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WHAT IS DISPERSION | MEASURES OF DISPERSION

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What is meant by Dispersion? The extent to which individual observations spread out from the central value is called dispersion. There are various measures which are used to express the degree of variation quantitatively. These measures are called measures of variation. Let’s discuss in detail measures of dispersion:-

MEASURES OF DISPERSION

There are two kinds of measures of dispersion:
(a)        Absolute Measures of Dispersion
(b)        Relative Measures of Dispersion
Measures of Absolute Dispersion       Q3 class
(i)         Range (ii)        Quartile Deviation
(iii)       Mean Deviation           (iv)       Standard Deviation and Variance

RANGE:

Range is the difference between the largest observation and the smallest observation in two lies between the set of data. Suppose X m is the largest observation and Xo is the smallest observation, the range denoted by R is given by:         Q3 = 69
Example: Find the range of following data:
8, 10, 5, 4, 12, 15 and 9.
Solution: I’m 15           and Xo = 4      MEAN
R = Xm -Xo= 15-4 = 11          abbrev;
Absolute measures are often presented within original data and can be expressed in units. It just tells us how much spread is there between the values from their mean. Absolute measures present the dispersion in same units or square of units in which the data is recorded. A relative measure is expressed in terms of absolute dispersion relative to the average. A relative measure is free from a unit of measurement.

QUARTILE DEVIATION:

Quartile Deviation abbreviated Q.D. is half of the difference between third and first quartiles.
Q.D  = Q3-Q1 / 2

MEAN DEVIATION:

Mean Deviation is the average of absolute deviations measured from the mean. It is abbreviated as M.D.

STANDARD DEVIATION AND VARIANCE:

Standard deviation is the square root of the mean of squared deviations taken from arithmetic mean. Standard deviation is denoted by the symbol S for sample data and c for population data. Variance is just the square of standard deviation.
S=(For ungrouped Data)

PROPERTIES OF VARIANCE:

Variance remains unchanged If we add or subtract a constant from each observation in a set of data i.e.
Var (X± a) = Var (X), where X is a variable and ” a” is a constant.
(ii)        When we multiply or divide each observation in a set of data by a constant, variance is multiplied or divided                   by the square of the constant i.e.
x          var ( i’ X!
Var (ax) =        Var (X) and Var ( 7 )
(iii)       Suppose we have k subgroups having a number of observations in each group respectively and their means are  X k and variances are respectively. We can find the mean of all subgroups with the help of following formula:

The following measures of dispersion are used to calculate the relative dispersion:
(i)         Coefficient of Range
(ii)        Coefficient of Quartile Deviation
(iii)       Coefficient of Mean Deviation
(iv)       Coefficient of Variation(C.V.)

Coefficient of Range = x,n-xo / xm-xo

Coefficient of Q. D.  Qa-Q1 / Qa-Q1

CENTRAL TENDENCY | MEASURES OF CENTRAL TENDENCY

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What is meant by Central Tendency? A measure of central tendency gives information about the central position of data. This information is usually very useful in understanding a set of data and is used widely in a number of problems. It has many drawbacks as well, although it is an effective tool of Statistics. For instance, there may be a number of sets of data having the same value of central tendency although the series is quite different from one another. Let us have a look at the following data sets and their arithmetic means:

(i)         50,50,50,50,50            X = 50

(ii)        40,45,50,55,60            X = 50

(iii)       30,40,50,60,70            X = 50

In all the above data sets, the value of arithmetic mean is 50 whereas each data set has different nature of individual values. It means that central tendency is only describing the location of each set of data and no other information is available other than the mean. Therefore it becomes necessary in such situations to see the manner in which the individual values in a set of data are spread away from the central location. The manner in which individual values are dispersed from the average is called dispersion and the measures which enable us to calculate the amount of dispersion, are called measures of dispersion.

MEASURES OF CENTRAL TENDENCY

A single value that describes a set of data by identifying the central position within that set of data is called the measure of central tendency or measure of central location. There are various measures which are used to find the central position of data. These measures are given as follows:
(1)Arithmetic Mean     (2) Geometric Mean
(3) Harmonic Mean    (4) Median       (5) Mode
We will explain these measures one by one.

