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Measures of Dispersion

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This term is used commonly to mean scatter, Deviation, Fluctuation, Spread or variability of data. The degree to which the individual values of the variate scatter away from the average or the central value, is called a dispersion. Types of Measures of Dispersions:
  • Absolute Measures of Dispersion: The measures of dispersion which are expressed in terms of original units of a data are termed as Absolute Measures.
  • Relative Measures of Dispersion: Relative measures of dispersion, are also known as coefficients of dispersion, are obtained as ratios or percentages. These are pure numbers independent of the units of measurement and used to compare two or more sets of data values.
http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur3.gif Absolute Measures
  • Range
  • Quartile Deviation
  • Mean Deviation
  • Standard Deviation
Relative Measure
  • Co-efficient of Range
  • Co-efficient of Quartile Deviation
  • Co-efficient of mean Deviation
  • Co-efficient of Variation.
The Range: 1.      The range is the simplest measure of dispersion.  It is defined as the difference between the largest value and the smallest value in the data: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur4.gif 2. For grouped data, the range is defined as the difference between the upper class boundary (UCB) of the highest class and the lower class boundary (LCB) of the lowest class. MERITS OF RANGE:-
  • Easiest to calculate and simplest to understand.
  • Gives a quick answer.
DEMERITS OF RANGE:-
  • It gives a rough answer.
  • It is not based on all observations.
  • It changes from one sample to the next in a population.
  • It can’t be calculated in open-end distributions.
  • It is affected by sampling fluctuations.
  • It gives no indication how the values within the two extremes are distributed
Quartile Deviation (QD): 1.      It is also known as the Semi-Interquartile Range.  The range is a poor measure of dispersion where extremely large values are present.  The quartile deviation is defined half of the difference between the third and the first quartiles: QD = Q3 – Q1/2 Inter-Quartile Range The difference between third and first quartiles is called the ‘Inter-Quartile Range’. IQR = Q3 – Q1 Mean Deviation (MD): 1.      The MD is defined as the average of the deviations of the values from an average: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur6.gif It is also known as Mean Absolute Deviation. 2.      MD from median is expressed as follows: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur7.gif 3.      for grouped data: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur8.gif http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur9.gif Mean Deviation:
  1. The MD is simple to understand and to interpret.
  2. It is affected by the value of every observation.
  3. It is less affected by absolute deviations than the standard deviation.
  4. It is not suited to further mathematical treatment.  It is, therefore, not as logical as convenient measure of dispersion as the SD.
The Variance:
  • Mean of all squared deviations from the mean is called as variance
  • (Sample variance=S2; population variance= σ2sigma squared (standard deviation squared). A high variance means most scores are far away from the mean, a low variance indicates most scores cluster tightly about the mean.
Variance2JPGFormula OR S2 = Calculating variance: Heart rate of certain patient is 80, 84, 80, 72, 76, 88, 84, 80, 78, & 78. Calculate variance for this data. Solution: Step 1: Find mean of this data measures-of-central-tendency-4 = 800/10 Mean = 80 Step 2: Draw two Columns respectively ‘X’ and deviation about mean (X- ). In column ‘X’ put all values of X and in (X- ) subtract each ‘X’ value with . Step 3: Draw another Column of (X- ) 2, in which put square of deviation about mean.
X (X- ) Deviation about mean (X- )2 Square of Deviation about mean
80 84 80 72 76 88 84 80 78 78 80 – 80 = 0 84 – 80 = 4 80 – 80 = 0 72 – 80 = -8 76 – 80 = -4 88 – 80 = 8 84 – 80 = 4 80 – 80 = 0 78 – 80 = -2 78 – 80 = -2 0 x 0 = 00 4 x 4 = 16 0 x 0 = 00 -8 x -8 = 64 -4 x -4 = 16 8 x 8 = 64 4 x 4 = 16 0 x 0 = 00 -2 x -2 = 04 -2 x -2 = 04
∑X = 800 = 80 ∑(X- ) = 0 Summation of Deviation about mean is always zero ∑(X- )2 = 184 Summation of Square of Deviation about mean
Step 4 Apply formula and put following values ∑(X- ) 2= 184 n = 10 Variance2JPG Variance = 184/ 10-1 = 184/9 Variance = 20.44 Standard Deviation
  • The SD is defined as the positive Square root of the mean of the squared deviations of the values from their mean.
  • The square root of the variance.
  • It measures the spread of data around the mean. One standard deviation includes 68% of the values in a sample population and two standard deviations include 95% of the values & 3 standard deviations include 99.7% of the values
  • The SD is affected by the value of every observation.
  • In general, it is less affected by fluctuations of sampling than the other measures of dispersion.
  • It has a definite mathematical meaning and is perfectly adaptable to algebraic treatment.
http://www.bmj.com/statsbk/2-20.gifFormula: OR S = Calculating Standard Deviation (we use same example): Heart rate of certain patient is 80, 84, 80, 72, 76, 88, 84, 80, 78, & 78. Calculate standard deviation for this data. SOLUTION: Step 1: Find mean of this data measures-of-central-tendency-4 = 800/10 Mean = 80 Step 2: Draw two Columns respectively ‘X’ and deviation about mean (X-). In column ‘X’ put all values of X and in (X-) subtract each ‘X’ value with. Step 3: Draw another Column of (X- ) 2, in which put square of deviation about mean.
