Descriptive Statistics
Epidemiology/Biostatistics
Kenneth Kwan Ho Chui, PhD, MPH
Department of Public Health and Community Medicine
kenneth.chui@tufts.edu
617.636.0853
Learning objectives in the syllabus
Distinguish between types of data
Know appropriate data presentation options for
various data types
Understand the strengths and limitations of various
descriptive statistics
Understand the concept of skewness and its
implications to discrete and continuous distribution
Appreciate the special aspects of the normal
distributions
Understand the calculation and application of z-scores
Population
Parameter
?
The true mean BMI of
Boston, Massachusetts
?
?
Researcher
Sample
Sample statistics
The mean BMI of
a sample from
Boston, Massachusetts
Population
Sample
Parameter
Sample statistics
Distribution of sample means
Know how to interpret and
calculate a confidence
interval for statistical
inference
Types of data
How to summarize data
Central tendency
Variability
Types of
data
Descriptive statistics
Tabulation
Attribute
15
16
17
18
19
20
21
22
23
24
25
Frequency
3
4
12
13
16
22
15
10
4
0
1
Graphical visualization
Mean = 19.43
Median = 20.00
Standard deviation = 2.01
Types of data: Nominal
Data representing attributes that are:



unordered
mutually exclusive
ideally exhaustive
Examples

Genders
Nominal variables with only two
possible attributes are also called
“dichotomous” or “binary”

Marital status
Census 2000, Long form
Graph for showing nominal data: Pie chart
Watson-Jones et. al. Effect of Herpes Simplex Suppression on Incidence of HIV among Women in Tanzania.
N Engl J Med 2008; 358:1560-71
Graph for showing nominal data: Bar chart
Horizontal axis is categorical
Watson-Jones et. al. Effect of Herpes Simplex Suppression on Incidence of HIV among Women in Tanzania.
N Engl J Med 2008; 358:1560-71
Graph for showing nominal data: Grouped bar chart
Watson-Jones et. al. Effect of Herpes Simplex Suppression on Incidence of HIV among Women in Tanzania.
N Engl J Med 2008; 358:1560-71
Types of data: Ordinal
Data representing attributes that are:


ordered
unequal difference between ranks
Examples

Language proficiency

Number of rooms, pay attention to the last option
Census 2000, Long form
Graph for showing oridinal data: Bar chart
US General Social Survey, 1991
Types of data: Discrete
Sometimes referred to as “count variable”
Data representing attributes that are:



ordered
equal difference between ranks
of finite amount of possible values, usually at the level
of integer (0, 1, 2, 3, 4, 5… etc.)
Example

Frequency of cooking dinner at home
NHANES 2007-08, Consumer Behavior section
Graph for showing discrete data: Histogram
No space between bars
Horizontal axis is a continuum
US General Social Survey, 1991
Types of data: Continuous
Data representing attributes that are:



ordered
equal difference between ranks
of infinite amount of possible values
Examples



