lec 5 measures of variation

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SUMMARIZATION OFDATA - II

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In last lecture . . . . .

Descriptive statistics

� Frequency tables

�Graphical techniques

Measures of central value

�Mean�Median

�Mode



MEASURES OF VARIATION

Range

Standard Deviation

Quartiles Percentiles

Coefficient of Variation



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



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.



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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



Example: Standard Deviation

�Mean = 6.35, n=20

� Standard Deviation =

7(X - x) = 106.55

n 20

SD = 2.31

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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



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



PERCENTILES :

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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.



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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.



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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



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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.



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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



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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



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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.



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THE NORMAL DISTRIBUTION



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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



MEASURES OF DISEASEMEASURES OF DISEASEFREQUENCYFREQUENCY

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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.

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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.

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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.

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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.

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Measurement of morbidity

Morbidity has been defined as ´any departure,

subjective or objective, from physical well-beingµ.

1. Prevalence

2. Incidence

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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)

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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.

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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.

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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

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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

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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.

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lec 5 measures of variation

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