X (X- ) Deviation about mean (X- )2 Square of Deviation about mean
80 84 80 72 76 88 84 80 78 78 80 – 80 = 0 84 – 80 = 4 80 – 80 = 0 72 – 80 = -8 76 – 80 = -4 88 – 80 = 8 84 – 80 = 4 80 – 80 = 0 78 – 80 = -2 78 – 80 = -2 0 x 0 = 00 4 x 4 = 16 0 x 0 = 00 -8 x -8 = 64 -4 x -4 = 16 8 x 8 = 64 4 x 4 = 16 0 x 0 = 00 -2 x -2 = 04 -2 x -2 = 04
∑X = 800 = 80 ∑(X- ) = 0 Summation of Deviation about mean is always zero ∑(X- )2 = 184 Summation of Square of Deviation about mean
Step 4 Apply formula and put following values ∑(X- )2 = 184 n = 10 sd calculation MERITS AND DEMERITS OF STD. DEVIATION
  • Std. Dev. summarizes the deviation of a large distribution from mean in one figure used as a unit of variation.
  • It indicates whether the variation of difference of a individual from the mean is real or by chance.
  • Std. Dev. helps in finding the suitable size of sample for valid conclusions.
  • It helps in calculating the Standard error.
DEMERITS-
  • It gives weightage to only extreme values. The process of squaring deviations and then taking square root involves lengthy calculations.
 Relative measure of dispersion: (a)    Coefficient of Variation, (b)   Coefficient of Dispersion, (c)    Quartile Coefficient of Dispersion, and (d)   Mean Coefficient of Dispersion. Coefficient of Variation (CV): 1.      Coefficient of variation was introduced by Karl Pearson.  The CV expresses the SD as a percentage in terms of AM: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur50.gif   —————- For sample data http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur51.gif   ————— For population data
  • It is frequently used in comparing dispersion of two or more series.  It is also used as a criterion of consistent performance, the smaller the CV the more consistent is the performance.
  • The disadvantage of CV is that it fails to be useful when  http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur52.gif   is close to zero.
  • It is sometimes also referred to as ‘coefficient of standard deviation’.
  • It is used to determine the stability or consistency of a data.
  • The higher the CV, the higher is instability or variability in data, and vice versa.
Coefficient of Dispersion (CD): If Xm and Xn are respectively the maximum and the minimum values in a set of data, then the coefficient of dispersion is defined as: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur53.gif Coefficient of Quartile Deviation (CQD): 1.      If Q1 and Q3 are given for a set of data, then (Q1 + Q3)/2 is a measure of central tendency or average of data.  Then the measure of relative dispersion for quartile deviation is expressed as follows: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur54.gif CQD may also be expressed in percentage. Mean Coefficient of Dispersion (CMD): The relative measure for mean deviation is ‘mean coefficient of dispersion’ or ‘coefficient of mean deviation’: http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur55.gif   ——————– for arithmetic mean http://www.freewebs.com/maeconomics/Statistics/measures_of_dispersion_files/measur56.gif   ——————– for median Percentiles and Quartiles The mean and median are special cases of a family of parameters known as location parameters. These descriptive measures are called location parameters because they can be used to designate certain positions on the horizontal axis when the distribution of a variable is graphed. Percentile:
  1. Percentiles are numerical values that divide an ordered data set into 100 groups of values with at the most 1% of the data values in each group. There can be maximum 99 percentile in a data set.
  2. A percentile is a measure that tells us what percent of the total frequency scored at or below that measure.
Percentiles corresponding to a given data value: The percentile in a set corresponding to a specific data value is obtained by using the following formula Number of values below X + 0.5 Percentile = ——————————————– Number of total values in data set Example: Calculate percentile for value 12 from the following data 13 11 10 13 11 10 8 12 9 9 8 9 Solution: Step # 01: Arrange data values in ascending order from smallest to largest
S. No 1 2 3 4 5 6 7 8 9 10 11 12
Observations or values 8 8 9 9 9 10 10 11 11 12 13 13
Step # 02: The number of values below 12 is 9 and total number in the data set is 12 Step # 03: Use percentile formula 9 + 0.5 Percentile for 12 = ——— x 100 = 79.17% 12 It means the value of 12 corresponds to 79th percentile Example2: Find out 25th percentile for the following data 6 12 18 12 13 8 13 11 10 16 13 11 10 10 2 14 SOLUTION Step # 01: Arrange data values in ascending order from smallest to largest
S. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Observations or values 2 6 8 10 10 10 11 11 12 12 13 13 13 14 16 18
Step # 2 Calculate the position of percentile (n x k/ 100). Here n = No: of observation = 16 and k (percentile) = 25 16 x 25 16 x 1 Therefore Percentile = ———- = ——— = 4 100 4 Therefore, 25th percentile will be the average of values located at the 4th and 5th position in the ordered set. Here values for 4th and 5th correspond to the value of 10 each. (10 + 10) Thus, P25 (=Pk) = ————– = 10 2 Quartiles These are measures of position which divide the data into four equal parts when the data is arranged in ascending or descending order. The quartiles are denoted by Q. quartile
Quartiles Formula for Ungrouped Data Formula for Grouped Data
Q1 = First Quartile below which first 25% of the observations are present Q1_ungroup
Q2 = Second Quartile below which first 50% of the observations are present. It can easily be located as the median value. Q2_ungroup Q2
Q3 = Third Quartile below which first 75% of the observations are present Q3_ungroup Q3
Symbol Key:

