lec 5 measures of variation
TRANSCRIPT
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 1/29
SUMMARIZATION OFDATA - II
1
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 2/29
In last lecture . . . . .
Descriptive statistics
� Frequency tables
�Graphical techniques
Measures of central value
�Mean�Median
�Mode
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 3/29
MEASURES OF VARIATION
Range
Standard Deviation
Quartiles Percentiles
Coefficient of Variation
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 4/29
Range:
It is defined as the difference between the highest
(maximum) and the lowest (minimum) observation e.g.
Heights of 7 women are
142, 141, 143, 144, 145, 146, 155 cm
Range= 155 ² 141
= 14 cm
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 5/29
Standard Deviation5
The STANDARD DEVIATION is a measure, which describes
how much individual measurements differ, on the average,
from the mean.
A large standard deviation shows that there is a wide
scatter of measured values around the mean, while a small
standard deviation shows that the individual values are
concentrated around the mean with little variation among
them.
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 6/29
6
STANDARD DEVIATION (SD)«..
SD =
Steps to calculate SD:1. Calculate mean of all observations
2. Calculate difference between each individual measurementand mean
3. Square all these differences
4. Take sum of all squared differences5. Divide this sum by number of measurements
6. Finally take the square root of value
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 7/29
Example: Standard Deviation
�Mean = 6.35, n=20
� Standard Deviation =
7(X - x) = 106.55
n 20
SD = 2.31
7
X X -x (X -x)²
3 -3.35 11.22
3 -3.35 11.22
4 -2.35 5.52
4 -2.35 5.52
4 -2.35 5.52
5 -1.35 1.82
5 -1.35 1.82
5 -1.35 1.82
6 -0.35 0.12
6 -0.35 0.12
6 -0.35 0.12
6 -0.35 0.12
7 0.65 0.42
7 0.65 0.42
8 1.65 2.728 1.65 2.72
9 2.65 7.02
10 3.65 13.32
10 3.65 13.32
11 4.65 21.62
Sum 0 106.55
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 8/29
QUARTILES8
The Points which divide the distribution of data into
four equal parts e.g.
If we want to find the points below which 25% and
50% values of the distribution lie, these are called
first and 2nd quartiles.
2nd quartile is also equal to median of the data
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 9/29
PERCENTILES :
9
Points, which divide all the measurements into 100
equal parts e.g.
3rd percentile (P3) ² value below which 3 % of
measurements lie.
50th percentile (P50) or median ² value belowwhich 50% of measurements lie.
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 10/29
10
COEFFICIENT OF VARIATION (C.V.)
Ratio of SD to the mean, expressed as a percentageµ
CV = SD/mean x 100 %
CV is used to compare variation of frequency distributions
measured in different units.
CV depicts the size of variation relative to the mean.
CV is independent of units of measurement.
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 11/29
11
EXAMPE:EXAMPE:
In two series of adults and children following values were
obtained for the height.
Find which series shows greater variation?
Persons Mean Height SD
Adults 160cm 10cm
children 60cm 5cm
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 12/29
12
EXAMPE: (cont·d ««)EXAMPE: (cont·d ««)
CV for adults = 10/160 x100 = 6.25%
CV for children = 5/60x100 = 8.33%
Conclusion: Thus, we find that heights in
children show greater variation than in
adults.
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 13/29
13
EXAMPE 2: (cont·d «..)EXAMPE 2: (cont·d «..)
In a sample of boys SBP and weight were measured as follows
Find which characteristic shows greater variation?
Characteristic Mean SD
SBP 120 kg 10
weight 60 kg 4
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 14/29
14
Solution:Solution:
CV of SBP = 10/120 x 100
= 8.33%
CV of height = 4/60 x 100
= 6.67 %
Conclusion: Thus, SBP is found to be a
more variable characteristic than height i.e. 8.33/6.67 = 1.25 times
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 15/29
15
THE NORMAL DISTRIBUTION
Many variables have a normal distribution. This is
a bell shaped curve with most of the values
clustered near the mean and a few values out
near the tails.
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 16/29
16
THE NORMAL DISTRIBUTION
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 17/29
17
The normal distribution is symmetrical around the
mean. The mean, median and the mode of a
normal distribution have the same value i.e.
mean = median = mode
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 18/29
MEASURES OF DISEASEMEASURES OF DISEASEFREQUENCYFREQUENCY
18
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 19/29
RATIO
It expresses a relation between two random quantities.
Obtained by simply dividing one quantity by another
without implying any specific relationship between the
numerator and denominator.
In ratio the numerator is not a part of denominator.
19
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 20/29
Example of Ratio
The ratio of white blood cells to red cells is 1:600 or
1/600 meaning that for each white cell there are 600
red cells.
Other examples are Sex-ratio, Doctor-population
Ratio etc.
20
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 21/29
PROPORTION
A proportion is a type of ratio in which those
who are included in the numerator must also be
included in the denominator.
For example: The number of children with
scabies out of the total number of children in
the village at the same time.
21
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 22/29
RATE:
A rate measures the occurrence of some particularevent in a population during a given time period.
There is a distinct relationship between the numeratorand denominator with a measure of time being apart of the denominator.
For example: the number of newly diagnosed cases
of breast cancer per 100,000 women during a givenyear.
22
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 23/29
Measurement of morbidity
Morbidity has been defined as ´any departure,
subjective or objective, from physical well-beingµ.
1. Prevalence
2. Incidence
23
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 24/29
PREVALENCE
The proportion of individuals in a population who have the
disease at a specific time.
It provides an estimate of the probability (risk) that an
individual will be ill at a point in time.
The formula for calculating the prevalenceformula for calculating the prevalence
Number Of Existing Cases Of A Disease
P = ---------------------------------------------------------------------
Total population (at a given point in time)
24
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 25/29
POINT PREVALENCE
It is defined as the number of all cases (old and new) of
a disease at one point of time, in a defined population.
This point of time may be a day, several days or even
weeks depending upon time it takes to examine the
population sample.
25
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 26/29
PERIOD PREVALENCE
It represents the proportion of cases that exist within a population at
any point during a specified period of time.
The numerator thus includes cases that were present at the start of
the period plus new cases that developed during this time.
E.g. Frequency of Hypertensive patients between May 31 ² Dec 01
2008.
26
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 27/29
INCIDENCE:
It is defined as ´The number of new cases occurring in
a defined population during a specified period of
time.µ it is calculated by
Number of new cases of
specific disease during a
specific time period
Incidence = ---------------------------------------- x 1000
Population at risk during that period
27
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 28/29
For Example
There had been 500 new cases of an illness in a
population of 30,000 in a year, the incidence would
be:
500
incidence = ------------------------ x 1000
30,000= 16.7 per 1000 per year
28
8/6/2019 Lec 5 Measures of Variation
http://slidepdf.com/reader/full/lec-5-measures-of-variation 29/29
MORTALITY RATE
It expresses the incidence of deaths in a particular
population during a period of time.
It is calculated by dividing the number of fatalities
during that period by the total population.
This can be further divided into cause specific or all
case mortality.
29