1 Background
Introduction
Methods Of Descriptive Statistics
2 Parameters In Descriptive Statistics
What Are Parameters?
Getting Practical
Location Parameters/Measures Of Central Tendency
Dispersion Parameters/Measures Of Spread
Making Life Easier in R
3 Exercise
Excel data
Parameters
Aarhus University Biostatistics - Why? What? How? 2 / 30
Background Introduction
Introduction
Descriptive statistics are used to summarize data.
The aim:
To describe a given set of data records n in regard to a certain
variable p
j
or set of variables p.
The procedure:
Using an adequately chosen set of methods to summarize or
visualize the data at hand.
Characteristics of variables are often expressed via parameters.
Aarhus University Biostatistics - Why? What? How? 4 / 30
Background Methods Of Descriptive Statistics
Methods & Quirks
Information is usually handed to descriptive statistics as n × p (row count ×
column count) data frames.
This information is used to calculate informative parameters:
Location Parameters (Measures Of
Central Tendency):
Arithmetic Mean
Mode
Median
Minimum, Maximum, Range
...
Dispersion Parameters (Measures Of
Spread):
Variance
Standard Deviation
Quantile Range
...
Descriptive statistics do not allow generalisation beyond the data!
Aarhus University Biostatistics - Why? What? How? 5 / 30
Parameters In Descriptive Statistics What Are Parameters?
Parameters And Their Meaning
What is a parameter?
In the case of descriptive statistics, a parameter presents some information on
the shape of the distribution of the values of a certain variable.
What’s the fuss?
Parameters can be used to summarise data properties and make large data
sets with a multitude of values per variable more accessible.
So?
To know which parameters to use one must know which ones there are and
how to calculate them.
Parameters are, more or less, digested data.
Aarhus University Biostatistics - Why? What? How? 7 / 30
Parameters In Descriptive Statistics Getting Practical
Creating Some Data
For the following computation of descriptive statistics parameters, we will need
the following data:
set.seed(42) # making the code reproducible
data_vec <- rnorm(mean = 20, sd = 2, n = 54)
matrix(sort(data_vec), nrow = 6)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## [1,] 14.69 17.22 18.55 19.28 19.73 20.64 21.01 21.52 22.89
## [2,] 15.12 17.26 18.72 19.39 19.79 20.73 21.27 22.07 23.02
## [3,] 15.17 18.30 18.78 19.43 19.81 20.81 21.27 22.43 23.15
## [4,] 16.44 18.38 18.87 19.44 19.87 20.87 21.29 22.61 23.79
## [5,] 16.47 18.43 19.14 19.49 20.07 20.91 21.31 22.64 24.04
## [6,] 16.57 18.43 19.14 19.66 20.41 20.92 21.41 22.74 24.57
→ Calculation of parameters of descriptive statistics is reserved almost
exclusively for numeric data records
Aarhus University Biostatistics - Why? What? How? 8 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Arithmetic Mean (Theory)
Definition:
Also called average, this metric is the mathematical average of
the given data values.
Non-resistant to outliers and asymmetric distributions.
Calculation:
x = µ =
1
n
n
P
i=1
x
i
x = µ Arithmetic mean
n
Number of samples (= number of values for the variable in
question)
i Index of variable values (i = 1, 2, .., n)
x
i
i
th
value of variable x
Aarhus University Biostatistics - Why? What? How? 9 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Arithmetic Mean (Calculation in R)
The arithmetic mean is calculated using the mean() function contained within
base R.
# calculation
mean(data_vec)
## [1] 20
15 20 25
0.00 0.05 0.10 0.15
Mean of data_vec
N = 54 Bandwidth = 0.7771
Density
Aarhus University Biostatistics - Why? What? How? 10 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Median (Theory)
Definition:
The median is the value separating the higher half of the data
values from the lower half.
Resistant to outliers and asymmetric distributions.
Calculation: median(x) = (
n+1
2
)
th
odd numbers of data values
median(x) =
(
n
2
)
th
+(
n
2
+1)
th
2
even numbers of data values
median(x) Median of the values available for variable x
n Number of observations available for x
Aarhus University Biostatistics - Why? What? How? 11 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Median (Calculation in R)
The median is calculated using the median() function contained within base
R.