Epidemiology MCQs

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1. Which of the following is NOT a part of continuum of natural history of the disease?
a) Stage of Susceptibility
b) Stage of preclinical
c) Stage of prevention
d) Stage of recovery

2. Which of the following is also known as retrospective studies?
a) Cohort studies
b) Descriptive studies
c) Experimental studies
d) Case control studies

3. Total number of deaths reported during a given time interval from estimated mid-interval population is called;
a) death rate
b) Crude death rate
c) mortality rate
d) proportional mortality

4. Number of live births reported during a given time interval from estimated mid-interval population is called;
a) Birth Rate
b) Growth Rate
c) Crude Fertility rate
d) Crude Birth Rate

5. Number of live births reported during a given time interval from estimated number of women age 15 to 44 years mid interval is known as;
a) Crude Fertility Rate
b) Birth Rate
c) Growth Rate
d) Sex ratio

6. Number of current cases(new and old) of specified disease identified over a given time interval from estimated population at mid interval is called;
a) Prevalence
b) Period Prevalence
c) Point Prevalence
d) Disease Prevalence

7. Use of statistics to analyze characteristics or changes to a population is termed as;
a) population Pyramid
b) vital statistics
c) Population statistics
d) Population dynamics

 

8. Which of the following term provides true representation of whole population?
a) Sampling
b) Random Sampling
c) Case reporting
d) Sample
9. Measure of the frequency of occurrence of death in a defined population during a specified interval is called;
a) Crude death rate
b) Mortality Rate
c) Death ratio
d) Mortality

10. Public health surveillance DOES NOT consists on the following step;
a) Systematic collection
b) Analysis
c) Planning
d) Interpretation

