27. Biostatistics I: stochastic variable and probability distribution, normal distribution and its parameters. Estimations of expected value and standard deviation from sample. Flashcards
What is experiment?
The way of collecting data or accessing data
What is data?
qualitative or quantitative properties.
What is qualitative data? Examples?
Categorical data that can be sorted into categories
→ for example, the names of the diseases, types of pathogens, or the severity of the condition
What is quantitative data?
Numerical data that can be characterized by a number
→ E.g, The size of the rash or the duration of the sickness can be expressed by a number
2 types of qualitative data
- Ordinal data (sortable)
- Nominal data (not sortable)
example of ordinal (sortable) data
The severity of the disease: modest, medium, strong
Example of Nominal (not sortable) data
the blood groups: A, B, AB, 0.
2 types of quantitative data
- Continuous data (e.g., weight, height, blood pressure).
- Discrete data (e.g., number of children in the family).
Classification of data
What is absolute frequency?
the number of times a particular piece of data or a particular value appears during a trial or set of trials.
What is relative frequency?
The frequency that equals the absolute frequency of the category divided by the total number of cases
Relative frequencies are always numbers between _ and _
between 0 and 1
Does measured data always have errors?
YES!
The final goal of the statistical methods is to draw __.
conclusions
Logical inference gives __
a statement that is 100 % sure
Statistical inference gives __
a statement of given probability (always less than 100 %).
(For example, if we state something with 95 % probability it means that in 5 cases out of 100 we were wrong, the inaccuracy is 5 %.)
What is the reason of inaccuracy?
in case of statistical inference
→ we are not able to take all of the circumstances into account.
What does Probability calculus give?
a mathematical description of laws of mass events in the material world that are not determined unambiguously by the circumstances.
(statistics are based on the principles of probability calculus.)
What is a continuous parameter? Give an example
a numeric parameter that can take any value in a specified interval.
→ E.g, The pulse rate, the frequency of heartbeat
What is population (fundamental ensemble)?
a set of all possible observations (from Multiple measurements)
→ The number of elements in a population is N
What is variable?
Observation of a specific feature of the population
→ i.e, a single general element x of the population
What is a sample?
An appropriate part of the population chosen for the examination → to draw the conclusions about the population
What does sampling mean?
It means choosing n elements, ideally randomly, from the population.
→ Sampling happens, for example, when we measure several times. Sampling makes sense only if n can be much smaller than N (see Fig. 3).
What is frequency distribution?
A (list) table of representation of frequencies or relative frequencies in classes
What is histogram?
The graph consists of a series of rectangles, each with an area proportional to the frequency of data in the corresponding class interval represented on the horizontal axis.
histogram.
Equal class widths are convenient, because in this case the frequency is proportional to the __ of the rectangle.
height
histogram.
Every small rectangle (or square) corresponds to (1)___
→ The total number of rectangle units equals ___
→ This is the total __
- one measured value
- the total number of measurements (n = 20).
- area under the frequency curve.
histogram.
If the data size is increased and at the same time the class width (1)___
→ then the rough steps of the envelope observed imitially gradually smooth into a (2)___ curve (Fig. 6.).
- decreased
- continuous
Distribution of the population, theoretical distribution curve
Let us have a closer look at the tendency shown in Fig. 6.
→ If the population consists of a finite number of elements (N ), then upon increasing the number of the sample elements (n) the sample size will eventually reach the (1)___
→ the sample will contain (2)____
- population size
- all the elements of the population (n = N ).
Distribution of the population, theoretical distribution curve
Let us have a closer look at the tendency shown in Fig. 6.
the distribution of a sample with N elements yields ___
- population size
- all the elements of the population (n = N ).
Distribution of the population, theoretical distribution curve
Let us have a closer look at the tendency shown in Fig. 6.
What is theoretical distribution?
- population size
- all the elements of the population (n = N ).
Distribution of the population, theoretical distribution curve
Let us have a closer look at the tendency shown in Fig. 6.
→ What does the population distribution provide?
the probabilities of all the possible values of the variable
Distribution of the population, theoretical distribution curve
Let us have an interval (a, b) on the coordinate axis.
