Tall Stories
or how a simple question doesn’t always have a simple answer
Neil Sheldon
Royal Statistical Society
Centre for Statistical Education
neilsheldon.net
Some people think
statistics is a branch of
mathematics ...
... but they’re wrong.
It’s more important
than that ...
... it’s a life skill
The purpose of statistics
is understanding,
not numbers
Understanding statistics
is understanding the
world around you
Understanding statistics
enables you take better
decisions
The statistics tell a
story ...
... but first you have to
understand the story
behind the statistics
Tall Stories
or how a simple question doesn’t
always have a simple answer
Tall stories
• How tall am I ...
Tall stories
• How tall am I ...
– in absolute terms?
What is my height
in feet and inches?
Tall stories
• How tall am I ...
– in absolute terms?
What is my height
in feet and inches?
– in relative terms?
Am I short, or tall,
or about average?
Tall stories
• What factors influence
the answers to
– the absolute question?
– the relative question?
Tall stories
• What factors influence
the answers to
– the absolute question?
– the relative question?
• Variation,
variation,
variation!
If only ...
Men, white
From the US National Health And Nutrition
Examination Survey ‘NHANES III’ 1988-94
Men, black
Men, Hispanic
Men, other
Women, white
Women, black
Women, Hispanic
Women, other
But where do the data come from?
And what are the implications of
sampling variation?
NormalSimulation.xlsx
Cross-sectional and longitudinal studies
Age differences in height derived from cross-sectional studies can be
the result of differential secular influences among the age cohorts.
To determine the magnitude of height loss that accompanies aging,
longitudinal studies are required. The authors studied 2,084 men and
women aged 17–94 years enrolled from 1958 to 1993 in the
Baltimore Longitudinal Study of Aging, Baltimore, Maryland. On
average, men's height was measured nine times during 15 years and
women's height five times during 9 years. The rate of decrease in
height was greater for women than for men. For both sexes, height
loss began at about age 30 years and accelerated with increasing
age. Cumulative height loss from age 30 to 70 years averaged about
3 cm for men and 5 cm for women; by age 80 years, it increased to 5
cm for men and 8 cm for women. Am J Epidemiol 1999;150:969-77.
Longitudinal data
Overlapping longitudinal data
They were shorter back then ...
• Judged by the height of the doorframes he built,
medieval man seems to have been short by today’s
standards.
• But evidence gathered from 3,000 skeletons reveals that
human height has varied little over the past 1,000 years.
• From the 10th century through to the 19th, the average
height of adult men was 5ft 7in or 170cm - just 2in below
today's average.
• Women were an average of 5ft 2in or 158cm - just over
an inch shorter than today.
All the bones in the
study came from the
medieval St Peter's
Church in Barton
upon Humber, North
East Lincolnshire.
... Or were they?
Based on a modest sample of skeletons from northern Europe, average heights
fell from 173.4 cm in the early Middle Ages to a low of roughly 167 cm during the
17th and 18th centuries. Taking the data at face value, this decline of
approximately 6.4 cm substantially exceeds any prolonged downturns found during
industrialization in several countries that have been studied. Significantly, recovery
to levels achieved in the early Middle Ages was not attained until the early 20th
century. It is plausible to link the decline in average height to climate deterioration;
growing inequality; urbanization and the expansion of trade and commerce,
which facilitated the spread of diseases; fluctuations in population size that
impinged on nutritional status; the global spread of diseases associated with
European expansion and colonization; and conflicts or wars over state building or
religion. Because it is reasonable to believe that greater exposure to
pathogens accompanied urbanization and industrialization, and there is evidence
of climate moderation, increasing efficiency in agriculture, and greater
interregional and international trade in foodstuffs, it is plausible to link the
reversal of the long-term height decline with dietary improvements.
Richard H Steckel
Variation in height during the day
Variation in height during the day
Did you know that astronauts are up to 2 inches taller
while they're in space? As soon as they come back to
Earth, though, they return to their normal height.
Imagine that the vertebrae in your back form a giant
spring. Pushing down on the spring keeps it coiled tightly.
When the force is released, the spring stretches out. In
the same way, the spine elongates by up to three percent
while humans travel in space.
To some degree, a similar stretching of the spine happens
to you every night. When you lie down, gravity isn't
pushing down on your vertebrae. Measure your height
carefully as soon as you get up or while you are still lying
down. You will find that you're about a centimeter or two
taller.
Variation with wealth
Mean height of Dutch adults
Time series data
Variation by sex
Men are, on average, taller than women.
But some women are taller than some
men.
How can we quantify this?
men and women heights overlap.xls
Data, data everywhere ...
Human height - Wikipedia, the free
encyclopedia.htm
Self-reported height
Genetic variation
• Children of tall parents are, on average,
– as tall as
– taller than
– shorter than
their parents ... ?
• Parents of tall children are, on average,
– as tall as
– taller than
– shorter than
their children ... ?
Regression to the mean
C
C=P
P
C=P
C
P
Individual heights are measured in standard deviations from male mean or female
mean as appropriate. Then P is the average of father’s height and mother’s height
• Variation, variation, variation
– A very simple question, ‘How tall am I’,
raises many issues to do with variation
– These issues go to the heart of many
statistical concepts
• Variation, variation, variation
–variation within groups
This is ‘the usual’ concept of variation: the
variability within a population or a sample is
measured by the standard deviation or the
inter-quartile range
• Variation, variation, variation
–variation between groups
Groups may differ from one another.
Sometimes the variation between groups is
more important, sometimes it is less
important, than the variation within groups.
(Analysis of variance treats this in fine detail.)
• Variation, variation, variation
–variation within individuals
Sometimes the attribute to be measured is not
constant:
 it may have a cyclical variation
 it may have a trend over time
 ... and it may have both
• Variation, variation, variation
–variation over time
Where an attribute is observed to vary with
time, the variation may be
 cross-sectional: “the older ones were like that
when they were young”
 longitudinal: “that’s what happens as you get
older”
 ... or a combination of the two
• Variation, variation, variation
–historical variation
A longer-term variation that may be quite
distinct from longitudinal or cross-sectional
variation
• Variation, variation, variation
–variation in definition
Any attribute being measured or counted has
first to be defined. It is very common for
definitions to vary from one situation to
another
• Variation, variation, variation
–sampling “error” variation
A misnomer, as it is in the nature of samples to
vary: it’s not a bug but a feature. We all know
that samples vary, but we are often tempted to
read more information into a sample than it
can actually offer
• Variation, variation, variation
–sampling bias variation
There are many ways in which a non-random
sample can be unrepresentative. Opportunity
sampling – measuring or counting whatever is
at hand – may be the most common and the
most dangerous
• Variation, variation, variation
–self-reporting variation
Lacks objectivity and so can be deeply
misleading. It’s like anecdotal evidence on a
large scale
• Variation, variation, variation
–variation and correlation
Strong correlations reduce variation: knowing
the value of one variable can reduce the
uncertainty about another
• Variation, variation, variation
–variation by error
And, underpinning everything else, there are
the errors we all make in counting and
measuring, recording and tabulating.
Even if all the other sources of variation are
controlled and understood, our own fallibility
ensures that there will always be variation in
the data
Tall Stories
how a simple question doesn’t always have a simple answer
Neil Sheldon
Royal Statistical Society
Centre for Statistical Education
neilsheldon.net