11. Surveillance system information cycles include;
a) Family and community
b) Public, Health care provider and Health agencies
c) None of the above
d) Public, Health care provider only

12. Epidemiology can be defined as follow EXCEPT;
a) Distribution of health related states
b) Community leaders and their family crises
c) Determinant of health related events
d) Apply to the control of health problems

13. A state of disorder that results from communication ONLY by direct contact is termed as;
a) Infectious disease
b) Contamination
c) Epidemic
d) Contagious disease

14. Which of the following is NOT a basic measurement in epidemiology;
a) Rate
b) Nominator
c) Ratio
d) Proportion

15. Which of the following is usually expressed as percentage;
a) Rate
b) Nominator
c) Ratio
d) Proportion

 

16. Measurement of disease, disability or death and converting this information in to rates and ratio is defined as;
a) Specificity
b) Screening
c) Frequency
d) Sensitivity

17. Measurement of current status of disease is termed as;
a) Prevalence
b) Incidence
c) Cumulative Incidence
d) Mid interval population

18. A person who harbors the microorganisms of a disease and excretes them without self suffering from symptoms is called;
a) Reservoir
b) Carrier
c) Host
d) Agent

19. The modes of transmission of infectious diseases are as follow EXCEPT;
a) Direct
b) Indirect
c) Physiological
d) Biological

20. The number of new cases occurring in a defined population during a specified period of time is called;
a) Prevalence
b) Incidence
c) a and b
d) Cumulative incidence

21. Epidemiological methods can be categorized as follow;
a) Descriptive, cohort and case control
b) Descriptive, cross sectional and experimental
c) Descriptive, prospective and experimental
d) Descriptive, Analytical and experimental

22. In descriptive epidemiology disease described in terms of;
a) What, Why and How
b) Host, Agent and Environment
c) Time, Place and Person
d) Agent, Place and Person

23. Which of the following is also known as prospective study;
a) Cohort studies
b) Descriptive studies
c) Experimental studies
d) Case control studies

24. In epidemiological triad environmental factors can be classified as;
a) Physical
b) Chemical
c) Social
d) Biological

25. Which of the following ratio provide us an estimate of risk in case control study;
a) Odd ratio
b) Sex ratio
c) Disease ratio
d) Dependency ratio

26. The entire group of people or elements that have at least one thing is common is known as;
a) Sample
b) Parameter
c) Hypothesis
d) Population

27. Sampling done on the basis of some pre determined ideas and its result can not be generalized is defined as follow;
a) Snow ball sampling
b) Purposive sampling
c) Probability sampling
d) Non-probability sampling

28. Tertiary prevention includes;
a) Disability limitation
b) Prompt treatment
c) Rehabilitation
d) a and c
e) a and b

29. Agents such as vitamins, protein, fat etc. are an examples of;
a) Physical Agents
b) Nutritive Agents
c) Chemical Agents
d) All of the above

30. Which of the following are key components of Epidemiological triangle,
a) Host, Agent and Physical Environment
b) Host, Genes and Physical Environment
c) Host, Agent and Environment
d) None of the above

31. Tertiary prevention Does not includes;
a) Disability limitation
b) Prompt treatment
c) Rehabilitation
d) a and c
32. Agents such as vitamins, protein, fat etc. are an examples of;
a) Physical Agents
b) Nutritive Agents
c) Chemical Agents
d) All of the above
33. Which of the following are not key components of Epidemiological triangle,
a) Host and Agent
b) Host and Environment
c) Host, Agent and Environment
d) Time, Place and Person
34. Which of the following is a part of continuum of natural history of the disease?
a) Stage of health promotion
b) Stage of prevention
c) Stage of Recovery
d) Stage of sampling
35. Which of the following are also known as retrospective studies?
a) Cohort studies
b) Descriptive studies
c) Experimental studies
d) Case control studies

36. A person who harbors the microorganisms of a disease and excretes them without self suffering from symptoms is called;
a) Reservoir
b) Carrier
c) Host
d) Agent
37. The modes of transmission of infectious diseases are as follow EXCEPT;
a) Direct
b) Indirect
c) Physiological
d) Biological

