Descriptive Statistics
We use “Statistics” to...
Describe/summarize data: descriptive statistics.
o Reduce the data to understandable pieces of information.
o Example: What proportion of Dutch adults has a driver’s licence?
o Example: What is the average delay across all train travels in the Netherlands
today?
Drawing inferences about populations: inferential statistics
o In science we often want to draw conclusions about populations
o Example: Are COVID-19 vaccines safe and effective in the general population?
o Problem: We can often only make observations on a selection of cases from a
population
o Solution: We can use inferential statistics to evaluate whether the results in the
sample are generalizable to the population.
Studying complex multivariate relationships: statistical modeling
o In research we are often interested in relationships between several variables.
o Example: To what extent does years of education predict healthy lifestyle,
controlled for income differences?
o Statistical modeling can help to uncover such complex relations.
Measurement Level
In the social sciences we often collect quantitative data using questionnaires
o Example: measurement of participant age, socio-economic status, attitudes, etc.
We make a basic distinction between four “types” of data, known as measurement levels:
o Nominal
o Ordinal
o Interval
o Ratio
They differ in how refined or exact the measurement is
Nominal is the lowest and ratio the highest level
o Measuring at a lower level is often easier but less informative
1) Nominal Data: numbers express different unordered categories or groups.
Example: Marital status:
1 = single
2 = married
3 = in a serious relationship, but no married
4 = not specified otherwise
Nominal variables classify cases into two or more categories. Categories must be exhaustive (all
possibilities should be covered) and mutually exclusive (i.e., every case fits into one category and
one category only).
1
,2) Ordinal Data: numbers express an ordering (less/more)
Example: Smoking intensity:
1 = never
2 = At least 1 cigarette per month.
3 = At least 1 cigarette per day.
4 = More than 5 cigarettes per day.
Numbers express more or less of a quantity, but the difference between 1 and 2 is not the
same in quantity than between 2 and 3, 3 and 4, and so on.
There should be a logical order
o Not logical: Never -> Occasionally -> Daily -> Often
3) Interval: numbers express differences in quantity using a common unit.
Example: IQ test score
o the difference between 70 and 80 IQ points is comparable to a difference between
100 and 110. Both span a difference of 10 units.
Example: Temperature
o If on Monday the temperature is 30 degrees, on Tuesday 25 degrees, and
Wednesday, 15 degrees, then we can say that the temperature drop between
Tuesday and Wednesday is twice as large as the drop between Monday and
Tuesday.
o Zero point is arbitrary: zero Fahrenheit ≠ zero Celsius
4) Ratio: numbers have a natural zero point
Example: Length, weight or income
A length, weight, or income of 0 can be meaningfully interpreted
This allows for relative comparisons: if Peter’s monthly income is 5000 euros and John’s is 50
euros, then Peter’s income is 100 times higher than John’s income
This comparison is not possible on interval level: 6 degrees is not twice as hot as 3 degree
Both interval and ratio-level data are referred to as scale data. The idea is simple: all variables
that are not nominal or ordinal are treated as scale-level variables. SPSS distinguishes between
nominal, ordinal and scale.
Measurement level is a property of the measurement values, it is not (!) an intrinsic property of
the thing you are measuring.
- Example: you cannot say that “intelligence” has interval level;
- Intelligence can be measured at different levels depending on the measurement
o Ordinal: variable indicating the highest completed education (primary, secondary,
etc.)
2
, o Interval: score resulting from an IQ test
o Ratio: skull circumference in centimeters
Measurement level and statistical analysis
- Measurement levels determine the kind of statistics and statistical analyses you can use
meaningfully.
Example: the mean of a nominal variable is meaningless (e.g., “the average eye color).
Hence, for the analyses you should always respect the measurement levels of the
variables you will use in statistical analyses.
- Many of the commonly used statistical techniques assume scale data.
