician is simply to
analyse the data, and with powerful computers available,
even complex studies with many variables can be easily
processed.
• However, analysis is only part of a statistician’s job, and
calculation of the ﬁnal ‘p-value’ a minor one at that!
• A far more important task for the medical statistician is to
ensure that results are comparable and generalisable.

Categorical or Qualitative data
• Nominal categorical data - Nominal or categorical data are data that one can name and put
into categories.
• They are not measured but simply counted.
• They often consist of unordered ‘either–or’ type observations which have two categories and
are often know as binary.
• For example: Dead or Alive; Male or Female; Cured or Not Cured; Pregnant or Not Pregnant.
• However, categorical data often can have more that two categories, for example: blood
group O, A, B, AB, country of origin, ethnic group or eye colour.
• Ordinal data - If there are more than two categories of classiﬁcation it may be possible to
order them in some way.
• For example, after treatment a patient may be either improved, the same or worse; a woman
may never have conceived, conceived but spontaneously aborted, or given birth to a live
infant.
• Ranks - In some studies it may be appropriate to assign ranks.
• For example, patients with rheumatoid arthritis may be asked to order their preference
for four dressing aids.
• Here, although numerical values from 1 to 4 may be assigned to each aid, one cannot treat
them as numerical values.
• They are in fact only codes for best, second best, third choice and worst

Numerical or quantitative data
• Count data – The table gives details of the number of pregnancies each woman had had, and
this is termed count data.
• Other examples are often counts per unit of time such as the number of deaths in a hospital
per year, or the number of attacks of asthma a person has per month. In dentistry, a common
measure is the number of decayed, ﬁlled or missing teeth (DFM).
• Measured or numerical continuous - Such data are measurements that can, in theory at least,
take any value within a given range. These data contain the most information, and are the
ones most commonly used in statistics.
• Examples of continuous data in the table are: age, years of menstruation and body mass
index.
• For simplicity, it is often the case in medicine that continuous data are dichotomised to make
nominal data.
• Thus diastolic blood pressure, which is continuous, is converted into hypertension (>90
mmHg) and normotension (≤90 mmHg).
• This clearly leads to a loss of information.
• There are two main reasons for doing this.
• It is easier to describe a population by the proportion of people affected (for example, the
proportion of people in the population with hypertension is 10%).
• Further, one often has to make a decision: if a person has hypertension, then they will get
treatment, and this too is easier if the population is grouped.
• One can also divide a continuous variable into more than two groups.

Interval and ratio scales
• One can distinguish between interval and ratio scales. In an interval scale,
such as body temperature or calendar dates, a difference between two
measurements has meaning, but their ratio does not.
• Consider measuring temperature (in degrees centigrade) then we cannot say
that a temperature of 20°C is twice as hot as a temperature of 10°C.
• In a ratio scale, however, such as body weight, a 10% increase implies the
same weight increase whether expressed in kilograms or pounds.
• The crucial difference is that in a ratio scale, the value of zero has real
meaning, whereas in an interval scale, the position of zero is arbitrary.
• One difﬁculty with giving ranks to ordered categorical data is that one cannot
assume that the scale is interval.
• Thus, as we have indicated when discussing ordinal data, one cannot assume
that risk of cancer for an individual educated to middle school level, relative
to one educated only to primary school level is the same as the risk for
someone educated to college level, relative to someone educated to middle
school level. Were Xu et al (2004) simply to score the three levels of
education as 1, 2 and 3 in their subsequent analysis, then this would imply in
some way the intervals have equal weight.

How a statistician can help?
• Sample size and power consideration
• Questionnaires
• Choice of sample and of control subjects
• Design of study
• Laboratory experiments
• Displaying data
• Choice of summary statistics and statistical analysis

Summarizing Categorical Data
• Binary data are the simplest type of data
• Each individual has a label which takes one of two values
• A simple summary will be to count the different types of
label
• However, a raw count is rarely useful
• Hence the first step to analyzing categorical data is to count
the number of observations in each category and express
them in proportions of the total sample size
• Proportions are a special example of ratio
• When time is involved it is known as a rate

Ratio, Proportions, Percentages, Risk and Rates
• A ratio is simply one number divided by another.
• Proportions are ratios of counts where the numerator (the
top number) is a subset of the denominator (the bottom
number).
• A proportion is known as a risk if the numerator counts
events which happen prospectively.

