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East Carolina University
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Biostatistics
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BIOS 1500
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Kevin O'brien
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Lecture 4

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Biostatistics

BIOS 1500

Kevin O'brien

Spring

Description

Distribution of Values
Chapters 1 and 2
Fall 2017
Distribution of Values
A primary concept in statistics is that of the distribution of the values for a variable.
The ‘distribution’ is the frequency or relative frequency with which each value occurs.
The relative frequency is the proportion of times a given value occurs. Recall the
frequentist idea of the proportion estimating the probability of a value occurring.
Distribution of
Values
The distribution can be viewed as a graph where the ‘X’ axis lists the possible values of the
variable, and the ‘Y’ axis gives the frequency or relative frequency with which each value
occurs.
Distribution of Values
One of the first things to consider is the type of variable: Continuous variable or Discrete
variable.
These two categories impact the graphical depiction of the distribution
Distribution of Values
If we have a variable which is discrete then the distribution for that variable will be
discrete: with gaps between the values.
A continuous variable will have a continuous distribution of values: no gaps (at least
theoretically). The Mode
An important characteristic of a distribution is that of a modal value or modal category.
The mode is the most frequently occurring value (or category).
If there are two modes we say a distribution is bi-modal.
Frequency Table
Consider the following data on the number of carious teeth from a sample of individuals.
Note this is a discrete, yet ratio, variable.
Data on Dental Carries
Frequency Distribution
The first column gives the values of the variable as: 0, 1, 2, 3, 4, 5, 6.
The frequency of each value for the particular sample is given in the second column. The
value 0 occurred 10 times while the value 6 occurred 7.
There were a total of 81 persons examined (sum of the frequencies).
Frequency Distribution
The relative frequency is the proportional distribution and is calculated by dividing each of
the frequency values by the total: 81.
Cumulative frequencies are found by adding the current frequency and those for all other
lower values.
Frequency Distribution
The CRF or cumulative relative frequency is obtained by dividing each cumulative
frequency by the total number of observations or 81 for these data. Note that relative frequencies all lie between 0 and 1 just like probabilities. The Frequentist
school of statistical inference interprets these as estimated probabilities for the occurrence
of the values.
Frequency Distribution
Those computations are the simplest of statistics that can be computed for a sample, yet
they underlie one of the most fundamental statistical concept, that of distribution.
Bar Chart of Frequencies
Bar Chart of Relative Frequencies Note on Shape
Note that the same shape is obtained by plotting either the frequencies or the relative
frequencies.
One rule to keep in mind, is that when comparing the distribution of a variable between two
or more groups, always use relative frequencies.
Distribution
Things we look for in a distribution are:
Central Tendency
Variability of values and their spread
Shape of the distribution.
Gaps and clumping of values
Distribution
Those questions about the distribution are the foundations of descriptive statistics.
Using descriptive statistics we try to describe, and bring out the salient features of the
distribution of values for a variable.
This can be done one variable at a time as a univariate analysis, or several variables at a
time: Multivariate Analysis.
Distribution
As mentioned previously, the type of variable (nominal, ordinal, interval or ratio) dictates
what type of descriptive statistical methods we should use: the graph and the numerical
summaries.
Shape
The aspect of shape has to do with whether the distribution is symmetrical or skewed. This
question only makes sense for variables that have an order in their values.
A symmetrical distribution is one which has a central value at which it can be folded over
on itself. Each half is the mirror image of the other.
Symmetric Distribution Shape
A skewed distribution is one where the values trail off to one side or the other. If they trail
off to the right side we say the distribution is right skewed or positively skewed.
A distribution with values tailing off to the left is left skewed or negatively skewed.
Right Skewed
Left Skewed Distributions
If we had a continuous interval or ratio variable, the graph for that distribution
would not have gaps between the values.
That aspect—no gaps- is the central idea behind ‘continuous’.
Continuous Distribution
Percentiles and Quantiles
A percentile or quantile is a value from the distribution which has a stated proportion or
percent of values that are less than or equal to it. th
If the value 29, was the 25 percentile, then 25% of the values are less than or equal to 29.
If 57 was the 85 percentile then 85% of the values are less than or equal to 57.
Relate to Probability
Deciles
Deciles and quartiles are special percentiles

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