Understanding centre and spread (year 13)
Centre and spread are the two most basic concepts used for descriptive and
comparative statistics. If we are describing one sample or comparing two samples,
we want to be able to identify the one best number that describes the whole group
(the centre) and how variable the individuals in that group are about that centre.
Other descriptors like skew and unusual features are also useful, but the most basic
information about the distribution is captured by describing centre and spread.
centre
spread
one best number to describe the group
how different members of the group are
from each other
position
variation
central tendency
dispersion
signal
noise
Centre
Median and mean are measures of central tendency or average. If you needed one
number to describe the whole group, this would be it. Centre describes position,
how far along a scale a group is.
Describe what you observe and use the median or mean as confirmation of your
observations. Demonstrate that you understand what the mean/median measures
in terms of the context (not the formula). Suitable words for demonstrating that
understanding include “on average” and “tend to”.
Spread
As well as describing the centre or position of our sample, we want to describe its
variability, how different the values are from each other, or how different the values
are compared to the centre. We need something to measure the variability of the
whole sample. IQR and SD are measures of variation or spread for a sample (or
population). They describe how different the values are from each other. The
discussion of spread should be separate from the discussion of centre, and should
not include any reference to position along the scale. Range is not useful as a
measure of spread, since it is determined only by extreme values.
Describe what you observe and use the IQR/SD as confirmation of your observations.
Give the IQR/SD with units in context and demonstrate that you understand what
the IQR/SD measures in terms of the context (not the formula, so not “width of
middle 50%”). The concept being described is the variability of the whole sample or
population. Large values of IQR/SD indicate a lot of variability in the sample or
population. What “large” means depends on the context. In manufacturing,
variability needs to be small. Some natural populations are very variable. Consider
whether the variation described by the IQR is large for the context you are
investigating.
If the data is approximately normally distributed, you could relate the SD to that
distribution (showing statistical insight). You could combine your description of
spread with your description and interpretation of the shape of the distribution.
Note that a measure of variability is a measure for the whole group, so we say that
“there is more variation in the heights of males than there is for females in my
sample”. We don’t say “tends to” or “on average” when we are talking about
variation.
Choosing your statistic
The mean is a more efficient measure of centre than the median (a sample mean
tends to be closer to the population mean, on average, than a sample median is to
the population median), so confidence intervals for means tend to be narrower than
confidence intervals for medians. However, means are more affected by extreme
values than medians, so for any context in which there are extreme values or a very
skewed distribution, the median is a better measure of centre than the mean. Your
exploratory data analysis will help you decide which measure you believe is best to
use for an investigation.
If using mean, use standard deviation with it. Both are affected by extreme values,
so do not use these if your sample has very extreme values or is very skewed.
Median and IQR are less efficient measures of centre and spread than the mean and
SD, but they are robust (unaffected by extreme values). If using median, use IQR
with it.
Shift and Overlap
Shift and overlap are comparisons of the centre relative to spread for two samples,
answering the questions “Which one is bigger?” and “How much bigger, relative to
the variation in each sample?”
Think, describe what you see and relate it to the real world. You will not get credit
for general sentences which do not relate to the context and could be taken from
the table of statistics without understanding eg It is not acceptable to say that “the
mean for females is 162.6cm which is about 5cm less than the mean for males at
167.5cm”, unless it is followed by further interpretation. Show understanding of the
context (eg “taller” or “shorter” showing an understanding that you are discussing
height) and understanding of the concept of average (eg “tend to”).
Example 1:
In my sample I notice that the year 9 boys tend to be taller than the year 9 girls.
The middle 50% of the boys heights is shifted further up the scale than the heights
of the girls. There is quite a lot of overlap between the middle 50% of boy and girl
height, indicating that there are a lot of boys and girls in my sample who are quite
similar in height. The mean height for boys in my sample was 167.5 cm, while the
mean height for girls was 162.6cm. This confirms that the boys in my sample tend
to be about 5cm taller than the girls. This makes sense because lots of year 9 boys
I know are taller than the year 9 girls I know.
In my sample I notice that the heights of year 9 boys have a similar spread to the
heights of year 9 girls. This is confirmed by the sample statistics. The SD of my
sample of girl heights is 10.7cm showing a reasonable amount of variation. The SD
of boy heights is 9.8cm, only slightly smaller. This means that there is less variation
in the heights of boys than girls in my sample. This makes sense because heights
tend to be normally distributed, and the distribution of heights of both sexes in my
sample are consistent with samples from a normal distribution with most heights in
the middle and fewer taller and shorter people. In a normally distributed population
almost everyone would be within 3SD of the mean. For males this would mean
almost everyone would be between 138cm and 197cm tall which is true in my
experience. For the females almost everyone would be between 130cm and 195cm
which would also be true.
Example 2
In my sample I notice that the males have driven at a faster maximum speed than
females, on average. The middle 50% of male speed is shifted toward higher speed
than females, and only part of the middle 50% of males and females overlaps. The
median maximum speed driven by males is 120km/hr while the median maximum
speed driven by females is 15km less at 105km/hr, which confirms my observations.
In my sample I notice that the males seem to be more variable in the maximum
speed driven compared to the female with more males driving at higher speeds.
The IQR of maximum speed driven for females in my sample is 77.5 km/hr, which is
a huge amount of variation in speed. For males the IQR was 40 km/hr, which is
quite a lot less. This means that the males in my sample are more similar to each
other in the maximum speed driven than the females are. This doesn’t at first make
sense when I compare it to my initial observations. This does make sense when I
look closer. The IQR being higher for females is because the female distribution is
more bimodal with lots of females never having driven (maximum speed of zero),
but most others clustered between 90 km/hr and 120 km/hr, so the statistical spread
is higher for females than males.