Statistics- 1: Description and Inferences

variable=

any characteristic, number or quantity that can be measured and can differ across entities or across time

variables examples

• hair colour, level of trust in the government, age, number of consecutive stairs climbed…

Types of variables:

• different levels of measurement: nature of information of the values assigned to variables

Categorical (level of measurement): nominal

• 2 or more exclusive categories

• no natural order

• no arithmetic operations possible

eye colour, marital status, hair colour, political party affiliation…

Categorical (level of measurement): ordinal

• clear ordering of the values (high to low)

• distance between values not the same across levels

education level, political interest, performance ratings, agreement to a statement…

Numerical (level of measurement): continuous

• can be measured to any level of precision

height, weight…

Numerical (level of measurement): Discrete

• only countable variables are possible

  • whole and positive numbers

• Can be measured in discrete terms= whole numbers

pets owned, point in an exam, number of car accidents…

Explanatory variables

= cause

often x

Independent variable

Response variable

Outcome

often y

Dependent variable

Organizing data in a dataset:

• column= variable

• row= a given record of the data set

• cell= one observation

Frequency distribution:

= how the values are distributed in relation to other values.

• display of the pattern of frequencies of a variable

• how often they occur in a data set.

Skewness:

negative (left) skew: mass concentrated on the right, left tail is longer.

positive (right) skew: mass concentrated on the left, tail longer on the right

how can we summarize/ describe distributions of variables?

1. visualize data

2. Calculate measures

   • measure of dispersion: how stretched or squeezed the distribution is.

Measures of central tendency/ level of measurement: nominal

mode

Measures of central tendency/ level of measurement: ordinal

median +mode

Measures of central tendency/ level of measurement: numeric

Mean + median+ mode

Mode=

• most frequent score in a data set

• data with 1 mode= uni modal

• can be several

Median=

• middle score of a set of data that has been arranges in order of magnitude:

  • when even numbers: divide by 2

Mean=

• arithmetic

• average of numbers.

• sensitive to extreme values (outliers)

• = robust statistic.

Measures of central dispersion/ level of measurement: nominal

not possible

Measures of central dispersion/ level of measurement: ordinal

range, inter-quartile range

Measures of central dispersion/ level of measurement: numeric

Range, inter-quartile range, variance/ standard deviation

Range=

difference between the lowest and highest values

Percentiles=

• split data into chunks

  • percentile= 100

  • deciles= 10

  • quintiles=5

  • quartiles=4

Inter-quartile range=

= range of the middle 50% if the data:

• calculate by subtracting 1st quartile from the 3rd quartile.

• robust statistic → not affected by outliers as covers middle 50% of the values

Problem with IQR?

• ‘robust’ → sensitive to outliers

  • only uses a selection of data

Measures using all data:

deviance= how much does each value deviate from the mean.

Deviance=

• calculate all deviances= value- mean

• and then the sum of them = total deviance

Problem with total deviance + solution=

• when added up=0 → not useful measure of spread

  • instead → calculate sum of squared errors (SS)

    • square the deviances

    • sum of the squared deviances.

Problem with squared errors + Solution=

• increase n -→ increase in SS= NOT useful to compare.

  • Solution= divide sum of squared errors by number of observations (N) minus 1.

    • =VARIANCE

calculate variance=

• divide sum of squared errors by number of observations (N) minus 1.

Standard deviation:

• calculated once we have the variance.

• Letter sigma (σ)

  = square root of the variance

• dependent on the scale.

Larger standard deviation=

bigger dispersion around the mean

Can you identify mode? (in levels of measurement)

• nominal= YES

• ordinal= YES

• numeric= YES

Can you identify median and percentiles? (Level of measurement)

• nominal= NO

• ordinal= YES

• numeric= YES

Can you add/ subtract? (Level of measurement)

• nominal = NO

• ordinal= NO

• numeric= YES

Can you identify the mean & standard deviation? (level of measurement)

• nominal = NO

• ordinal= NO

• numeric= YES