MRM1
Data
Consists of:
- Variables (number of properties)
- Unit (measured from a set of things/people/etc.)
Levels of Measurement
Variables
Categorical:
- Binary (two outcome; e.g., yes/no; dead/alive) Amount of
- Nominal (different types of groups; e.g. omnivore, vegetarian, vegan) information
- Ordinal (e.g. likert scale) ! order is important
Numerical:
- Discrete (counts; e.g., number of defects) ! not an inbetween number, e.g. 1.5
- Continuous (distinct scores; e.g., temperature; body length)
Variables can be converted, e.g. from continuous to ordinal ! loss of information (irreversible)
The lower the amount of information in data, the larger sample needs to be
Measurement Error
Systematic (difference between the average measurement result and the true value)
! Easy to solve by shifting the repetitive error
Random (unsystematic deviations due to imprecision of the measurement system)
! Things that are not defined (e.g. retaking IQ test at a different time)
EXAMPLE:
We have reference material at our disposal that has a ‘true’ value of 5.0.
Measuring device 1 produces the following outcomes: 3.8, 4.4, 4.2, 4.0
Measuring device 2 produces the following outcomes: 6.5, 4.0, 3.2, 6.3
QUESTIONS: Bias: difference between measured
- Which method has the largest bias? average and true value
Device 1: 5.0-4.1=0.9 * (overstating/understating true value)
Device 2: 5.0-5.0=0
- Which one has the largest measurement spread?
Device 1: 4.4-3.8=0.6 Measurement spread: range of
Device 2: 6.5-3.2=3.3 * measurements (how similar/varied
- Which method do you prefer? Why? are the observations?)
Device 1 has largest bias (0.9 vs. 0).
Device 2 has largest measurement spread (3.3 vs. 0.6).
Describing Data
- Median: the middle score when data is ordered (second quartile)
, - Mean: the sum of the data divided by the amount of data (average)
- Range: maximum value minus minimum value
! sensitive to outliers
- Interquartile range: the range of the middle 50% of the data (so Q3-Q1)
First find the median and then define lower and upper quartile
- Variance(Abweichnung): the average squared distance between each point and the mean of the data
Sum of squares
(!! !!)!
variance = 𝑠! =
!!! Number of scores
- Standard deviation(Standardabweichnung): the square root of the variance
𝑠𝑑 = 𝑠 = 𝑠!
- Dispersion: mean might be the same, but data set can differ in dispersion
Based on range and variance
!The higher the range the more dispersed the data set
! The lower the variance, the lower the dispersion of data set
- Confidence Interval: probability of how close sample is to total population
! 95%
e.g. If repeated samples were taken and the 95% confidence interval was computed for each sample, 95%
of the intervals would contain the population mean
! 99%
! !
𝑥−2 <𝜇<𝑥+ 2
! !
- Skew: asymmetry/symmetry of the distribution
Can be analysed by dataset only when
calculating mean and median
!Symmetric
mean ≈ median
balanced left and right tails
Data
Consists of:
- Variables (number of properties)
- Unit (measured from a set of things/people/etc.)
Levels of Measurement
Variables
Categorical:
- Binary (two outcome; e.g., yes/no; dead/alive) Amount of
- Nominal (different types of groups; e.g. omnivore, vegetarian, vegan) information
- Ordinal (e.g. likert scale) ! order is important
Numerical:
- Discrete (counts; e.g., number of defects) ! not an inbetween number, e.g. 1.5
- Continuous (distinct scores; e.g., temperature; body length)
Variables can be converted, e.g. from continuous to ordinal ! loss of information (irreversible)
The lower the amount of information in data, the larger sample needs to be
Measurement Error
Systematic (difference between the average measurement result and the true value)
! Easy to solve by shifting the repetitive error
Random (unsystematic deviations due to imprecision of the measurement system)
! Things that are not defined (e.g. retaking IQ test at a different time)
EXAMPLE:
We have reference material at our disposal that has a ‘true’ value of 5.0.
Measuring device 1 produces the following outcomes: 3.8, 4.4, 4.2, 4.0
Measuring device 2 produces the following outcomes: 6.5, 4.0, 3.2, 6.3
QUESTIONS: Bias: difference between measured
- Which method has the largest bias? average and true value
Device 1: 5.0-4.1=0.9 * (overstating/understating true value)
Device 2: 5.0-5.0=0
- Which one has the largest measurement spread?
Device 1: 4.4-3.8=0.6 Measurement spread: range of
Device 2: 6.5-3.2=3.3 * measurements (how similar/varied
- Which method do you prefer? Why? are the observations?)
Device 1 has largest bias (0.9 vs. 0).
Device 2 has largest measurement spread (3.3 vs. 0.6).
Describing Data
- Median: the middle score when data is ordered (second quartile)
, - Mean: the sum of the data divided by the amount of data (average)
- Range: maximum value minus minimum value
! sensitive to outliers
- Interquartile range: the range of the middle 50% of the data (so Q3-Q1)
First find the median and then define lower and upper quartile
- Variance(Abweichnung): the average squared distance between each point and the mean of the data
Sum of squares
(!! !!)!
variance = 𝑠! =
!!! Number of scores
- Standard deviation(Standardabweichnung): the square root of the variance
𝑠𝑑 = 𝑠 = 𝑠!
- Dispersion: mean might be the same, but data set can differ in dispersion
Based on range and variance
!The higher the range the more dispersed the data set
! The lower the variance, the lower the dispersion of data set
- Confidence Interval: probability of how close sample is to total population
! 95%
e.g. If repeated samples were taken and the 95% confidence interval was computed for each sample, 95%
of the intervals would contain the population mean
! 99%
! !
𝑥−2 <𝜇<𝑥+ 2
! !
- Skew: asymmetry/symmetry of the distribution
Can be analysed by dataset only when
calculating mean and median
!Symmetric
mean ≈ median
balanced left and right tails