Problem: in the social sciences, it is not evident that variables have an interval level
o Example: attitude towards a governmental policy on a scale of 0 to 10
o Therefore, it is common practice to simply assume that we have acquired interval
data, without worrying too much if this is really true and this turns out to be very
useful.
Data Inspection
Every analysis starts with data inspection (“getting to know your data”): its goal is to get a clear
picture of the data by examining one variable at the time (univariate), or pairs of variables
(bivariate). In general, we want to inspect:
o Central tendency: What are the most common values?
o Variability: How large are the differences between the subjects? Are there
extreme values in the sample?
o Bivariate Association: for each pair of variables, do they associate/covary (i.e., do
low/large values on one variable go together with low/large values on the other
variable
- To accomplish this goal, we use graphs and statistics. Which statistics and graphs are
most appropriate depends on the measurement level (i.e., whether the data are nominal,
ordinal, or scale level).
Three common graph types for VISUAL data inspection
Bar charts nominal & ordinal data u 1+ variable
Histograms scale data 1+ variable
Scatterplots scale data 2+ variables
3
, NUMERICAL data inspection
Three common statistical approaches
Frequency tables
o Counts & percentages
o Nominal & ordinal data
Central tendencies
o What is the centre of scores on a variable?
o Nominal, ordinal & scale data
Variability measures
o How much variation is there in variable scores?
o Ordinal & scale data
Numerical data inspection Central Tendency – Mode, Median and Mean
- Mode: the score that is observed most frequently
o For nominal, ordinal or scale data
- Median: the score that separates the higher half of data from the lower half Example1:
(𝑁=unequal): 4,5,6,7,8,9,9=>medianis7
Example 2: (𝑁 = equal): 4, 5, 6, 8, 9, 9 => median 7
(arithmetic mean of the two middle values 6 and 8).
o For ordinal or scale data that are not normally distributed
- Mean
M= =
ΣX ∑ of all scores
N total number of scores
o For ordinal or scale data that are normally distributed
o X IS THE SET OF NUMBERS
o N IS THE NUMBER OF SCORES
4
We use “Statistics” to...
Describe/summarize data: descriptive statistics.
o Reduce the data to understandable pieces of information.
o Example: What proportion of Dutch adults has a driver’s licence?
o Example: What is the average delay across all train travels in the Netherlands
today?
Drawing inferences about populations: inferential statistics
o In science we often want to draw conclusions about populations
o Example: Are COVID-19 vaccines safe and effective in the general population?
o Problem: We can often only make observations on a selection of cases from a
population
o Solution: We can use inferential statistics to evaluate whether the results in the
sample are generalizable to the population.
Studying complex multivariate relationships: statistical modeling
o In research we are often interested in relationships between several variables.
o Example: To what extent does years of education predict healthy lifestyle,
controlled for income differences?
o Statistical modeling can help to uncover such complex relations.
Measurement Level
In the social sciences we often collect quantitative data using questionnaires
o Example: measurement of participant age, socio-economic status, attitudes, etc.
We make a basic distinction between four “types” of data, known as measurement levels:
o Nominal
o Ordinal
o Interval
o Ratio
They differ in how refined or exact the measurement is
Nominal is the lowest and ratio the highest level
o Measuring at a lower level is often easier but less informative
1) Nominal Data: numbers express different unordered categories or groups.
Example: Marital status:
1 = single
2 = married
3 = in a serious relationship, but no married
4 = not specified otherwise
Nominal variables classify cases into two or more categories. Categories must be exhaustive (all
possibilities should be covered) and mutually exclusive (i.e., every case fits into one category and
one category only).
1
,2) Ordinal Data: numbers express an ordering (less/more)
Example: Smoking intensity:
1 = never
2 = At least 1 cigarette per month.
3 = At least 1 cigarette per day.
4 = More than 5 cigarettes per day.
Numbers express more or less of a quantity, but the difference between 1 and 2 is not the
same in quantity than between 2 and 3, 3 and 4, and so on.