Ratio, Proportions, Percentages, Risk and Rates
• Rates always have a time period attached.
• If 600 000 people in the South Africa die in one year, out of
a population of 60 000 000, the death rate is 600 000/60
000 000 or 0.01 deaths per person per year.
• This is known as the crude death rate (crude because it
makes no allowance for important factors such as age).
• Crude death rates are often expressed as deaths per
thousand per year, so the crude death rate is 10 deaths per
thousand per year, since it is much easier to imagine 1000
people, of whom 10 die, than it is 0.01 deaths per person!

Example: Special Baby Care Unit
Types of delivery for 98 babies admitted to a special care
baby unit (Simpson, 2004)
Type of delivery Frequency Percentage
Standard vaginal delivery 38 39
Assisted vaginal delivery 10 10
Elective caesarean section 8 8
Emergency caesarean section 13 13
Emergency caesarean section/
not in labour
29 30
Total 98 100

Example: Special Baby Care Unit
Types of delivery and gender for 98 babies admitted to a special care baby unit
(Simpson, 2004)
Type of delivery Gender
Male Female
n (%) n (%)
Standard vaginal delivery 15 (33) 23 (43)
Assisted vaginal delivery 4 (9) 6 (11)
Elective caesarean section 4 (9) 4 (8)
Emergency caesarean
section
6 (13) 7 (13)
Emergency caesarean
section/not in labour
16 (36) 13 (25)
Total 45 (100) 53 (100)

Labelling Binary Outcomes
• It is common to call the outcome ‘an event’ and ‘a non-
event’ in binary data
• So having a car accident in Pretoria may be an event
• We often score an ‘event’ as 1 and a ‘non-event’ as 0
• These may also refer to a ‘positive’ or ‘negative’ outcome or
‘success’ and ‘failure’
• It is important to realize that these terms are merely labels
and the main outcome of interest might be a success in one
context and a failure in another
• Thus in a study of a potentially lethal disease the outcome
might be death, whereas in a study about a disease that can
be cured it might be being alive

Comparing Outcomes for Binary Data
• Many studies involve a comparison
of two groups.
• We may wish to combine simple
summary measures to give a
summary measure which in some
way shows how the groups differ.
• Given two proportions one can
either subtract one from the other,
or divide one by the other.
• Suppose the results of a clinical trial,
with a binary categorical outcome
(positive or negative), to compare
two treatments (a new test
treatment versus a control) are
summarised in a 2 x 2 contingency
table as in the table.
• Then the results of this trial can be
summarised in a number of ways.
Example of 2 x 2 contingency
table with a binary outcome and
two groups of subjects
Outcome Treatment group
Test Control
Positive a b
Negative c d
Total a + c b + d

Example of 2 x 2 contingency table with a binary
outcome and two groups of subjects
Outcome Treatment group
Test Control
Positive a b
Negative c d
Total a + c b + d

Ways of Summarising the Data
• From the table, the proportion of subjects with positive outcome
under the active or test treatment is 𝑝𝑝𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 =
𝑎𝑎
𝑎𝑎+𝑐𝑐
and under the
control treatment is 𝑝𝑝𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 =
𝑏𝑏
𝑏𝑏+𝑑𝑑
• The difference in proportion is given by 𝑑𝑑𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑝𝑝𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑝𝑝𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
• In prospective studies the proportion is also known as a risk.
• When one ignores the sign, the above quantity is known as the
absolute risk different (ARD), that is 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑝𝑝𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑝𝑝𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
• Where the symbol . mean to take the absolute value.
• If we anticipate that the treatment to reduce some bad outcome
(such as deaths) then it may be known as the absolute risk
reduction (ARR).
• The risk ratio, or relative risk (RR), is given by 𝑅𝑅𝑅𝑅 =
𝑝𝑝𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
𝑝𝑝𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶
• A further summary measure, used only in clinical trials is the
number needed to treat/harm.
• This is defined as the inverse of ARD.

