Sheffield
Hallam University
Faculty of Health and Wellbeing
Professional Development 1
and Methods of Enquiry 1 Quantitative Analysis (for level 2 click here) Glossary of Statistics
Using the Glossary >>> Barchart: >>> Box-plot: >>> Correlation: >>> Data types: >>> Decimals, Fractions and Percentages >>> Dependent and Independent Variables: >>> Descriptive Statistics: >>> Explanatory and Response Variables: >>> Histogram: >>> Inter-quartile Range: >>> Mean (Arithmetic mean): >>> Mean (Geometric mean): >>>
Median: >>> Mode: >>> Normal Distribution: >>> Outlier: >>> Paired Data: >>> Population: >>> Range: >>> Sample: >>> Scatterplots :
>>> Standard Deviation: >>> Variance: >>> X and Y axes and co-ordinates: >>>
Using the Glossary
This does not set out to tell you everything about the topics listed.
Nor does it require you to learn and understand everything in it! It is
hoped that what is included will help you to make sense of the concepts
you are meet in your course. It should also be useful for reference when
you read articles.
You will be directed to read certain parts as you work through the
course. You will probably want to read, do an activity, and then read again
with more understanding. It would be useful to skim through it all before
you start, to get an idea of what you already know and what you are hoping
to understand better by the end of this course.
Barchart:
Similar to a Histogram but the bars don’t touch each other and the
x-axis usually does not have a continuous scale.
The example shows a bar chart of the colour of car owned by ten people.
Box-plot: (also known
as box and whisker plot)
A Boxplot divides the data into quarters. The middle line shows the
median (the value that divides the data in half), the box shows the range
of the two middle quarters, and the whiskers show the range of the rest
of the data. The values at the ends of the box are called the quartiles,
(SPSS refers to these as the 25th and 75th percentiles)
The distance between them is called the interquartile range (IQR).
The more sophisticated version (which SPSS uses) marks outliers with
circles, counting anything more than one and a half times the interquartile
range away from the quartiles as an outlier, those over three times the
interquartile range away from the quartiles are called extremes and marked
with asterisks. The length of the box is equal to the interquartile range
(IQR).
Boxplots are most often used for comparing two or more sets of data.
They allow you to compare level (the median), spread (the interquartile
range) at a glance, as well as showing the minimum and maximum.
The graph on the left compares the heights of men with different kinds
of housing. You can see at a glance that the men who own their own houses
tend to be smaller, and that there is less variation among them than among
those with mortgages or in council housing. You can also see that the tallest
and the smallest subjects both have mortgages.
Correlation:
A measure of the relationship between two paired sets of data. This
can be seen by eye from a scattergram.
Strong positive correlation: The points cluster about
a line that slopes upwards from bottom left to top right. Large values
of one variable tend to be associated with large values of the other. Example:
Height and shoe-size exhibit a high positive correlation. Tall people tend
to wear large shoes and small people tend to wear small shoes.
No Correlation:The points are spread out in a way that
doesn’t seem to slope up or down from left to right. Example: The number
of visits to a doctor in the last six months is unlikely to be correlated
with shoe-size. People with small shoes do not tend to visit the doctor
more or less than people with large shoes.
Strong negative correlation: The points cluster about
a line that slopes downward from left to right. Large values of one variable
tend to be associated with small values of the other. Example: Percentage
of patients on waiting list treated in less than 6 months and percentage
of patients on waiting list treated after more than 6 months. Regions where
the first is small the second will be large and vice-versa.
Weak positive or negative correlation: A definite slope
can be seen in the pattern of the points, but they are not so close to
the line, making a shape more like an ellipse.
Non-linear correlation: The points cluster about a curve,
not a line.
The correlation coefficient (Pearson’s product-moment correlation coefficient)
is a way of assigning a number to these situations. It is 1 for perfect
positive correlation (all the points exactly on a line sloping up from
bottom left to top right), 0 for no correlation and -1 for perfect negative
correlation (all the points exactly on a line sloping down from top left
to bottom right). It takes in-between values for in-between situations.
It should be noted that a high correlation coefficient on a small sample
may not indicate real correlation in the background population, and that
a fairly low correlation coefficient on a large sample may still indicate
background correlation.
There is another correlation coefficient, known as Spearman’s correlation
coefficient. It is similar to Pearson’s but calculated slightly differently,
and less affected by extreme values. It is used in tests for correlation
in circumstances where Pearson’s cannot be used.