→ The probability that a randomly chosen value falls within the interval (a, b) equals __
the area that lies under the distribution curve in this interval (from a to b)
Distribution of the population, theoretical distribution curve
If in the interval (a, b) the distribution curve has small values, the area under the curve is (1)___
→ the corresponding probability of incidence of these values of the variable will be (2)___ (Fig. 7/1)
- small
- low
Distribution of the population, theoretical distribution curve
if in the interval (a, b) the distribution curve has large values
→ the (1)___ and the (2)___ will be high (Fig. 7/2).
- area
- corresponding probability
Distribution of the population, theoretical distribution curve
If the width of the interval is (1)___, the area under the curve increases, which means (2)___ probability of incidence for these values (Fig. 7/3).
- increased
- higher
Distribution of the population, theoretical distribution curve
Similarly to the histograms, the total area under the curve equals ___
→ because the “interval” (a,b) which in this case spans ___ contains any randomly chosen value for sure.
- 1
- from –∞ to +∞
Distribution of the population, theoretical distribution curve
Consequently, in case of continuous variables probability that a randomly chosen value exactly matches a given number is ___.
zero
Difference between theoretical distribution and histograms?
- The theoretical distribution → describes all the possible data (i.e., the population)
- the histogram concerns only the elements of a sample taken from the population (i.e., the data of the specific measurement).
Can the theoretical distribution have different shapes?
Yes
What is Gaussian distribution?
A type of theoretical distribution which has a symmetric bell-shaped curve with one peak
→ It has following parameters
- Expected value (µ)
- Theoretical standard deviation
What is this equation?
The mathematical expression of the Gaussian distribution function
Is the normal or Gaussian distribution a single distribution function? Why?
No
→ Due to its parameters it describes a whole family of them: the shape of the curves is similar, but their position, width and height may vary
Gaussian distributions
the curve descends at the extremes toward the horizontal axis
→ Does the curve touch the horizontal axis?
Although the curve descends at the extremes toward the horizontal axis, it never actually touches it, no matter how far out one goes
What are the parameters of Gaussian distribution?
- Expected value (µ)
- Theoretical standard deviation
What is GAUSSIAN CURVE?
Histogram envelope of a population with Gaussian distribution if N = ∞ and x → 0
→ if the population contains an infinite number of elements and the class width approaches 0.
Fig. 9. The bell-shaped Gaussian distribution and its parameters.
Identify 1. Give the definition
Theoretical standard deviation
→ measure of the width of the Gaussian curve.
→ Width of the Gaussian curve at half height is approximately 2 times of Theoretical standard deviation
Fig. 9. The bell-shaped Gaussian distribution and its parameters.
Identify 2
Inflexion points
Fig. 9. The bell-shaped Gaussian distribution and its parameters.
Identify 3. Give the definition
Expected value
→ one of the parameters () of the distribution, a value that corresponds to the maximum of the Gaussian curve.
→ In other words, it is the value above or below which half the cases fall, and it is also the point that divides the area under the curve in half.
Is the height of the Gaussian curve an independent parameter?
No
What is the standard normal distribution?
(second curve from the left on the Fig. 8).
Among the infinite number of possible normal distributions there is a special one, for which …. (image)
Estimation of the parameters, statistical properties of the sample
The expected value is estimated most often by ___
the mean
What is the mean?
the arithmetic mean (average*) of the data (elements of the sample)
→ the most stable central tendency measure of the distribution
→ the least sensitive to the change of the sample
What makes the mean of high importance?
the sum of all deviations from this number equals zero (because the sum of the negative deviations will be equal to the sum of positive deviations):
If we imagine the data spread across a board according to their values
→ the mean corresponds to ___
the position of the balance point of the distribution.
What does this expression mean?
the empirical standard deviation (s),
What is the empirical standard deviation (s)?
Estimate of the theoretical standard deviation (from the squares of the deviation of the points from the mean.)
What is this expression defined as?
variance
→ the square of the empirical standard deviatio
What is variance?
The square of the empirical standard deviation
What is this expression?
the degree of freedom.
What is the degree of freedom?
the number of independent members or sample elements.
→ By subtracting the number of related elements from the total sample size we obtain the degree of freedom, that is, the number of independently chosen elements.
( Evidently, every element cannot be independent if there is a relationship between them. )
as the number of sample elements approaches infinity,
→ the mean approaches the ___
→ the empirical standard deviation approaches ___
- expected value
- the theoretical standard deviation with higher and higher accuracy.