ARITHMETIC MEAN

Arithmetic mean is a value obtained by dividing the sum of observations by the number of observations. Suppose we have n observations such as, so the Arithmetic Mean denoted by the symbol is X:
Weighted Arithmetic Mean:
The simple arithmetic mean is calculated where individual observations of the data same importance, but in real life, we observe that different commodities have different as these commodities are not purchased in the same quantity for e.g. we purchase more quantity of flour from the market as compared to salt. It means that we attach more importance to flour than Salt.
weighted arithmetic mean
Example7: A student obtained 40, 50, 80 and 70 marks in the subjects of Economics, Statistics Mathematics, and Urdu respectively. The weights assigned to these subjects were 2, 2, 2 and l’ respectively. Calculate the weighted arithmetic mean for marks in these subjects. Solution:
Subject            x          w
Economics      40        2          80
Statistics          80        2          160
Mathematics   60        2          120
Urdu                70        1          70
7          430
X= 61.43
PROPERTIES OF ARITHMETIC MEAN
The arithmetic mean has the following properties:
1) Sum of the deviations taken from arithmetic mean is always zero i.e.
2) Sum of the squared deviations taken from the arithmetic mean is minimum than a sum of the squared deviations taken from any other observation i.e. XI, R.
3) If we have k subgroups having means Xk with observations in each subgroup respectively, the combined mean for all n observations denoted by the symbol Xc is given by the formula:
MERITS OF ARITHMETIC MEAN:
1)         It is easy to calculate.
2)         It is easy to understand.
3)         It is based on all observations of data.
4)         It is capable of further algebraic treatment.
5)         It is a stable average as it is not much affected by changes in the sample.

DEMERITS OFARITHMETIC MEAN:
l)          It is affected by extreme observations.
2)         It is not a suitable average when open end classes are present in the data.
3)         The answer an f arithmetic mean is sometimes unrealistic .e.g. 7.3 men or 9.5 eggs etc.

GEOMETRIC MEAN:

The geometric mean of n positive observations is the nth root of their product. Suppose we have n observations such as X1′ X1……. Xn, the geometric mean of these n observations is given by.
G M. = Antilog 1/n logx

MERITS OF GEOMETRIC MEAN
1)         It is based on all observations of data.
2)         It is not affected by extreme observations.
3)         It is capable of further algebraic treatment.
4)         It is a stable average as it is not much affected by changes in the sample.

5)         Geometric mean is a suitable average when data consist of percentages.
DEMERITS OF GEOMETRIC MEAN
1)         It is not easy to calculate.
2)         It is not easy to understand.
3)         It is not a suitable average when open end classes are present in the data.

4) It cannot be determined if any observation is zero in the data.

HARMONIC MEAN:

Harmonic mean is the reciprocal of the arithmetic mean of reciprocal of the observations. Suppose we have n observations such X1′ X1……. Xn, the harmonic mean of these n observations is given by:
H M. = n/ Sum 1/x.

MERITS OF HARMONIC MEAN

1)         It is based on all observations of data.
2)         It is not affected by extreme large observations.
3)         It is capable of further algebraic treatment.
4)         It is a stable average as it is not much affected by changes in the sample.
DEMERITS OF HARMONIC MEAN:
1)         It is not easy to calculate.
2)         It is not easy to understand.
3)         It is not a suitable average when open end classes are present in the data.
4)         It cannot be determined if any observation is zero in the data.
Relation Between A.M., G.M., and H.M.

MEDIAN

Median is the middle value of arranging a set of data in ascending or descending order. When data contains odd number of values, a median is exactly the middle value. But when data contains even number of value, the median is the mean of two middle values.

X=n/2 th

MERITS OF MEDIAN:
1)         It is easy to understand.
2)         It is not affected by extreme observations.
3)         It is a suitable average when open-end classes are present in the data.
4)         It can also be determined from a diagram.
5)         It is possible to find median when data are of qualitative nature.
DEMERITS OF MEDIAN:
1)         It is not based on all observations of data.
2)         It is not easy to calculate. Arranging the observations in ascending or descending order is a very difficult task.

3)         It is not a stable average as it is affected by changes in the sample.

4)         It is not capable of further algebraic treatment.

QUARTILES, QUINTILES, DECILES AND PERCENTILES


Quartiles: Quartiles consist of a set of three values QI.Q2 and Qa that divide the arranged set of data into four equal parts. The data are arranged in ascending order only.

Quintiles: Quintiles consist of a set of four values QI, Q2, and Qa that divide the arranged set of data into five equal parts. The data are arranged in ascending order only.

Deciles: Deciles consist of a set of nine values DI, D2, . … .D’ that divide the arranged set of data into ten equal parts. The data are arranged in ascending order only.

Percentiles: Percentiles consist of a set of ninety-nine values PI, P-, and P99 that divide the arranged set of data into hundred equal parts. The data are arranged in ascending order only.

MODE:

The most repeated observation in a set of data is called mode. The symbol used for mode is     X.