Data (Different types)
Nominal Data: These are data which give classes which
have no real connection with numbers and can’t be ordered meaningfully.
Examples: Male or female, Town of residence.
Ordinal Data: These are data that can be put in an order,
but don’t have a numerical meaning beyond the order. So for instance, a
distance of 2 between two numbers would not be meaningfully equivalent
to a distance of 2 between two others.
Level of pain felt in joint rated on a scale from 0 (comfortable)
to 10 (extremely painful).
Social class coded by number.
Interval Data: These are numerical data where the distances
between numbers have meaning, but the zero has no real meaning. With interval
data it is not meaningful to say than one measurement is twice another,
and might not still be true if the units were changed.
Examples: Temperature (Centigrade), Year, adult shoe size (In all
examples the zero point has been chosen conventionally, as the freezing
point of water or the year of Christ’s birth, or to make 1 smallest size
of shoes adults were expected to wear.
If my shoe size is twice yours in British sizes, this will not also be
true in Continental sizes.
Ratio Data: These are data that are numerical data where
the distances between data and the zero point have real meaning. With such
data it is meaningful to say that one value is twice as much as another,
and this would still be true if the units were changed.
Examples: Heights, Weights, Salaries, Ages.
Note that if someone is twice as tall as someone else in inches, this
will still be true in centimetres.
Percentage Data: Data expressed as percentages.
Example: Percentage of patients on waiting list operated on within
5 months.Decimals,
Fractions and Percentages
It is useful to be able to convert between these. If you are not happy
with converting between fractions, decimals and percentages it is worth
reminding yourself of the following and working out a few for yourself,
so you don’t panic if you meet something in an unfamiliar form.
Percentages to decimals: divide by 100. e.g. 7% = 0.07
or 50% = 0.5
Decimals to percentages: multiply by 100. e.g. 0.003 = 0.3%
or 0.25 = 25%
Fractions to decimals: divide the top by the bottom. e.g.
3/8 = 3 ¸ 8 = 0.375
Decimals to fractions: Put the decimal places over 10, 100, or
1000 etc. depending on how many there are. e.g. 0.3 = 3/10, 0.04 = 4/100,
0.007= 7/1000. You can then often simplify these by dividing the top
and the bottom by a common factor, or using a calculator that does this
for you: e.g. 4/100 =1/25.
Percentages to Fractions: If it is a simple whole number put
100 underneath it and simplify if necessary. Otherwise turn it into a decimal
first.
e.g. 5% = 5/100 = 1/20, 3.7% = 0.037 = 37/1000
Fractions to Percentages: If there’s 100 on the bottom, leave
it off. Otherwise turn it into a decimal first. e.g. 3/100 = 3%, 7/200
= 7 ¸ 200 = 0.035 = 3.5%
A general term for ways of describing a sample without attempting to
draw conclusions about the background population. The mean, median, standard
deviation and inter-quartile range are examples of descriptive statistics,
as are graphs.
Explanatory
and Response Variables:
In a situation where we have a hypothesis that changes in one variable
explain changes in another, we call the first the explanatory variable
and the second the response variable (because it responds to changes in
the first). A scattergram should always have the explanatory variable on
the x-axis and the response variable on the y-axis.
Example: the hypothesis is that your heart rate increases the longer
you exercise. You control the time of exercise by taking measurements of
heart rate after 0, 5, 10, 15 minutes etc. Time is the explanatory variable
and heart rate is the response variable.
A hypothesis that changes in one variable explain changes in another is
best tested in a situation where the explanatory variable can be controlled,
as in the above example.
In medical statistics, situations where one variable is controlled can
be difficult to set up ethically. (How would patients react in your
discipline if they were told the length of their treatment would be decided
at random as part of an experiment?)
This means we often cannot choose people at random to give different
treatments, but must use the treatments they were given for other reasons.
This may mean that the explanation for the response variable comes not
from the different treatments, but from other different factors that determined
the treatments.
Example: it was argued for a long time that heavy smokers did not
die of lung cancer because they were heavy smokers, but because of other
lifestyle factors which drove them to become heavy smokers.
There are many situations where variables are correlated but neither is
explanatory.
Example: Areas where more households have two cars also report
less deaths from lung cancer. Both variables are at least partly explained
by the variable money.