Height
Consider a reported height of 165.5 cm. In reality it
could be 165.4810550654211381380… cm, so fine that we
can never exactly measure it.
Blood pressure
Age
Graph for showing continuous data: Histogram
US General Social Survey, 1991
The “relationship diagram”
Nominal
Collectively referred
to as “categorical data”
Ordinal
Discrete
Continuous
Also called “rank”
Also called “count”
Share similar statistical properties.
Techniques good for continuous
data are often good for discrete
data. In fundamental level, it is
safe to group them together.
(Until you learn analysis that is
specific for discrete data, and that
is out of our syllabus.)
The hierarchy of data types
Once the data are collected…
Continuous
Discrete
you can them
aggregate down
…but you cannot
go back up
Ordinal
Nominal
The hierarchy of data types, cond.
Continuous
Birthday
Discrete
Age in years
Ordinal
<20
20-29
30-39
40-49
50-59
≥60
Nominal
Below 21 vs. above 21
Downward
aggregation
If you ever end up designing a study or collect your own data,
always strive for the highest type in the hierarchy, within reason.
Central
tendency
Central tendency
The tendency of quantitative data to cluster around
some central value
Three major types:
Mean (also called average)
Median (also called 50th percentile)
Mode
Central tendency: Mean
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
Central tendency: Median
A median is the numeric value separating the higher
half of a sample from the lower half
Median can be found by:
1.
2.
3.
Arranging all the observations in ascending or
descending order
Picking the middle number as the median
If there is an even number of observations, then the
median is the mean of the two middle values
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
Since we have even number of cases, the median is
then the mean of the two middle values, which is
(4+4)/2 = 4
Central tendency: Mode
A mode is a data value with the highest frequency compared
to the other values’ frequencies
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
If we compile a frequency table with the numbers, we get:
Value
Frequency
1
1
2
1
3
2
4
3
5
2
6
1
Because value “4” has the highest
frequency (3), “4” is the mode.
A variable can only have one mean and one median.
However, it can have more than one mode.
Which one is the right central tendency?
Mean
Median
Mode
Nominal
No
No
Yes
Ordinal
No
Yes
Yes, but
uncommon**
Discrete
Yes
Yes, esp. if
skewed*
Yes, but
uncommon**
Continuous
Yes
Yes, esp. if
skewed*
No
* Skewness will be explained shortly in this lecture
** Numbers of possible responses in ordinal and discrete variables tend to be much
more than that of nominal, causing the inconvenience of reporting too many
modes
Variability
Variability
The magnitude of dispersion of the data around their
own central value
Four major expressions:
Range
Interquartile range (IQR)
Variance
Standard deviation
Variability: Range
A range is the difference between the smallest and the
largest values in a variable
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
The range is (6 – 1) = 5
No conventional way of pairing with any particular
central tendency measure
Variability: Interquartile range
Quartile is a set of three numbers that breaks the
variables into four groups of equal sample size
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6



The first one is 3, it’s also called the lower quartile or
25th percentile
The middle one is (4+4)/2=4. It’s the median or
50th percentile
The last one is 5, it’s also called the upper quartile or
75th percentile
Interquartile range is simply:
75th percentile – 25th percentile = 5 – 3 = 2
Often paired with median in data reporting
A little caveat about quartiles
The median is well defined, but there has not been a
universal agreement on how the upper and lower
quartiles should be derived.
Two examples:
1, 2, 3, 4
1, 2
Lower quartile: 1.5
3, 4
Upper quartile: 3.5
1, 2, 2.5
Lower quartile: 2
2.5, 3, 4
Upper quartile: 3
1, 2, 3, 4
Graph for showing quartiles: Boxplot
A variable
e.g. height
Outlier
Highest data point
within (75th percentile +
1.5  IQR)
1.5  IQR
Upper quartile
75th percentile
IQR
Median
Lower quartile
25th percentile
1.5  IQR
Lowest data point
within (25th percentile –
1.5  IQR)
Variability: Variance
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
To get the variance:
1.
2.
3.
4.
5.
Compute the mean, which is 3.7 (we did this already)
Subtract the mean from each value:
-2.7, -1.7, -0.7, -0.7, 0.3, 0.3, 0.3, 1.3, 1.3, 2.3
Square them:
7.29, 2.89, 0.49, 0.49, 0.09, 0.09, 0.09, 1.69, 1.69, 5.29
Add them up:
20.1
Divided the sum by (number of cases – 1):
20.1/(10 – 1) = 2.23
Fortunately, computer can now do all these for us!
Variability: Standard deviation
Standard deviation is the square root of variance
Consider a variable with data:
1, 2, 3, 3, 4, 4, 4, 5, 5, 6
We calculated the variance in the previous slide (2.23)
The standard deviation (SD) is then:
Often paired with mean in data reporting
Which one is the right variability?
Range
IQR*
Nominal
No
No
No
Ordinal
Yes
Yes
No
Discrete
Yes
Yes, esp. if
skewed**
Yes
Continuous
Yes
Yes, esp. if
skewed**
Yes
* IQR: Interquartile range; SD: Standard deviation
** Skewness will be explained shortly in this lecture
Variance
SD*
Normal
distribution
Mean ± SD
You will see “Mean ± SD” a lot. Most continuous data
are summarized with mean ± standard deviation
(± is pronounced as “plus-minus ”)
E.g. In the Aspirin study*, the BMI data in Table 1 for
the two groups are:


