Statistics for the Health Scientist: Basic Statistics I

Topic 1
An Introduction to Statistics
Dr Luke Kane
April 2014
Topic 1: An Introduction to Statistics 1

OK… Rules – Very serious!
• YOU MUST ASK QUESTIONS
• If you dont understand - let's work it out!
• Otherwise – no rules

What is Statistics?
• Statistics is the science of learning from data,
and of measuring, controlling, and
communicating uncertainty;
• it thereby provides the navigation essential for
controlling the course of scientific and societal
advances

Outline
• Describing variables and data
• Descriptive statistics
– Tables
– Charts
– Shapes

Objectives
• Define variable
• Define data
• Classify variables in quantitative or categorical
• Sub-classify quantitative variables into discrete or
continuous
• Sub-classify categorical variables into nominal or
ordinal
• Use the type of variable to determine which table and
chart to display it
• Understand the normal distribution and other shapes

What is a Variable?
• A variable is something whose value can vary
• Examples (many!):

What is Data?
• Data are the values you get when you
measure variables
• Example:

Types of Variable
• Lots of different ways of thinking about
variables:
– Categorical vs. Metric
– Continuous vs. Categorical
I like this one:
Quantitative Vs Categorical

Categorical Variables - "What type?”
• "Categories"
• Nominal:
– Unordered, order not important
– Male or female, dead/alive, Blood group A B AB O
• Ordinal:
– Ordered, order is important
– type of breast cancer, agree  neither agree nor
disagree disagree

Categorical Variables
– Types of houses
– Days of the week
– Opinions/viewpoints
– Hair colour
– Malaria positive or negative

Quantitative Variables - " How much?"
• Also known as metric
• Quantitative variables can be:
• Continuous:
– The variables come from measuring
– Have units of measurement
– Good for analysis
• Discrete:
– The variables come from counting
– The values are usually integer (whole number)

Quantitative Variables
• Weight, Height
• Number of cigarettes per day
• Blood pressure
• How many malaria parasites in the blood
• Number of workers with malaria

Variables: A Summary Table
Quantitative
Continuous Discrete
Blood pressure, height, weight, age Number of children
Number of asthma attacks per child
Categorical
Ordinal Nominal
Grade of breast cancer
Better, same, worse
Disagree, neutral, agree
Sex – male or female
Alive or dead
Blood group

Variables – more!
• It is easy to summarise categorical variables
• You can convert quantitative variables into categorical variables
– For example, in diabetes it is dangerous when sugar is very low
– So a blood sugar of 1.6mmol/l is the quantitative measurement
– You can place this in a low, normal or high range (which makes it a categorical
variable)
– 1.6 is low - patient needs treatment (sugar!)
• Continuous variables allow better analysis as they are the real numbers
• Tests have more power if used on continuous variables
• So it is better to use continuous variables for statistical analysis
• Better to use categorical variables for summarising results and
presentation

Descriptive Statistics
• This is taking the raw data and consolidating it
into a table or chart

Descriptive Statistics
• Frequency tables
• Relative frequency tables
• Grouping the data
• Open ended groups
• Cumulative frequency
• Cross tabulation

Frequency Tables
• Nominal Categorical Variables
• Start with largest
• Tell reader what the total number is (n = X)
Category
Hair Colour
Frequency (number of adults)
n=116
Black 85
Brown 17
Blonde 8
Red / Ginger 4
Other (e.g. blue, green) 2

Relative Frequency
Category
Hair Colour
Frequency (number of adults)
n=116
Relative Frequency (%)
Black 85 73.3
Brown 17 14.7
Blonde 8 6.8
Red / Ginger 4 3.5
Other (e.g. blue, green) 2 1.7

Ordinal Categorical Variables
• Hair colour is a nominal categorical variable so
does not need to be ordered.
• Satisfaction is an ordinal categorical variable so
you can make a frequency table but you must put
the categories in order.
• For example: How would you put these in order?
• Unsatisfied
• Very satisfied
• Satisfied
• Extremely unsatisfied

Continuous Data
• Not practical to display all of
the raw data
• The table is too big even with
the small sample size.
• Easier to group the data
• Then make a frequency table
Pig Number (n =
21)
Weight of pigs at market / Kg
1 120
2 210
3 110
4 209
5 205
6 164
7 145
8 177
9 185
10 184
11 180
12 183
13 182
14 190
15 198
16 134
17 140
18 156
19 154
20 201

Grouped Frequency Table
• So if we group the data into groups of equal
width you get a grouped frequency
distribution
Weight of pigs at market / kg Number of pigs (Frequency)
n =21
110-130 2
131-150 3
151 - 170 3
171- 190 7
191-210 6

Outliers
• This is fine if all the data is close
together
– i.e. if all the pigs weigh about the
same
– But what do you do if there are
some giant pigs and some tiny pigs?
– Like if you added two extra pigs to
our data set:
• a pig weighing 54kg
• one big one weighing 327kg
Weight of pigs
at market / kg
Number of pigs
(Frequency) n =21
51-70 1
71-90 0
91-110 0
111-130 2
131-150 3
151 – 170 3
171- 190 7
191-210 0
211-230 0
231-250 0
251-270 0
271 – 290 0
291 – 310 0
311 – 330 1