In situations where the explanatory variable is controlled experimentally
it is often known as the independent variable, and the response variable
as the dependent variable (as you can decide independently what
the independent one will be, and the other depends on it).
Histogram:
A kind of barchart where each bar represents the frequency of a group
of data between certain values. The bars touch each other and the x-axis
has a continuous scale. (Not the case in other types of bar chart, where
the data does not need to be continuous.)
Histograms are usually used to examine the distribution of data: whether
they are evenly spread out along the range, or bunched together more at
some points. In particular, a histogram is one way of checking whether
data are roughly normally distributed.
Inter-quartile Range:
A measure of spread or variability, similar to the standard deviation.
It
is most often used to compare the variability of different samples.
It is the difference between the lower quartile and the upper quartile.
These are the values that a quarter of the data lies below, and that three
quarters of the data lie below, so the inter-quartile range is the range
of the middle half of the data.
Example: A group of 12 patients has ages 18, 18, 19, 19, 19, 20,
21, 23, 30, 33, 45, 81. The lower quartile is 19 and the upper quartile
is 31.5. The interquartile range is 12.5. (31.5 - 19 = 12.5)
Another group of 12 patients has ages 18, 19, 19, 19, 19, 19, 20,
21, 21, 22, 22, 85. The lower quartile is 19 and the upper quartile is
21.5. The interquartile range is 2.5. The first group has more variability
in age.
Box-plots show the quartiles.
SPSS will calculate the quartiles and the inter-quartile range can be
calculated easily from these by subtracting the lower quartile from the
upper one.
(There is some disagreement in different books about the exact method
of calculating quartiles - all different methods come out pretty close
and we are not concerned here with the details.)
Mean
(Arithmetic mean):
A measure of level or central tendency, the mean gives a number somewhere
in the middle of your data set. The Mean is often referred to as the average,
but this can cause confusion as the Median and the Mode are also kinds
of averages.
The mean is calculated by adding up all the data and dividing by how
many there are. SPSS will do this for you on the computer. Most scientific
calculators will also give you means directly.
Example: A sample of 5 patients have ages 18, 23, 20, 18, 81. The
mean is (18+23+20+18+81) ¸ 5 = 32. Note
that this mean is considerably larger than 4 of the ages in the set. If
the 81 had in fact been mistyped for 18 your result would be seriously
affected by this.
The mean has the advantage over the median that it takes into account all
the data, and the disadvantage that very large or very small values can
have a distorting effect on it.
Mean (Geometric mean):
Another measure of level or central tendency but much more difficult
to calculate than the Arithmetic mean! Rather than adding the numbers together
and dividing by the number of numbers, the numbers are multiplied together
and for "N" numbers the Nth route of the result is taken. When
people refer to the mean they usually mean the Arithmetic mean, so don’t
worry about the geometric mean. I Include it here mainly for completeness.
Median:
Another measure of level or central tendency. The median is found by
ranking the data set in order and taking the middle one (or the mean of
the two middle ones if there are two).
Example: A sample of 5 patients have ages 18, 23, 20, 18, 81. In
order this is 18, 18, 20, 23, 81. The median is 20, the middle value. If
a patient’s age lies below the median they are in the bottom half of the
set, and if above the median they are in the top half.
The median has the advantage over the mean that it is often easier to see
by eye for very small data sets, and is not unduly affected by extreme
values. It can be calculated on SPSS and some calculators. It is useful
when you want to know whether a particular result lies in the top or bottom
half of a data set.
Box-plots show the median.
In a symmetrical distribution, the mean and the median will be close.
Differences between the mean and median indicate asymmetry.
Mode:
The most frequent data value. It is often the easiest to pick out by
eye.
Example: A sample of 5 patients have ages 18, 23, 20, 18, 81. The
mode is 18, since this age occurs most often.
In a roughly normal distribution the mode will be close to the mean and
the median.
It is possible for a data set to have several modes. The presence of
several modes in a large dataset can indicate that different populations
have been combined.
Normal Distribution:
The name of a specific distribution with a lot of data values near the
mean, and gradually less further away, symmetrically on both sides. A lot
of biological data fit this pattern closely.
The histogram for a large number of normal data has a bell-shaped curve.
Outlier:
A data value, which is very big or very small, compared with the others.