PRESENTATION OF DATA | FREQUENCY DISTRIBUTION

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What is mean by a presentation of Data? There are various methods of collecting statistical raw data. Whatsoever method is used in the collection of data, the question arises; what is the procedure for presentation of data? Everyone is aware of the fact that raw data is in great bulk and it is very difficult to understand and handle data in this form. There are various statistical tools which are used to present data in a meaningful manner. These tools are helpful in classification, tabulation and graphical presentation of statistical data.
When statistical data are arranged according to some common characteristic or similarities, it is known as classified data. Data classification is the categorization of data for its most effective and efficient use. The statistical data may be classified according to geographical location, chronological order etc. The following are the various methods which are used for presentation of data.

FREQUENCY DISTRIBUTION| PRESENTATION OF DATA

A representation of statistical data in a tabular format is called a frequency distribution. A frequency distribution displays the number of observations a particular class. The intervals must be mutually exclusive and exhaustive.

CUMULATIVE FREQUENCY DISTRIBUTION:
A table showing the cumulative frequencies is called a cumulative frequency distribution. A cumulative frequency distribution is a summary of a set of data showing the number of items less than or equal to the upper class boundary of each class. A cumulative frequency distribution is also called less than frequency distribution.

DECUMULATIVE FREQUENCY DISTRIBUTION:
A decumulative frequency distribution is a summary of a set of data showing the number of items more than or equal to the lower class boundary of each class. A decumulative frequency distribution is also called more than frequency distribution.

RELATIVE FREQUENCY DISTRIBUTION

A relative frequency distribution is a tabular summary of a set of data showing the relative frequency of items in each of several non-overlapping classes. The relative frequency of a class is obtained by dividing the frequency of the class by the total of frequencies of the whole data. The relative frequency is the fraction or proportion of the total number of items belonging to a class.

PERCENTAGE RELATIVE FREQUENCY DISTRIBUTION

The relative frequency distribution expressed in percentage form is called a percentage frequency distribution.

GRAPHIC AND DIAGRAMMATIC REPRESENTATION OF DATA:


BAR CHART:
A bar chart is a diagram showing a number of bars of equal width and each bar is followed by a reasonable space. The height of each bar is taken according to the size of that class. A bar chart is usually used to express the data of geographical nature.

PIE CHART:
A pie chart or a circular diagram is represented by drawing a circle of 360 degrees and subdividing the circle into various sectors, where each sector is proportional to the quantity it represents.

HISTOGRAM

A histogram is a graph of frequency distribution. The graph is represented by a set of adjacent rectangles whose width is equal to the size of a class and whose area is proportional to the frequency of the corresponding class.

Frequency Curve:

Year    HDI
2001    0.4454
2002    0.4542
2003    0.4630
2004    0.4718
2005    0.4807
2006    0.4860
2007    0.4910
2008    0.4940

A frequency curve is a freehand smoothed curve which is plotted by taking mid points of classes on the horizontal axis and the frequencies along vertical axis for drawing a freehand curve through these mid points. It can also be obtained by joining the mid points of the rectangles of a histogram by means of a smooth curve. The frequency polygon is different from a frequency curve which is obtained by joining the mid points of class intervals with the help of straight liars.

HISTORIGRAM:

The graph of time series data is called historigram. To construct a historigram, the horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring.

WHAT IS DATA COLLECTION | METHODS OF DATA COLLECTION

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What is Data Collection? We know that Data Collection is an important element in Survey method. data are collected in various ways.Different methods can be adopted for the data collection e.g. questionnaire method, interview method including personal interviews and telephone interview. Mail survey method is also utilized. Let’s discuss in detail various ways of data collection;-

METHODS OF DATA COLLECTION

MAIL SURVEY

Mail surveys are used to distribute self-administered questionnaires that respondents fill out on their own.
Advantages
One advantage of mail surveys is that they usually can be completed quickly. They are the best for dealing with highly personal or embarrassing topics, especially when the anonymity of respondents is preserved. Mailed questionnaires have been very useful in social research. It is useful for highly selective respondents with a strong interest in the subject matter and with greater education. When respondents are widely widespread geographically it seems very useful.
Disadvantages
This data collection technique has some disadvantages also. As the respondents are not able to ask questions, the questionnaire must be self-explanatory. A researcher has less control over the order in which the respondents answer the questions. Respondent’s bias is also a serious problem with mail surveys. Low response rate, whether a failure to complete or failure to return the survey, is a major problem in mail surveys. Cost of mailing, waiting for time and degree of response must be carefully considered. Mail questionnaires have serious defects that include possible lack of response and the inability to check the responses given. Responses to mail questionnaires are generally poor. As a result of low returns, valid generalizations cannot be made. To choose between interview schedules or mail the questionnaire is highly important; their relative advantage for the purpose and conditions of the project must be considered. The best advice would be to avoid mail questionnaire if a better method can possibly be used. In this method, every effort should be made to obtain returns of at least 80 to 90 percent or more. Then its results can be reliable.