Open Ended Groups
• The big and small pig are called Outliers.
• To make things easier you can use open ended
groups at the top and bottom
Weight of pigs at market / kg Number of pigs (Frequency) n =21
≤110 1
111-130 2
131-150 3
151 - 170 3
171- 190 7
191-210 6
≥211 1

Symbols
• > More than
• < Less than
• ≥ Equal to or more than
• ≤ Equal to or more than

Cumulative Frequency
• Adding up (cumulate) the frequencies as you
go along
• Enables you to make a nice chart - see later
• For example, the lengths of snakes below
Length of snake / cm Frequency (number of
snakes) n = 61
Cumulative frequency
of snakes
<30 10 10
31-60 17 27 (=10+17)
61-90 19 46 (=10+17+19)
91-120 12 58 (=10+17+19+12)
>121 3 61 (10+17=+19+12+3)Topic 1: An Introduction to Statistics 25

Cross-Tabulation
• Everything so far has been a table of a single
variable
• Sometimes you want to look at how two
variables influence one sample
• Crosstab - is the combination of two variables

Cross Tab Example
• Does drinking alcohol affect the number of
accidents people have on motorbikes?
• What are the two variables?
• The two variables are accidents and drinking
• If there was a big party and you breathalysed
500 people leaving, you could determine if
they were above or below the drink-drive
limit. You could then ask them the next day if
there was an accident on their way home.

Cross Tab Example
Accident on way
home?
Above the alcohol limit?
Yes No
Yes 40 2
No 116 342
Total 156 344

Cross Tab Example
• You can then convert this into percentages by adding up
the columns and rows.
• It is very easy to see that over 99% of the sober drivers did
not have accidents and more than 1 in 4 of the drunk
drivers had accidents
• Dont drink and drive!
Accident on way
home?
Above the alcohol limit?
Yes No
Yes 25.6% 0.6%
No 74.4% 99.4%

Charts
• Charts are a good way of describing data
• Categorical data is easily plotted as:
– Pie chart
– Bar chart
– Clustered bar chart
– Stacked bar chart

Pie Charts
• Good: for categorical nominal data, easy to
make, easy to understand
• Bad: Can only use one variable - need
separate pie chart for each variable, confusing
if many categories used

Simple Bar Chart
• Good: for
categorical
nominal data, easy
to make, easy to
understand
• Bad: only one
variable
• Note must have
spaces between
bars, equal bars

Clustered Bar Chart
• Very similar to a simple bar chart but allows
you to compare sub-groups, e.g. boys and girls
• Good for comparing category sizes between
groups, e.g. blonde boys and blonde girls

Stacked Bar Chart
• Good for comparing total number of subjects
in each group, e.g. all boys and all girls

Quantitative Charts
• Bar Charts can also be used to graph discrete
quantitative data
• But for continuous quantitative data it is
better to use a histogram
• Cumulative quantitative data can be charted
with a step chart or a frequency curve

Histograms
• Frequency Histogram
• Uses data that is
grouped together to
save space
• There are no gaps
between the bars - it is
a continuous variable
• Bad: only use one
variable at a time

Frequency Curve
• For cumulative data you can
make a frequency curve
• Continuous quantitative data
is assumed to have a smooth
continuum of values
• This should make a nice,
smooth curve - the cumulative
frequency curve
• This is also known as an ogive

Frequency Curve - Snakes
• So if we take the snakes:
Length of snake / cm Frequency (number of
snakes) n = 61
Cumulative frequency
of snakes
% Cumulative
frequency of snakes
<30 10 10 10/61 = 16.4%
31-60 17 27 27/61 = 44.3%
61-90 19 46 75.4%
91-120 12 58 95.1%
>121 3 61 100%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
<30 31-60 61-90 91-120 >121Topic 1: An Introduction to Statistics 38

Shapes
• OK now we have charts
– how do you describe data from the shape of the
graph?
• A uniform distribution is evenly distributed
– "A normal curve represents perfectly symmetrical
distribution"
– Also known as a "bell shape"
• Then you have "skews" to the left or right
– Left skews are negatively skewed
– Right skews are positively skewed
• Bimodal distributions have two humps

Normal distribution

Skew
• A measure of the asymmetry

Bimodal distribution

Summary so far…
• Types of data and variables
• Ways to put this data in tables
• Ways to put this data in charts
• Ways to examine the shape of the data
• Next: TOPIC 2
– Using numbers to summarise the data
– Prevalence and Incidence

References
• This lecture is based on David Bowers
“Medical statistics from Scratch: An
introduction for health professionals”
• Bowers, D. (2008) Medical Statistics from
Scratch: An Introduction for Health
Professionals. USA: Wiley-Interscience.

Statistics for the Health Scientist: Basic Statistics I

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