INTRODUCTION TO STATISTICAL ANALYSIS
WL 1
3 categories of statistic:
A. Univariate: only measure one property that affected many people > ex: what was the
average grade of the ISA last year? > consider the grade
B. Bivariate: two properties related to one another > ex: did males and females differ in their
grades? > consider gender to influence the grade
C. Multivariate: different properties that relate to another variable > was the grade dependent
on initial motivation, the time spent on reading and gender? > consider motivation, time
spent, gender to influence the grade
Statistics is the study of how we describe and make inferences from data (Sirkin)
Distinction between descriptive & inferential statistics
• Descriptive statistic: taking direct measurement of what you have (data) > you
describe your data
• Inferential statistic: we are able to make statement and draw conclusion based on our
data. Inference = conclusion reached on the basis of evidence and reasoning
Units of analysis = the what or who that is being studied > the unit that you will be able to draw
conclusions about. Typically, all units are the same type of “thing” in a single data set
Variable = Measured property of each of the units of analysis
Levels of measurement (1 & 2 = qualitative variables; 3 & 4 = quantitative variables) > always
know the level of measurement in order to know which statistical technique we may use for the
given variable(s)
1. Nominal level of measurement: group classification; no meaningful ranking possible;
numerical coding arbitrary.
2. Ordinal level of measurement: meaningful ranking/ordering (3 is more than 2); distance
between categories is unknown/not equal (difference between 1 and 2 not equal to
difference between 2 and 3)
3. Interval level of measurement: ex: meaningful ranking; distance are equal (ex:temperature
in degrees Celsius > difference between 15 and 17 is equal to difference between 20 and 22
> NO meaningful zero point since 0° doesn’t mean absence > we can go below 0)
4. Ratio level of measurement: meaningful ranking; distance are equal; absolute and
meaningful zero point (ex: age)
Continuous and discrete variables
Continuous variable = measured along a continuum > has fractional parts (Interval/Ratio)
,Discrete (= categorical) variable = measured in whole units or categories > no fractional parts
(Nominal/Ordinal)
Measures of Central Tendency (CT); (3 appropriate measures)
Used to (univariately) describe the distribution of variables on different levels of measurement
1. Mean: (vedi formula: M= Σ x/n) > can only be used for interval/ratio variables; most useful for
describing (more or less) normally distributed variables
• changing any score will change mean;
• adding or removing a score will change mean;
• adding, subtracting, multiplying, dividing each score by a given value causes the mean to
change accordingly;
• sum of differences form the mean is zero [Σ (x-M) = 0]
• Sum of squared (= al quadrato) differences from the mean is minimal > the value (=
risultato) is also called the Sum of Squares (SS). A larger SS means that scores deviate more
from the mean
• outliers = means a value far away from our distribution (very high or very low) > are not
really useful to describe the group
2. Median, also called 50th percentile (the one in the middle) > can be used for ordinal or
interval/ratio variables; often used for interval/ratio variables that have skewed distributions
• To find the median:
1. Sort all cases based on their value on x
2. The value of the ‘middle case’ equals the median (equal amount of cases below
and above)
3. !! Whenever n is an even number, the median is the mean (=average) value of
the two middle cases (p.74/76)
• outliers = means a value far away from our distribution (very high or very low) > are not
really useful to describe the group > median seems more useful here (not sensitive to
outlier)
• To determine the median from a frequency table, we need to identify the first category that
exceeds 50% in the ‘cumulative percent’ column
3. Mode = the category with the largest amount of cases > can be used for nominal, ordinal or
interval/ratio variables
Types of Distributions (visibile su istogrammi)
A. Normal distribution > when the graphic is symmetrical and mean, median and mode have all
the same value
, B. Skewed distribution > when the mean is shifted to the right, in a negatively skewed distribution
or to the left, in a positively skewed distribution
In real life, most distributions will look normal
WL 2
News article: “coffee may prevent depression, scientists say” > it is not clear why it might have this
effect., but the authors believe caffeine in coffee may alter the brain’s chemistry. Decaffeinated
coffee did not have the same effect.
Measures of variability
We consider two different groups each with n = 20:
Group 1: ten people 20 y/o and ten people 60 y/o
Group 2: ten people 39 y/o and ten people 41 y/o
Observations:
• the mean (average) does not differ between the two groups (M = 40)
• BUT the variability(/dispersion) differs.
Measures of CT alone do not carry enough information to adequately describe distributions of
variables, we need a second type of measures: Measures of variability
Measure of Variability 4 types:
1. Range (ordinal, interval/ratio) = the difference between highest and lowest score (highest -
lowest). It is always reported with a maximum & minimum score.
2. Interquartile range (IQR) (ordinal, interval/ratio) > (instead of splitting the group in two
half, as range) it is based on “quartiles” that split our data into four equal groups (=quarters)
of cases. To find the quartile we order the numbers we have, find the median quartile (=
the number that divides in two half; Q2) and then we find the lower quartile (Q1) and the
(Q3) upper quartile (= the numbers that divides each half in other two half) > the IQR is
based on distance between Q1 and Q3 > IQR = Q3 - Q1.
3. Variance (interval/ratio)
= based on the sum of squares.
For the calculation of the
variance, it matters whether we
have sample data or population
data (typically: sample data).