Importance of considering both absolute risk
reduction and relative risk
• Women aged 15–45 not on the contraceptive pill have a risk of deep vein
thrombosis (DVT) of about 20 per 100 000 women per year.
• Consider a contraceptive pill which increases the risk to 40 per 100 000 per year.
• The relative risk of DVT for the new pill is 2, which would seem a large risk.
• However, the increase in risk is 20/100 000 = 0.00002, or an additional 2 women
with DVTs in 10 000 years of exposure.
• This risk is very small and hence may be considered worthwhile, when balanced
against other factors such as cost or convenience.
• Also, it is worth mentioning that a pregnant woman has a risk of a DVT of about 80
per 100 000 per year.
• When one reads in the papers about a new risk to health that has been discovered,
often only the relative risk is quoted, but one should ask about the absolute risk
difference, which is often negligible.
• If you are at very low risk, then you will remain at very low risk even when exposed
to a hazard, unless the relative risk for the hazard is enormous!

Smoking cessation rates at one year in patients
with coronary heart disease
Stopped Smoking
Intervention Control
n (%) n (%)
Yes 57 (57) 44 (37)
No 43 (43) 74 (63)
Total 100 (100) 118 (100)

Summarising results from a clinical trial – smoking
cessation
• The table shows the results of a randomised controlled trial conducted by Quist-
Pauslen and Gallefoss (2003) to determine whether a nurse-led smoking cessation
intervention can improve smoking cessation rates in patients admitted for coronary
heart disease.
• There are two study groups: the control group (
randomised to receive usual care)
and the experimental or intervention group (
randomised to receive a booklet and
which emphasised the health beneﬁts of quitting smoking after a coronary event).
• The main outcome measure was smoking cessation rates at 1year determined by
self-report and biomedical veriﬁcation.
• The proportion of patients who stopped smoking in the Intervention group is 57
/ 100
= 0
. 57and in the Control group 44
/ 118= 0
. 37or a difference of 0
. 20or 20
%. If we
started with 100women in each arm we would expect 20fewer patients smoking in
the intervention arm compared to the control at the end of the study.
• The Relative Risk is 0.57/0.37 = 1.5.
• This is the risk of stopping smoking (
a good thing )with the intervention compared to
the control group.
• Thus patients with coronary heart disease are 1
. 5times more likely to stop smoking
in the intervention group than the control group.

Summarising binary data – odds and odds ratios
• A further method of summarizing the results is to use the odds
of an event rather than the probability.
• The odds of an event are defined as the ratio of the probability
of occurrence of the event to the probability of non-occurrence,
that is,
𝒑𝒑
𝟏𝟏−𝒑𝒑
.
• Using the notations of the 2x2 contingency table before, we can
see that the odds of an outcome for the test group to the odds of
an outcome of control group is the ratio for the test group to the
odds for control group:
• The odds ratio (OR) is
𝒑𝒑𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻
𝟏𝟏−𝒑𝒑𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻
𝒑𝒑𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
𝟏𝟏−𝒑𝒑𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪
• The odds ratio from the example of contingency table is
𝑶𝑶𝑶𝑶𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻/𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪𝑪 =
𝒂𝒂
𝒄𝒄
𝒃𝒃
𝒅𝒅
=
𝒂𝒂𝒂𝒂
𝒃𝒃𝒃𝒃

Summarising binary data –
odds and odds ratios
• When the probability of an
event happening is rare, the
odds and probabilities are
close, because then a is much
smaller than c and so a/(a + c)
is approximately a/c and b is
much smaller than d and so
b/(b + d) is approximately b/d.
• Thus the OR approximates the
RR when the successes are rare
(say with a maximum incidence
less than 10% of either pTest or
pControl)
• Sometime the odds ratio is
referred to as ‘the approximate
relative risk’.
• The approximation is
demonstrated in the table on
the right.
Comparison of RR and OR for
differet baseline
RR OR
RR
and
OR
0.05 0.1 0.5 0.47 Close
0.1 0.2 0.5 0.44 Close
0.2 0.4 0.5 0.38
Not
close
0.4 0.2 2 2.66
Not
close
0.2 0.1 2 2.25 Close
0.1 0.05 2 2.11 Close