Sometimes these are due to mistakes in entering the data and should always
be checked.
Outliers which are not mistakes can occur. It is worth examining your
data carefully and trying to explain why certain items stand out.
There are different rules for deciding exactly what to count as an outlier.
In SPSS a circle on a boxplot is used to mark outliers with values between
1.5 and 3 box lengths from the upper or lower edge of the box. (The box
length is the interquartile range.)
In SPSS an asterisk on a boxplot represents an extreme outlier (just
called an extreme in SPSS documentation but I feel the term extreme outlier
is more helpful) which is a value more than 3 times the interquartile range
from a quartile.
Paired
Data:
Data are paired if the entries in each row are connected with each other.
Examples:
Paired:
the ages and weights of a group of gymnasts
the weights of a group of gymnasts before and after a training session
Non-paired:
the weights of a group of gymnasts and a group of non-gymnasts
the changes in weight of two groups of gymnasts given different kinds
of training session
If you are not sure whether two columns of data are paired or not, consider
whether rearranging the order of one of the columns would affect your data.
If it would, they are paired.
Paired data often occur in ‘before and after’ situations. They are also
known as ‘related samples’. Non-paired data can also be referred to as
‘independent samples’.
Scatterplots (also called scattergrams) are only meaningful for paired
data.
Population:
The background group that we are using the sample to find out about.
Example: A group of 20 patients with anxiety problems are used
to draw conclusions about how any patients with anxiety problems would
respond to treatment. The population could be: patients in Sheffield with
similar problems, patients in England, patients all over the world, patients
from similar ethnic groups etc.
Conclusions may be more or less valid depending on how wide the population
they are supposed to apply to is, and how representative of that population
the sample is. Strictly, a sample should be drawn at random from its population.
Range:
The difference between the smallest and largest value in a data set.
It is a measure of spread or variability, but only depends on the two
extreme values, and does not tell anything about how spread out the rest
are.
It can be distorted by one extreme value.
Example: a group of patients are aged 18, 20, 23, 18, 81. The range
is 63. The 81 year old has a huge effect on this: if it were a mis-typing
for 18 the result would be very distorted.
It is useful as a very quick measure of variability, but the inter-quartile
range or the standard deviation are to be preferred for more precise comparisons
between different data sets.
Sample:
The group ofpeople, (or things, or places,) that the data have
been collected from. In most situations it is important to pick a representative
sample, which is not biased e.g. mainly women, mainly from particular
age or income bands or with particular educational qualifications.
There is a range of methods for doing this.
Scatterplots
(Also known as x-y plots and Scattergrams):
A graph used to show how paired data are related.
Each point represents a pair of data values, one given by its x co-ordinate
and the other by the y co-ordinate. They are used to look for correlation.
They can also be used to look for increases or decreases after a treatment,
by plotting before and after values and seeing whether most of the points
lie above or below the y = x line.
See the graphs used to illustrate correlation for examples of scattergrams.
Standard
Deviation:
A measure of the spread or variability of a data set.
The larger the standard deviation, the more spread out about the mean
the data are.
Like the mean, the standard deviation takes all values into account
and can be very affected by an extreme value. The Inter Quartile Range
is less effected.
You can find how to calculate it in any standard statistics book but
you do not need to, as SPSS will calculate it for you. Most scientific
calculators will also calculate it from the raw data if you do not have
access to a computer.
Example: Two groups of 5 patients have the following ages: Group
A: 18, 24, 30, 36, 42, Group B: 18, 19, 20, 38, 55, . Both groups have
the same mean, 30. The standard deviations are 8.5 for Group A and 14.5
for Group B, showing the ages in Group B are more spread out from the mean.Variance:
The square of the standard deviation.
It is used a lot in statistical calculations, but you won’t need it
to use and interpret statistics. The standard deviation is the square root
of the Variance.
X
and Y axes and co-ordinates:
The x-axis is the horizontal line along the bottom of a graph and the
y-axis is the vertical line up the side, (except where negative values
are involved, when the axes will be in the middle of the graph). Any point
on a graph has an x co-ordinate, which is the number on the x-axis level
with it, and a y co-ordinate, which is the number on the y-axis level with
it.
The point where both co-ordinates are zero is called the origin.
The diagonal line which goes through all the points whose x and y co-ordinates
are the same is called the line y = x.
Sheffield
Hallam University