WL 1
3 categories of statistic:
A. Univariate: only measure one property that affected many people > ex: what was the
average grade of the ISA last year? > consider the grade
B. Bivariate: two properties related to one another > ex: did males and females differ in their
grades? > consider gender to influence the grade
C. Multivariate: different properties that relate to another variable > was the grade dependent
on initial motivation, the time spent on reading and gender? > consider motivation, time
spent, gender to influence the grade
Statistics is the study of how we describe and make inferences from data (Sirkin)
Distinction between descriptive & inferential statistics
• Descriptive statistic: taking direct measurement of what you have (data) > you
describe your data
• Inferential statistic: we are able to make statement and draw conclusion based on our
data. Inference = conclusion reached on the basis of evidence and reasoning
Units of analysis = the what or who that is being studied > the unit that you will be able to draw
conclusions about. Typically, all units are the same type of “thing” in a single data set
Variable = Measured property of each of the units of analysis
Levels of measurement (1 & 2 = qualitative variables; 3 & 4 = quantitative variables) > always
know the level of measurement in order to know which statistical technique we may use for the
given variable(s)
1. Nominal level of measurement: group classification; no meaningful ranking possible;
numerical coding arbitrary.
2. Ordinal level of measurement: meaningful ranking/ordering (3 is more than 2); distance
between categories is unknown/not equal (difference between 1 and 2 not equal to
difference between 2 and 3)
3. Interval level of measurement: ex: meaningful ranking; distance are equal (ex:temperature
in degrees Celsius > difference between 15 and 17 is equal to difference between 20 and 22
> NO meaningful zero point since 0° doesn’t mean absence > we can go below 0)
4. Ratio level of measurement: meaningful ranking; distance are equal; absolute and
meaningful zero point (ex: age)
Continuous and discrete variables
Continuous variable = measured along a continuum > has fractional parts (Interval/Ratio)
,Discrete (= categorical) variable = measured in whole units or categories > no fractional parts
(Nominal/Ordinal)
Measures of Central Tendency (CT); (3 appropriate measures)
Used to (univariately) describe the distribution of variables on different levels of measurement
1. Mean: (vedi formula: M= Σ x/n) > can only be used for interval/ratio variables; most useful for
describing (more or less) normally distributed variables
• changing any score will change mean;
• adding or removing a score will change mean;
• adding, subtracting, multiplying, dividing each score by a given value causes the mean to
change accordingly;
• sum of differences form the mean is zero [Σ (x-M) = 0]
• Sum of squared (= al quadrato) differences from the mean is minimal > the value (=
risultato) is also called the Sum of Squares (SS). A larger SS means that scores deviate more
from the mean
• outliers = means a value far away from our distribution (very high or very low) > are not
really useful to describe the group
2. Median, also called 50th percentile (the one in the middle) > can be used for ordinal or
interval/ratio variables; often used for interval/ratio variables that have skewed distributions
• To find the median:
1. Sort all cases based on their value on x
2. The value of the ‘middle case’ equals the median (equal amount of cases below
and above)
3. !! Whenever n is an even number, the median is the mean (=average) value of
the two middle cases (p.74/76)
• outliers = means a value far away from our distribution (very high or very low) > are not
really useful to describe the group > median seems more useful here (not sensitive to
outlier)
• To determine the median from a frequency table, we need to identify the first category that
exceeds 50% in the ‘cumulative percent’ column
3. Mode = the category with the largest amount of cases > can be used for nominal, ordinal or
interval/ratio variables
Types of Distributions (visibile su istogrammi)
A. Normal distribution > when the graphic is symmetrical and mean, median and mode have all
the same value
, B. Skewed distribution > when the mean is shifted to the right, in a negatively skewed distribution
or to the left, in a positively skewed distribution
In real life, most distributions will look normal
WL 2
News article: “coffee may prevent depression, scientists say” > it is not clear why it might have this
effect., but the authors believe caffeine in coffee may alter the brain’s chemistry. Decaffeinated
coffee did not have the same effect.
Measures of variability
We consider two different groups each with n = 20:
Group 1: ten people 20 y/o and ten people 60 y/o
Group 2: ten people 39 y/o and ten people 41 y/o
Observations:
• the mean (average) does not differ between the two groups (M = 40)
• BUT the variability(/dispersion) differs.
Measures of CT alone do not carry enough information to adequately describe distributions of
variables, we need a second type of measures: Measures of variability
Measure of Variability 4 types:
1. Range (ordinal, interval/ratio) = the difference between highest and lowest score (highest -
lowest). It is always reported with a maximum & minimum score.
2. Interquartile range (IQR) (ordinal, interval/ratio) > (instead of splitting the group in two
half, as range) it is based on “quartiles” that split our data into four equal groups (=quarters)
of cases. To find the quartile we order the numbers we have, find the median quartile (=
the number that divides in two half; Q2) and then we find the lower quartile (Q1) and the
(Q3) upper quartile (= the numbers that divides each half in other two half) > the IQR is
based on distance between Q1 and Q3 > IQR = Q3 - Q1.
3. Variance (interval/ratio)
= based on the sum of squares.
For the calculation of the
variance, it matters whether we
have sample data or population
data (typically: sample data).