Why use odds ratio?
• The calculation for an odds ratio (OR) may seem rather
perverse, given that we can calculate the relative risk
directly from the 2 x 2 table and the odds ratio is only an
approximation of this.
• However, the OR appears quite often in the literature, so it
is important to be aware of it.
• It has certain mathematical properties that render it
attractive as an alternative to the RR as a summary
measure.
• Some statisticians argue that the odds ratio is the natural
parameter and the relative risk merely an approximation.
• The OR features in logistic regression and as a natural
summary measure for case–control studies.

Example
Cohort study – psychosis and
cannabis
• Henquet et al (2005) took a
random sample of 2437 normal
adolescents and questioned
them about their use of
cannabis.
• They followed them up 4 years
later and asked about psychotic
symptoms.
• The results are summarised in
the table on the right.
• Calculate:
• The risk of psychosis for non-
cannabis
• The relative risk of psychosis for
cannabis smokers
• The odds ratio of psychosis for
cannabis smokers
Psychosis
Cannabis Use Total
Yes No
Yes 82 342 424
No 238 1775 2013
Total 320 2117 2437

Why use odds ratio?
• One point about the OR that can be seen immediately from
the formula is that the OR for Failure as opposed to the OR
for Success in contingency table is given by OR = bc/ad.
• Thus the OR for Failure is just the inverse of the OR for
Success.
• Thus in the cannabis and psychosis study, the odds ratio of
not developing psychosis for the cannabis group is 1/1.79 =
0.56.
• In contrast the relative risk of not developing psychosis is (1
- 0.26)/(1 - 0.16) = 0.88,
• Which is not the same as the inverse of the relative risk of
developing psychosis for the cannabis group which is
1/1.625 = 0.62.
• This symmetry of interpretation of the OR is one of the
reasons for its continued use.

Displaying Categorical Data
0
5
10
15
20
25
30
35
40
45
Standard vaginal
delivery
Assisted vaginal
delivery
Elective
caesarean section
Emergency
caesarean section
Emergency
caesarean
section/ not in
labour
Percent
Type of Delivery

Standard vaginal delivery
39%
Assisted vaginal
delivery
10%
Elective
caesarean
section
8%
Emergency caesarean
section
13%
Emergency caesarean
section/ not in labour
30%
Type of delivery

Standard
vaginal delivery
Assisted vaginal
delivery
Elective
caesarean
section
Emergency
caesarean
section
Emergency
caesarean
section/ not in
labour

0
10
20
30
40
50
Standard
vaginal
delivery
Assisted
vaginal
delivery
Elective
caesarean
section
Emergency
caesarean
section
Emergency
caesarean
section/not
in labour
Percent
Type of delivery
Male (n=45)
Female (n=53)

Summarizing Continuous Data
• A quantitative measurement contains more information than
a categorical one, and so summarizing these data is more
complex.
• One chooses summary statistics to condense a large amount
of information into a few intelligible numbers, the sort that
could be communicated verbally.
• The two most important pieces of information about a
quantitative measurement are ‘where is it?’ and ‘how
variable is it?’
• These are categorized as measures of location (or sometimes
‘central tendency’) and measures of spread or variability.

Measures of Central Tendency
• Mean or average: The arithmetic mean or average of n
observations ̅
𝑥𝑥 (pronounced x bar) is simply the sum of the
observations divided by their number; thus
�
𝒙𝒙 =
𝒔𝒔𝒔𝒔𝒔𝒔 𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗
𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔 𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
=
∑𝒊𝒊=𝟏𝟏
𝒏𝒏
𝒙𝒙𝒊𝒊
𝒏𝒏
• In case of the above equation, 𝑥𝑥𝑖𝑖 represents the individual
sample value and ∑𝑖𝑖=1
𝑛𝑛
𝑥𝑥𝑖𝑖 their sum.
• The Greek letter ‘Σ’ (sigma) is the Greek capital ‘S’ and
stands for ‘sum’ and simply means ‘add up the n
observations 𝑥𝑥𝑖𝑖 from the first to the last (nth)’.

