Transcript Average quadratic deviation

Average Arithmetic and
Average Quadratic
Deviation
Average Arithmetic and
Average Quadratic Deviation
 The
average values, which give the
generalized quantitative
description of certain characteristic
in statistical totality at the certain
terms of place and time, are the
most widespread form of statistical
indices. They represent the typical
lines of variation characteristic of
the explored phenomena.
Average Arithmetic and
Average Quadratic Deviation

Because of that quantitative description of
characteristic is related to its high-quality
side, it follows to examine average values
only in light of terms of high-quality
analysis. Except of summarizing estimation
of certain characteristic the necessity of
determination of changeable quantitative
average values for the totality arises up
also, when two groups which high-quality
differ one from other are compared.
The use of averages in health
protection

for description of work organization of
health protection establishments (middle
employment of bed, term of stay in
permanent establishment, amount of visits
on one habitant and other);
The use of averages in
health protection

for description of indices of physical
development (length, mass of body,
circumference of head of new-born and
other);
The use of averages in
health protection

for determination of medical-physiology
indices of organism (frequency of pulse,
breathing, level of arterial pressure and
other);
The use of averages in
health protection

for estimation of these medical-social and
sanitary-hygienic researches (middle
number of laboratory researches, middle
norms of food ration, level of radiation
contamination and others).
Averages

Averages are widely used for comparison in
time, that allows to characterize the major
conformities to the law of development of the
phenomenon. So, for example, conformity to
the law of growth increase of certain age
children finds the expression in the
generalized indices of physical development.
Conformities to the law of dynamics (increase
or diminishment) of pulse rate, breathing,
clinical parameters at the certain diseases
find the display in statistical indices which
represent the physiology parameters of
organism and other.
Average Values
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Mean:  the average of the data
 sensitive to outlying data
Median:  the middle of the data
 not sensitive to outlying data
Mode:  most commonly occurring value
Range:  the difference between the largest observation and
the smallest
Interquartile range:  the spread of the data
 commonly used for skewed data
Standard deviation:  a single number which measures how much
the observations vary around the mean
Symmetrical data:  data that follows normal distribution
 (mean=median=mode)
 report mean & standard deviation & n
Skewed data:  not normally distributed
 (meanmedianmode)
 report median & IQ Range
Average Values
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Limit is it is the meaning of edge variant
in a variation row
lim = Vmin Vmax
Average Values
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Amplitude is the difference of edge
variant of variation row
Am = Vmax - Vmin
Average Values
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Average quadratic deviation
characterizes dispersion of the variants
around an ordinary value (inside
structure of totalities).
Average quadratic deviation
σ=
d
2
n 1
simple arithmetical method
Average quadratic deviation
d=V-M
genuine declination of variants from the true
middle arithmetic
Average quadratic deviation
d

σ=i
n
2
p
  dp 


 n 


method of moments
2
Average quadratic deviation
is needed for:
1. Estimations of typicalness of the middle
arithmetic (М is typical for this row, if σ is less
than 1/3 of average) value.
2. Getting the error of average value.
3. Determination of average norm of the
phenomenon, which is studied (М±1σ), sub
norm (М±2σ) and edge deviations (М±3σ).
4. For construction of sigmal net at the
estimation of physical development of an
individual.
Average quadratic deviation
This dispersion a variant around of
average characterizes an average
quadratic deviation (  )
2
nd


n
 Coefficient
of variation is the
relative measure of variety; it
is a percent correlation of
standard deviation and
arithmetic average.
Terms Used To Describe The
Quality Of Measurements
Reliability is variability between subjects
divided by inter-subject variability plus
measurement error.
 Validity refers to the extent to which a test
or surrogate is measuring what we think it
is measuring.
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Measures Of Diagnostic Test
Accuracy
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Sensitivity is defined as the ability of the test to identify
correctly those who have the disease.
Specificity is defined as the ability of the test to identify
correctly those who do not have the disease.
Predictive values are important for assessing how
useful a test will be in the clinical setting at the individual
patient level. The positive predictive value is the
probability of disease in a patient with a positive test.
Conversely, the negative predictive value is the
probability that the patient does not have disease if he
has a negative test result.
Likelihood ratio indicates how much a given diagnostic
test result will raise or lower the odds of having a disease
relative to the prior probability of disease.
Measures Of Diagnostic Test
Accuracy
Expressions Used When
Making Inferences About Data
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Confidence Intervals
- The results of any study sample are an estimate of the true value
in the entire population. The true value may actually be greater or
less than what is observed.
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Type I error (alpha) is the probability of incorrectly
concluding there is a statistically significant difference in
the population when none exists.
Type II error (beta) is the probability of incorrectly
concluding that there is no statistically significant
difference in a population when one exists.
Power is a measure of the ability of a study to detect a
true difference.
Multivariable Regression
Methods
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Multiple linear regression is used when the
outcome data is a continuous variable such as
weight. For example, one could estimate the
effect of a diet on weight after adjusting for the
effect of confounders such as smoking status.
Logistic regression is used when the outcome
data is binary such as cure or no cure. Logistic
regression can be used to estimate the effect of
an exposure on a binary outcome after adjusting
for confounders.
Survival Analysis
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Kaplan-Meier analysis measures the ratio of
surviving subjects (or those without an event)
divided by the total number of subjects at risk for
the event. Every time a subject has an event, the
ratio is recalculated. These ratios are then used
to generate a curve to graphically depict the
probability of survival.
Cox proportional hazards analysis is similar to
the logistic regression method described above
with the added advantage that it accounts for
time to a binary event in the outcome variable.
Thus, one can account for variation in follow-up
time among subjects.
Kaplan-Meier Survival Curves
Why Use Statistics?
Cardiovascular Mortality in Males
1.2
1
0.8
SMR 0.6
0.4
0.2
0
'35-'44 '45-'54 '55-'64 '65-'74 '75-'84
Bangor
Roseto
Descriptive Statistics
Identifies patterns in the data
 Identifies outliers
 Guides choice of statistical test

Percentage of Specimens Testing
Positive for RSV (respiratory syncytial virus)
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun
South 2
2
5
7
20
30
15
20
15
8
4
3
North- 2
east
West 2
3
5
3
12
28
22
28
22
20
10
9
2
3
3
5
8
25
27
25
22
15
12
2
2
3
2
4
12
12
12
10
19
15
8
Midwest
Descriptive Statistics
Percentage of Specimens Testing Postive for
RSV 1998-99
35
30
25
20
15
10
5
0
South
Northeast
West
Midwest
Jul
Sep
Nov
Jan
Mar
May
Jul
Distribution of Course Grades
14
12
10
Number of 8
Students 6
4
2
0
A
A- B+ B
B- C+ C
Grade
C- D+ D
D-
F
Describing the Data
with Numbers
Measures of Dispersion
•
•
•
RANGE
STANDARD DEVIATION
SKEWNESS
Measures of Dispersion
• RANGE
highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
•
•
•
Measures of Dispersion
• RANGE
highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
•
•
•
Measures of Dispersion
•
•
•
RANGE
• highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
The Normal Distribution
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Mean = median =
mode
Skew is zero
68% of values fall
between 1 SD
95% of values fall
between 2 SDs
Mean, Median, Mode
.
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2