# calculation
median(data_vec)
## [1] 19.84
15 20 25
0.00 0.05 0.10 0.15
Median of data_vec
N = 54 Bandwidth = 0.7771
Density
Aarhus University Biostatistics - Why? What? How? 12 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Mode (Theory)
Definition:
The mode of a set of data values is the value that is the most
common.
Resistant to outliers but the shape of the distribution
might be crucial.
Calculation: mode(x) = max
k=1
I
i=1
(x
i
= x
k
)
mode(x) Mode of the values available for variable x
max
k=1
() Maximising argument for k in 1 to p
I
i
()
Identifier that returns 1 if the internal statement is true with i in
1 to n
Aarhus University Biostatistics - Why? What? How? 13 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Mode (Calculation in R)
One may wish to use the max() and table() function contained within the
base
R
or through
mlv(..., method="mfv")
within the
modeest
package:
# counts of values in rounded vector
table <- table(round(data_vec))
table # counts
##
## 15 16 17 18 19 20 21 22 23 24 25
## 3 2 3 4 11 7 12 3 6 2 1
# most common appearance
max <- max(table)
max # maximum appearances
## [1] 12
# position of maximum in table
pos <- which(table == max)
pos # mode position
## 21
## 7
# value at maximum position
mode <- names(table)[pos]
as.numeric(mode) # mode
## [1] 21
Aarhus University Biostatistics - Why? What? How? 14 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Minimum, Maximum, Range (Theory)
Sometimes, one may want to use the following, simple information on data
values:
Maximum: The highest value available for a given variable.
Minimum: The lowest value available for a given variable.
Range:
The span of values that the data distribution defined by
minimum and maximum extends over.
Aarhus University Biostatistics - Why? What? How? 15 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Minimum, Maximum, Range (Calculation in R)
# calculation
min(data_vec)
## [1] 14.69
max(data_vec)
## [1] 24.57
range(data_vec)
## [1] 14.69 24.57
15 20 25
0.00 0.05 0.10 0.15
Minimum and Maximum of data_vec
N = 54 Bandwidth = 0.7771
Density
Aarhus University Biostatistics - Why? What? How? 16 / 30
Parameters In Descriptive Statistics Location Parameters/Measures Of Central Tendency
Which Location Parameter Do I Use?
All measures of central tendency describe the central position of a frequency
distribution of values of a given variable in the data set at hand.
The arithmetic mean is only really useful when concerned with symmetric
distributions of data values.
The median exhibits robust behaviour when faced with asymmetric distributions
of data values.
The mode is most applicable to the classification setting and rarely used.
→ The median will usually do.
Aarhus University Biostatistics - Why? What? How? 17 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Variance (Theory)
Definition:
Variance measures how much data values are spread out from
their average value.
Non-resistant to outliers and asymmetric distributions.
Calculation:
s
2
=
1
n−1
n
P
i=1
(x
i
− x)
2
s
2
Variance
n
Number of samples (= number of values for the variable in
question)
i Index of variable values (i = 1, 2, .., n)
x
i
i
th
value of variable x
x Arithmetic mean
Aarhus University Biostatistics - Why? What? How? 19 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Variance (Calculation in R)
The variance is calculated using the var() function contained within base R.
# calculation
var(data_vec)
## [1] 5.181
15 20 25
0.00 0.05 0.10 0.15
Variance of data_vec
N = 54 Bandwidth = 0.7771
Density
Note that his plot shows the span of the variation around the mean.
Aarhus University Biostatistics - Why? What? How? 20 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Standard Deviation (Theory)
Definition:
The standard deviation quantifies the amount of variation or
dispersion of a set of data values.
Non-resistant to outliers and asymmetric distributions.
Calculation: SD = s =
√
s
2
SD = s Standard Deviation
s
2
Variance
Aarhus University Biostatistics - Why? What? How? 21 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Standard Deviation (Calculation in R)
The standard deviation is calculated using the sd() function contained within
base R.