Example: Calculation of the mean – birthweights
• Consider the following ﬁve birthweights in kilograms
recorded to 1 decimal place selected randomly from the
study of low birthweight babies: 1.2, 1.3, 1.4, 1.5, 2.1
• The sum of these observations is (1.2 + 1.3 + 1.4 + 1.5 + 2.1)
= 7.5.
• Thus the mean ̅
𝑥𝑥 =
7,5
5
= 1,50 kg.
• It is usual to quote 1 more decimal place for the mean than
the data recorded.

• The major advantage of the mean is that it uses all the data
values, and is, in a statistical sense, efficient.
• The mean also characterizes some important statistical
distributions.
• The main disadvantage of the mean is that it is vulnerable to
what are known as outliers.
• Outliers are single observations which, if excluded from the
calculations, have noticeable influence on the results.
• For example if we had entered ‘21’ instead of ‘2.1’ in the
calculation of the mean, we would find the mean changed
from 1.50 kg to 7.98 kg.
• It does not necessarily follow that outliers should be
excluded from the final data summary, or that they result
from an erroneous measurement

• If the data are binary, that is nominal data that can only have
two values which are coded 0 or 1, then �
𝒙𝒙 is the proportion
of individuals with value 1, and this can also be expressed as
a percentage.
• Thus, in Simpson’s data, if 15 out of 98 babies were very low
birthweight (<1 kg) and if this is coded as a ‘1’ in the data
set, and a ‘0’ coded for those above 1 kg, then the mean of
this variable is 0.15 or 15%.

• Median: The median is estimated by ﬁrst ordering the data
from smallest to largest, and then counting upwards for half
the observations.
• The estimate of the median is either the observation at the
centre of the ordering in the case of an odd number of
observations, or the simple average of the middle two
observations if the total number of observations is even
• The median has the advantage that it is not affected by
outliers, so for example the median in the data would be
unaffected by replacing ‘2.1’ with ‘21’.
• However, it is not statistically efﬁcient, as it does not make
use of all the individual data values

• Consider the following ﬁve birthweights in kilograms
selected randomly from the Simpson (2004) study.
• If we had observed an additional value of 3.5 kg in the
birthweight the median would be the average of the 3rd and
the 4th observation in the ranking, namely the average of
1.4 and 1.5, which is 1.45 kg.
Rank Order Weight (kg)
1 1.2
2 1.3
3 1.4
4 1.5
5 2.1
Median

• Mode: A third measure of location is termed the mode.
• This is the value that occurs most frequently, or, if the data are grouped, the grouping
with the highest frequency.
• It is not used much in statistical analysis, since its value depends on the accuracy
with which the data are measured;
• It is useful for categorical data to describe the most frequent category.
• However, the expression ‘bimodal’ distribution is used to describe a distribution with
two peaks in it.
• This can be caused by mixing two or more populations together.
• For example, height might appear to have a bimodal distribution if one had men and
women in the population.
• Some illnesses may raise a biochemical measure, so in a population containing
healthy individuals and those who are ill one might expect a bimodal distribution.
• However, some illnesses are deﬁned by the measure of, say obesity or high blood
pressure, and in these cases the distributions are usually unimodal with those above
a given value regarded as ill.

Measures of dispersion or variability
• Range and interquartile range: The range is given as
the smallest and largest observations.
• This is the simplest measure of variability.
• For some data it is very useful, because one would
want to know these numbers,
• For example in a sample the age of the youngest and
oldest participant.
• However, if outliers are present it may give a
distorted impression of the variability of the data,
since only two of the data points are included in
making the estimate.