Aspirin: 26.1 ± 5.1
Placebo: 26.0 ± 5.0
For both groups to be comparable, both means and
SDs have to be similar
If we are willing to make an assumption, we can even
infer more about the data! This magical assumption is
“normal distribution”
* See course reading “A Randomized Trial of Low-Dose Aspirin in the Primary
Prevention of Cardiovascular Disease in Women”
Normal distribution
Some variables, when plotted in the form of a
histogram, look like this:
Reasonably symmetric
More values at the center
Decreasing number
of values towards the
two ends
Looks like a bell
When this happens, we can say a lot more with the
mean and standard deviation!
Feature #1: The 68-95-99 rule
99% of samples are within ± 3SD
95% of samples are within ± 2SD
68% of sample are within ± 1SD
# of SD:
Percentile:
0.5th
2.5th
16th
50th
84th
97.5th
99.5th
Application of the 68-95-99 rule (I)
The mean (±SD) of the daily caloric intake of a certain
group is 1200 ± 150
ASSUME THE VARIABLE DAILY CALORIC
INTAKE IS NORMALLY DISTRIBUTED*, then:



68% of the participants have caloric intakes ranging
from 1050 to 1350 kcal (– 1 SD to 1 SD)
95% of the participants have caloric intakes ranging
from 900 to 1500 kcal (– 2 SD to 2 SD)
99% of the participants have caloric intakes ranging
from 750 to 1650 kcal (– 3 SD to 3 SD)
* This assumption is needed for the 68-95-99 rule to work. The
distribution can be checked with histogram or other statistics
(not covered in this class)
Application of the 68-95-99 rule (II)
The mean (±SD) of the daily caloric intake of a certain
group is 1200 ± 150
ASSUME THE VARIABLE DAILY CALORIC
INTAKE IS NORMALLY DISTRIBUTED*, then:



The data point at the 84th percentile is about
(1200 + 150) = 1350 kcal
The data point at the 99.5th percentile is about
(1200 + 450) = 1650 kcal
A subject with kcal = 1200 is likely to be the 50th
percentile in this sample
* This assumption is needed for the 68-95-99 rule to work. The
distribution can be checked with histogram or other statistics
(not covered in this class)
Feature #2: Standardized comparison with z-score
Consider an imaginary sample


Height: Mean = 160 cm, SD = 15 cm
Weight:Mean = 95 lb, SD = 10 lb
How is someone who is 180 cm tall and 107 lb heavy
doing relative to the rest? The different units are
impeding direct comparison, but z-score can help
i.e. z-score is simply how many SDs a value is away
from the mean
z-score for the person’s height: (180 – 160)/15 = 1.33
z-score for the person’s weight: (107 – 95)/10 = 0.70
Continuous/discrete variables are (mostly) not normal
Problems with skewed distribution
Mean & Median
Problems with skewed distributions
Median
Mean
Positively skewed/Right skewed
Median is more or less the same
IQR is more or less the same
Mean becomes larger
SD is inflated
Mean
Median
Negatively skewed/Left skewed
Median is more or less the same
IQR is more or less the same
Mean becomes smaller
SD is inflated
For variables with a skewed distribution, median &
interquartile range is a better representation of the
central tendency and variability, respectively
Tell-tale signs of skewness
When the mean and median of the variable are very
different
When you try to reconstruct the histogram of the
normal distribution for the variable, a good part of the
curve falls into an illogical or biologically implausible
domain:


In a study with an entry criteria of age ≥ 45, the mean
and standard deviation of the age is 52.0±7.0
A study on eating out reported that an average family
makes dinner at home on 5.2±2.0 nights/week
When the authors reported only median with/without
quartiles for the variable
So what if it’s skewed? (Advanced teaser)
Skewness distorts the means, and hence distorts
analyses that heavily rely on the sample means
Ask if the skewness is relevant

For some variables in statistical analysis, we don’t care
as much if they are skewed or not
Ask if the skewness is serious

Some analyses are robust enough to tolerate some
skewness
Check if the authors employed solutions such as:



Transformation (e.g. logarithmic, square root, etc.)
Aggregating down the data type hierarchy
Using analyses that have relaxed requirement on the
sample’s distribution (e.g. non-parametric procedures)