# calculation
sd(data_vec)
## [1] 2.276
15 20 25
0.00 0.05 0.10 0.15
Standard Deviation of data_vec
N = 54 Bandwidth = 0.7771
Density
Note that his plot shows the span of one standard deviation above and below
the mean.
Aarhus University Biostatistics - Why? What? How? 22 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Quantile Range (Theory)
Definition:
Quantiles are cut points dividing the range of a distribution of
data values into adjacent intervals with equal probabilities. You
will always receive one cut-point less than quantiles are
produced.
Resistant to outliers and asymmetric distributions.
Most often, one uses the following quantiles:
Quantile 50: This is basically the median.
Quantile 25 and 75: These are also known as quartiles.
Aarhus University Biostatistics - Why? What? How? 23 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Quantile Range (Calculation in R)
Quantiles are calculated using the
quantile()
function contained within base
R. A second argument, within this function can be specified to call certain
quantiles.
# quantiles we want
q <- c(0.25, 0.5, 0.95, 0.99)
# calculation
quantile(data_vec, q)
## 25% 50% 95% 99%
## 18.74 19.84 23.38 24.29
15 20 25
0.00 0.05 0.10 0.15
Quantiles of data_vec
N = 54 Bandwidth = 0.7771
Density
Aarhus University Biostatistics - Why? What? How? 24 / 30
Parameters In Descriptive Statistics Dispersion Parameters/Measures Of Spread
Which Dispersion Parameter Do I Use?
All measures of spread describe the spread of a frequency distribution of
values of a given variable in the data set at hand.
The variance is only really useful when concerned with symmetric distributions
of data values.
The standard deviation is only really useful when concerned with symmetric
distributions of data values.
The quantiles exhibit robust behaviour when faced with asymmetric
distributions of data values.
→ The quantiles will usually do.
Aarhus University Biostatistics - Why? What? How? 25 / 30
Parameters In Descriptive Statistics Making Life Easier in R
The summary() Function
# calculation
summary(data_vec)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 14.7 18.7 19.8 20.0 21.3 24.6
The summary()function can be called
on a vector object in R to return some
of the most useful information on
measures of central tendency and
measures of spread.
15 20 25
0.00 0.05 0.10 0.15
Summary of data_vec
N = 54 Bandwidth = 0.7771
Density
Aarhus University Biostatistics - Why? What? How? 26 / 30
Exercise Excel data
Loading Excel data into R
Excel is a valuable tool for data accquisition but almost useless when it
comes to statistical analyses or data visualisation in biological sciences.
So how do you get your excel data into R?
Loading procedure depends on file format:
.csv - I recommend using this format
as it allows for less alteration and is
compressed.
Functions: read.csv() and
read.table() (also works on .txt
files)
.xls, .xlsx, etc.
- Go for this if you need
to alter your data by hand (which you
shouldn’t. EVER!).
Functions: read.xlsx() (included in
xlsx package)
You can also use R to save data in excel format.
Aarhus University Biostatistics - Why? What? How? 28 / 30
Exercise Excel data
Inspecting Data
The most common form of data is the data frame which you can:
Inspect using functions such as:
- dim() to access the dimensions
- str() to access types and modes
- colnames()/rownames() to asses
column and row names
- head()/tail() to show only the top
or bottom five rows of the data set
- table() to show a count of items in
a vector
Subset using the different sub-setting
methods:
- [r,c] can be used to index rows (r)
and columns (c)
-
$
can be used to index column names
→ This is also how you extract data from a data frame (and most objects
within R).
Aarhus University Biostatistics - Why? What? How? 29 / 30
Exercise Parameters
Calculating parameters of descriptive statistics
Your ToDo-List for this exercise:
Load the file DescriptiveData.csv into R
Identify what kind of information it contains
Calculate the location parameters and parameters of spread for any of the
variables contained within the data set that catch your interest.
Question the validity of your findings and the data
The solution file will deal with all of the variables contained within the data set
so don’t worry about which one to pick and just have fun.
Aarhus University Biostatistics - Why? What? How? 30 / 30