• Quartiles: The qu a
rtiles, n a
mely the lower qu a
rtile, the me d
i a
n a
n d
the upper quartile, divide the data into four equal parts;
• Th a
t is there will b
e a
p proxim a
tely equ a
l numb
ers of o b
serv a
tions in
the four se ctions ( a
n dex ac
tly equ a
l if the s a
mple size is divisi b
le b
y
four and the measures are all distinct).
• The qu a
rtiles a
re ca
l c
ul a
te din asimil a
r w a
y to the me d
i a
n; ﬁrst or d
er
the data and then count the appropriate number from the bottom.
• The interqua
rtile r a
nge is auseful me a
sure of v a
ri ab
ility a
n dis given b
y
the difference of the lower and upper quartiles.
• The interqua
rtile r a
nge is not vulner ab
le to outliers, a
n dwha
tever the
d
istri b
ution of the da
t a
, we know th a
t 50% of them lie within the
interquartile range.

Calculation of the range, quartiles and interquartile
range
• Suppose we had 10 birthweights arranged in increasing order from the Simpson
(2004) study.
• The range of birthweights in these data is from 1.51 kg to 2.85 kg (simply the
smallest and largest birthweights).
• The median is the average of the 5th and 6th observations (2.18 + 2.22)/2 = 2.20 kg.
• The first half of the data has five observations so the first quartile is the 3rd ranked
observation, namely 1.79 kg, and similarly the third quartile would be the 8th ranked
observation, namely 2.40 kg.
• So the interquartile range is from 1.79 to 2.40 kg.
Order
Birthweight
(kg)
1 1.51
2 1.55
3 1.79
4 2.10
5 2.18
6 2.22
7 2.37
8 2.40
9 2.81
10 2.85
Lower Quartile (25th percentile)
Median (50th percentile)
Upper Quartile (75th percentile)
Inter Quartile
Range

• Standard deviation and variance: The standard deviation (SD or s) is calculated as follows:
𝑆𝑆𝑆𝑆 = 𝑠𝑠 =
∑𝑖𝑖=1
𝑛𝑛
𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
𝑛𝑛 − 1
• The expression ∑𝑖𝑖=1
𝑛𝑛
𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2may look complicated, but it is easier to understand when
thought of in stages.
• From each x value subtract the mean ̅
𝑥𝑥, square this difference, then add each of the n squared
differences.
• This sum is then divided by (n - 1).
• This expression is known as the variance.
• The variance is expressed in square units, so we take the square root to return to the original
units, which gives the standard deviation, s.
• Examining this expression it can be seen that if all the x’s were the same, then they would
equal ̅
𝑥𝑥 and so s would be zero.
• If the x’s were widely scattered about ̅
𝑥𝑥, then s would be large.
• In this way s reﬂects the variability in the data.
• The standard deviation is vulnerable to outliers, so if the 2.1 was replaced by 21 we would get
a very different result.

Calculation of the standard deviation
Consider the ﬁve birthweights (in kg): 1.2, 1.3, 1.4, 1.5, 2.1.
The calculations to work out the standard deviation are given
in the following table
Subject
Weight (kg)
Mean
Weight
(kg)
Differences
from mean
Square of
differences
from mean
𝑥𝑥𝑖𝑖 𝑥𝑥 𝑥𝑥𝑖𝑖 − 𝑥𝑥 𝑥𝑥𝑖𝑖 − 𝑥𝑥 2
1 1.2 1.5 –0.30 0.09
2 1.3 1.5 –0.20 0.04
3 1.4 1.5 –0.10 0.01
4 1.5 1.5 0.00 0.00
5 2.1 1.5 0.60 0.36
Totals (Sum) 7.5 0 0.50 kg2
Mean 1.50 Variance 0.13 kg2
n 5 Standard
deviation
0.35 kg
n - 1 4
Variance = 0,50/4
SD = square root
of the Variance

Means or Medians?
• Me a
ns a
n d me d
i a
ns c
onvey d
ifferent impressions of the lo ca
tion of
da
t a
, a
n done ca
nnot give apresc
ription a
s to whi c
h is prefer ab
le;
often both give useful information.
• If the d
istri b
ution is symmetri c
, then in gener a
l the me a
n is the b
etter
summ a
ry sta
tisti c
, a
n dif it is skewedthen the me d
i a
n is less inﬂuen c
ed
by the tails.
• If the da
t a a
re skewed
, then the me d
i a
n will reﬂe c
t a ‘typi ca
l’
individual better.
• For ex a
mple if in a c
o untry me d
ia
n in c
ome is £20 000 a
n d me a
n
in c
ome is £24 000, most people will rel a
te b
etter to the former
number.