38. Total number of deaths reported during a given time interval from estimated mid-interval population is called;
a) death rate
b) Crude death rate
c) mortality rate
d) proportional mortality
39. Number of live births reported during a given time interval from estimated mid-interval population is called;
a) Birth Rate
b) Growth Rate
c) Crude Fertility rate
d) Crude Birth Rate
40. Number of live births reported during a given time interval from estimated number of women age 15 to 44 years mid interval is known as;
a) Crude Fertility Rate
b) Birth Rate
c) Growth Rate
d) Sex ratio

41. Number of current cases(new and old) of specified disease identified over a given time interval from estimated population at mid interval is called;
a) Prevalence
b) Period Prevalence
c) Point Prevalence
d) Disease Prevalence
42. Use of statistics to analyze characteristics or changes to a population is termed as;
a) population Pyramid
b) vital statistics
c) Population statistics
d) Population dynamics
43. Measure of the frequency of occurrence of death in a defined population during a specified interval is called;
a) Crude death rate
b) Mortality Rate
c) Death ratio
d) Mortality
44. Public health surveillance DOES NOT consists on the following step;
a) Systematic collection
b) Analysis
c) Planning
d) Interpretation
45. Surveillance system information cycles include;
a) Family and community
b) Public, Health care provider and Health agencies
c) None of the above
d) Public, Health care provider only

46. A state of disorder that results from communication ONLY by direct contact is termed as;
a) Infectious disease
b) Contamination
c) Epidemic
d) Contagious disease
47. Which of the following is NOT a basic measurement in epidemiology;
a) Rate
b) Nominator
c) Ratio
d) Proportion
48. Measurement of current status of disease is termed as;
a) Prevalence
b) Incidence
c) Cumulative Incidence
d) Mid interval population
49. The number of new cases occurring in a defined population during a specified period of time is called;
a) Prevalence
b) Incidence
c) a and b
d) Cumulative incidence
50. Which of the following is also known as prospective study;
a) Cohort studies
b) Descriptive studies
c) Experimental studies
d) Case control studies

51. Which of the following ratio provide us an estimate of risk in case control study;
a) Odd ratio
b) Sex ratio
c) Disease ratio
d) Dependency ratio
52. The entire group of people or elements that have at least one thing is common is known as;
a) Sample
b) Parameter
c) Hypothesis
d) Population
53. Sampling done on the basis of some pre determined ideas and its result can not be generalized is defined as follow;
a) Snow ball sampling
b) Purposive sampling
c) Probability sampling
d) Non-probability sampling

54. Graphical illustration that shows the distribution of various age groups in population is known as;
a) Dependency Ratio
b) Age Ratio
c) Population Pyramid
d) Population Dynamics
55. Ratio of population who are economically not active to those who are economically active can be defined as;
a) Dependency Ratio
b) Age Ratio
c) Population Ratio
d) Risk benefit ratio
56. In which of the following sampling there is a minimum chance of bias and equally chances of being selected for study.
a) Accidental Sampling
b) Simple Random Sampling
c) Purposive Sampling
d) Snow ball Sampling
57. In study if we are selecting every seventh subject it comes under which of the following sampling method?
a) Stratified Sampling
b) Quota Sampling
c) Systematic Sampling
d) Purposive Sampling
58. Systematic errors produced by your sampling procedure is known as;
a) Sampling bias
b) Sampling errors
c) Non sampling errors
d) Random error
59. The profile of single patient is reported in detail by one or more clinicians is called as follow;
a) Case control study
b) Case Series
c) Investigation
d) Case Report
60. In which of the following study we compare one group among whom the problem is present and another group where problem is absent?
a) Case control study
b) Case Series
c) Cohort study
d) Case Report

Answer key:
1. C
2. D
3. B
4. D
5. A
6. B
7. C
8. D
9. B
10. C
11. B
12. B
13. D
14. B
15. D
16. C
17. A
18. B
19. C
20. B

21. D
22. C
23. A
24. B
25. A
26. D
27. B
28. D
29. B
30. C
31. B
32. B
33. D
34. C
35. D
36. B
37. C
38. B
39. D
40. A
41. B
42. C
43. B
44. C
45. B
46. D
47. B
48. A
49. B
50. A
51. A
52. D
53. B
54. C
55. A
56. B
57. C
58. A
59. D
60. A