INTERVIEW SCHEDULE

When personal interviews are used to collect survey data, respondents are usually contacted in their homes or in a shopping mall. Trained interviewers administer the questionnaire. The personal interview allows greater flexibility in asking questions then does the mail survey. In a personal interview, the respondent can obtain clarification, when questions are unclear and the trained interviewer can follow up incomplete or ambiguous answers to open-ended questions. Interviewers are oriented’ trained and sent out with complete instructions as to whom to interview and how the interview is to be handled.

The interviewer controls the order of questions and can ensure that all respondents complete the questions in the same order. The personal interview can be very helpful in learning a respondent’s own estimate of his reasons for doing or believing something. When asked reasons for his actions, intentions or attitudes, a person may say he has done’ something, intends to do something, or feels a certain way about something. He may specify the details,

A significant disadvantage of conducting personal interviews is the cost. The use of trained interviewers is expensive in terms of both money and time, a potential for interviewer bias is the most critical disadvantage of this technique. Interviewer bias occurs when the interviewer records only selected of a Portions of the respondent’s answers or try to adjust the wording question to fit the respondent.

TELEPHONE INTERVIEWS

For brief surveys, telephone interviews have become the method of choice. Telephone interviewing also provide better access to the dangerous neighborhood, locked buildings and respondents available only during evening hours.
Telephone surveys have little to recommend them beyond speed and low cost; especially when the interviewer is unknown to the respondents. They are limited by possible non-response, un-cooperativeness and by a reluctance to answer more than simple, superficial questions. This technique also has some drawbacks. It is a matter of question that how long respondents are willing to stay on the phone and individuals may respond differently when talking on a phone. Many people are less willing to be interviewed. Sometimes it becomes difficult to approach the people because options have made it easier for people to avoid unwanted calls.

WHAT IS QUESTIONNAIRE | TOOL OF SURVEY

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What is questionnaire? To collect information about the problem under study different can be utilized. It can be the questionnaire or any other prescribed test to scale. The main task is to translate the research question into an interview or other instrument constructed for the survey. Selection of tool is determined according to the nature Of sample. If the sample consists of illiterate persons the interview schedule or performance tests may be recommended.

TOOL OF SURVEY | QUESTIONNAIRE


For a good tool of a survey, the questionnaire must be flawless and according to the problem under study.  In general, the word form refers to a tool for securing answers to queries by employing a type that and therefore the stuffed respondents in by the enquirer fill in by themselves. however once a form is asked during a face to face scenario with a subject, it’s known as interview schedule. Constructing a form involve deciding what info ought to be sought-after and therefore the style of form writing a draft of form, ‘protesting the form and last with specifying the procedure for its use.The wording of questionnaire should be clear and specific using simple, direct and familiar vocabulary. The order in which questions are asked on a questionnaire needs to be considered seriously because the order can affect respondents answer. Usually, two general types of questions are chosen.

  • Structured items (close-ended).
  • Unstructured items (open-ended).

In open-ended items, the subject is more likely to tell what is more important to him/her. Close-ended items may be more difficult to construct, but they are more readily scored. Open-ended items can be prepared easily but not the scoring. Some types of questions are automatically structured because of the categories of answer e.g. when subject is asked married, divorced, single etc,
Monthly income;
Education                       Metric, F.A, B.A, M.A
Number of simpling        O, 1, 2, 3, 4, 5, 6, etc.
Likewise, many questions cannot be structured easily; it happens when responses to the item cannot be anticipated in detail e.g. Due to which factors a young person should join the Party Politics?
A questionnaire must be limited in its length and scope. In general, an interview should not extend much beyond half an hour. Likewise, a self-administering questionnaire should not require more than 30 minutes to complete, and an even shorter period is desirable. Before formulation of a questionnaire, it is the moral obligation of the researcher to learn as much as possible about the subject matter. Moreover, every item of questionnaire must be logically related to the central problem. The researcher should also consult his colleagues and friends to get their thinking on this problem. Related literature may also be consulted, After taking all these steps a big item pool may be constructed, then each item must be judged carefully upon its own merit. Order of questions must be observed carefully. Introducing items must be attention catching. The beginning should have the power to evoke interest, without arousing strong controversial response simpler items should be placed first and withholding the more complex one. Respondents should never be asked to give an answer which could be embarrassing. All questions should be in logical order; respondents should be brought as smoothly as possible from one frame of reference to another. Pilot Study; After the formulation of questionnaire pilot study or pre-test helps the researcher to remove the mistakes. If a questionnaire is poorly constructed then results of the survey will become useless. A questionnaire must yield reliable and valid measures of demographic variables and of individual differences on the self-report scale.