Displaying continuous data
• A picture is worth a thousand words, or numbers, and there
is no better way of getting a ‘feel’ for the data than to display
them in a ﬁgure or graph.
• The general principle should be to convey as much
information as possible in the ﬁgure, with the constraint that
the reader is not overwhelmed by too much detail.

Presentation: Graphs
• In any graph there are clearly certain items that are
important.
• For example, scales should be labelled clearly with
appropriate dimensions added.
• The plotting symbols are also important; a graph is used to
give an impression of pattern in the data, so bold and
relatively large plotting symbols are desirable.
• By all means identify the position of the point with a ﬁne
pen but mark it so others can see.
• This is particularly important if it is to be reduced for
publication purposes or presented as a slide in a talk.

• A graph should never include too much clutter; for example, many overlapping groups each
with a different symbol.
• In such a case it is usually preferable to give a series of graphs, albeit smaller, in several panels.
• The choice of scales for the axes will depend on the particular data set.
• If transformations of the axes are used, for example, plotting on a log scale, it is usually better
to mark the axes using the original units as this will be more readily understood by the reader.
• Breaks in scales should be avoided. If breaks are unavoidable under no circumstances must
points on either side of a break be joined.
• If both axes have the same units, then use the same scale for each.
• If this cannot be done easily, it is sensible to indicate the line of equality, perhaps faintly in
the ﬁgure.
• False impressions of trend or lack of it, in a time plot can sometimes be introduced by omitting
the zero point of the vertical axis.
• This may falsely make a mild trend, for example a change from 101 to 105, into an apparently
strong trend (seemingly as though from 1 to 5).
• There must always be a compromise between clarity of reproduction that is ﬁlling the space
available with data points and clarity of message.
• Appropriate measures of variability should also be included.
• One such is to indicate the range of values covered by two standard deviations each side of a
plotted mean.

• It is important to distinguish between a bar-chart and a histogram.
• Bar-charts display counts in mutually exclusive categories, and so the
bars should have spaces between them.
• Histograms show the distribution of a continuous variable and so
should not have spaces between the bars.
• It is not acceptable to use a bar-chart to display a mean with standard
error bars.
• These should be indicated with a data point surrounded with errors
bars, or better still a 95% conﬁdence interval.
• With currently available graphics software one can now perform
extensive exploration of the data, not only to determine more carefully
their structure, but also to ﬁnd the best means of summary and
presentation.
• This is usually worth considerable effort.

Presentation: Tables
• Although graphical presentation is very desirable it should
not be over- looked that tabular methods are very
important.
• In particular, tables can give more precise numerical
information than a graph, such as the number of
observations, the mean and some measure of variability of
each tabular entry.
• They often take less space than a graph containing the same
information.
• Standard statistical computer software can be easily
programmed to provide basic summary statistics in tabular
form on many variables.

Points when reading the literature
1
. Is the number of study subjects involved clearly stated?
2
. Are appropriate measures of location and variation used in the paper?
• F o
r e
x am
p le
, if t
he distrib uti on o
f th e data is ske
w ed, t
h enh as th e m
edi an rath er th e m
eanb een
quoted?
• Is it sensible to quote a standard deviation, or would a range or interquartile range, be better?
• In general do not use SD for data which have skewed distributions.
3
. On graphs, are appropriate axes clearly labelled and scales indicated?
4
. Do the titles adequately describe the contents of the tables and graphs?
5
. Do the graphs indicate the relevant variability?
• F o
r e
x am
p le
, if th e m
ai n o
bj ect of t
h e stud
y is a wit
h in
-su
bj ect com
p arison
, h as w ithin
-su
bj ect
variability been illustrated?
6
. Does the method of display convey all the relevant information in a study?
• For example if the data are paired, is the pairing shown?
• Can one assess the distribution of the data from the information given?

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