There should be a logical order
o Not logical: Never -> Occasionally -> Daily -> Often
3) Interval: numbers express differences in quantity using a common unit.
Example: IQ test score
o the difference between 70 and 80 IQ points is comparable to a difference between
100 and 110. Both span a difference of 10 units.
Example: Temperature
o If on Monday the temperature is 30 degrees, on Tuesday 25 degrees, and
Wednesday, 15 degrees, then we can say that the temperature drop between
Tuesday and Wednesday is twice as large as the drop between Monday and
Tuesday.
o Zero point is arbitrary: zero Fahrenheit ≠ zero Celsius
4) Ratio: numbers have a natural zero point
Example: Length, weight or income
A length, weight, or income of 0 can be meaningfully interpreted
This allows for relative comparisons: if Peter’s monthly income is 5000 euros and John’s is 50
euros, then Peter’s income is 100 times higher than John’s income
This comparison is not possible on interval level: 6 degrees is not twice as hot as 3 degree
Both interval and ratio-level data are referred to as scale data. The idea is simple: all variables
that are not nominal or ordinal are treated as scale-level variables. SPSS distinguishes between
nominal, ordinal and scale.
Measurement level is a property of the measurement values, it is not (!) an intrinsic property of
the thing you are measuring.
- Example: you cannot say that “intelligence” has interval level;
- Intelligence can be measured at different levels depending on the measurement
o Ordinal: variable indicating the highest completed education (primary, secondary,
etc.)
2
, o Interval: score resulting from an IQ test
o Ratio: skull circumference in centimeters
Measurement level and statistical analysis
- Measurement levels determine the kind of statistics and statistical analyses you can use
meaningfully.
Example: the mean of a nominal variable is meaningless (e.g., “the average eye color).
Hence, for the analyses you should always respect the measurement levels of the
variables you will use in statistical analyses.
- Many of the commonly used statistical techniques assume scale data.
Problem: in the social sciences, it is not evident that variables have an interval level
o Example: attitude towards a governmental policy on a scale of 0 to 10
o Therefore, it is common practice to simply assume that we have acquired interval
data, without worrying too much if this is really true and this turns out to be very
useful.
Data Inspection
Every analysis starts with data inspection (“getting to know your data”): its goal is to get a clear
picture of the data by examining one variable at the time (univariate), or pairs of variables
(bivariate). In general, we want to inspect:
o Central tendency: What are the most common values?
o Variability: How large are the differences between the subjects? Are there
extreme values in the sample?
o Bivariate Association: for each pair of variables, do they associate/covary (i.e., do
low/large values on one variable go together with low/large values on the other
variable
- To accomplish this goal, we use graphs and statistics. Which statistics and graphs are
most appropriate depends on the measurement level (i.e., whether the data are nominal,
ordinal, or scale level).
Three common graph types for VISUAL data inspection
Bar charts nominal & ordinal data u 1+ variable
Histograms scale data 1+ variable
Scatterplots scale data 2+ variables
3
, NUMERICAL data inspection
Three common statistical approaches
Frequency tables
o Counts & percentages
o Nominal & ordinal data
Central tendencies
o What is the centre of scores on a variable?
o Nominal, ordinal & scale data
Variability measures
o How much variation is there in variable scores?
o Ordinal & scale data
Numerical data inspection Central Tendency – Mode, Median and Mean
- Mode: the score that is observed most frequently
o For nominal, ordinal or scale data
- Median: the score that separates the higher half of data from the lower half Example1:
(𝑁=unequal): 4,5,6,7,8,9,9=>medianis7
Example 2: (𝑁 = equal): 4, 5, 6, 8, 9, 9 => median 7
(arithmetic mean of the two middle values 6 and 8).
o For ordinal or scale data that are not normally distributed
- Mean
M= =
ΣX ∑ of all scores
N total number of scores
o For ordinal or scale data that are normally distributed
o X IS THE SET OF NUMBERS
o N IS THE NUMBER OF SCORES
4
