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Q1. What are the characteristics of a good measure of central tendency? [5Marks](b) What are the uses of averages? [5 Marks]Answer:a).
The characteristics of a good measure of central tendency are:
Present mass data in a concise form
The mass data is condensed to make the data readable and to use it for further analysis.•
Facilitate comparison
It is difficult to compare two different sets of mass data. But we can compare those twoafter computing the averages of individual data sets.While comparing, the same measure of average should be used. It leads to incorrectconclusions when the mean salary of employees is compared with the median salary of the employees.•
Establish relationship between data sets
The average can be used to draw inferences about the unknown relationships betweenthe data sets. Computing the averages of the data sets is helpful for estimating theaverage of population.•
Provide basis for decision-making
In many fields, such as business, finance, insurance and other sectors, managerscompute the averages and draw useful inferences or conclusions for taking effectivedecisions.The following are the requisites of a measure of central tendency:•It should be simple to calculate and easy to understand•It should be based on all values•It should not be affected by extreme values•It should not be affected by sampling fluctuation•It should be rigidly defined•It should be capable of further algebraic treatment b) Appropriate Situations for the use of Various Averages
1. Arithmetic mean is used when:a. In depth study of the variable is neededb. The variable is continuous and additive in naturec. The data are in the interval or ratio scaled. When the distribution is symmetrical2. Median is used when:a. The variable is discreteb. There exists abnormal valuesc. The distribution is skewedd. The extreme values are missinge. The characteristics studied are qualitativef. The data are on the ordinal scale3. Mode is used when:a. The variable is discreteb. There exists abnormal valuesc. The distribution is skewedd. The extreme values are missinge. The characteristics studied are qualitative4. Geometric mean is used when:a. The rate of growth, ratios and percentages are to be studiedb. The variable is of multiplicative nature5. Harmonic mean is used when:a. The study is related to speed, timeb. Average of rates which produce equal effects has to be found

(1) Plural Sense: In plural sense, the word statistics refer to numerical facts and figures collected in a systematic manner with a definite purpose in any field of study. In this sense, statistics are also aggregates of facts which are expressed in numerical form. For example, Statistics on industrial production, statistics or population growth of a country in different years etc. (3) Plural of Word “Statistic”: The word statistics is used as the plural of the word “Statistic” which refers to a numerical quantity like mean, median, variance etc…, calculated from sample value.

Quota sampling is a non-probability sampling technique wherein the assembled sample has the same proportions of individuals as the entire population with respect to known characteristics, traits or focused phenomenon. by Joan Joseph Castillo (2009)
In addition to this, the researcher must make sure that the composition of the final sample to be used in the study meets the research’s quota criteria.
EXAMPLE OF QUOTA SAMPLES
In a study wherein the researcher likes to compare the academic performance of the different high school class levels, its relationship with gender and socioeconomic status, the researcher first identifies the subgroups.
Usually, the subgroups are the characteristics or variables of the study. The researcher divides the entire population into class levels, intersected with gender and socioeconomic status. Then, he takes note of the proportions of these subgroups in the entire population and then samples each subgroup accordingly.
WHEN TO USE QUOTA SAMPLES * The main reason why researchers choose quota samples is that it allows the researchers to sample a subgroup that is of great interest to the study. If a study aims to investigate a trait or a characteristic of a certain subgroup, this type of sampling is the ideal technique. * Quota sampling also allows the researchers to observe relationships between subgroups. In some studies, traits of a certain subgroup interact with other traits of another subgroup. In such cases, it is also necessary for the researcher to use this type of sampling technique.

Secular Trend:
Secular Trend is also called long term trend or simply trend. The secular trend refers to the general tendency of data to grow or decline over a long period of time. For example the population of India over years shows a definite rising tendency. The death rate in the country after independence shows a falling tendency because of advancement of literacy and medical facilities. Here long period of time does not mean as several years. Whether a particular period can be regarded as long period or not in the study of secular trend depends upon the nature of data. For example if we are studying the figures of sales of cloth store for 1996-1997 and we find that in 1997 the sales have gone up, this increase can not be called as secular trend because it is too short period of time to conclude that the sales are showing the increasing tendency. On the other hand, if we put strong germicide into a bacterial culture, and count the number of organisms still alive after each 10 seconds for 5 minutes, those 30 observations showing a general pattern would be called secular movement.
Mathematically the secular trend may be classified into two types 1. Linear Trend 2. Curvi-Linear Trend or Non-Linear Trend. If one plots the trend values for the time series on a graph paper and if it gives a straight line then it is called a linear trend i.e. in linear trend the rate of change is constant where as in non-linear trend there is varying rate of change.
Down ward linear trend
Upward linear trend

Non-linear trend
Value
of variable Value of variable

Non linear trend
Linear trend Time period Time period

The following methods are generally used to determine trend in any given time series. 1) Free hand curve method or eye inspection method 2) Semi average method 3) Method of moving average 4) Method of least squares 1) Free hand curve method or eye inspection method Free hand curve method is the simplest of all methods and easy to under stand. The method is as follows. First plot the given time series data on a graph. Then a smooth free hand curve is drawn through the plotted points in such a way that it represents general tendency of the series. As the curve is drawn through eye inspection, this is also called as eye-inspection method. The free hand curve method removes the short term variations to show the basic tendency of the data. The trend line drawn through the free hand curve method can be extended further to predict or estimate values for the future time periods. As the method is subjective the prediction may not be reliable. There is another method which is adopted while drawing a free hand curve called as method of selected points. By this method we select points on the graph of the original data and draw a smooth curve through these points. For example if we want to draw a straight line trend, two characteristic points are selected on the graph and a line is drawn through these points.

Example: year | production of cotton | 1971 | 91 | 1972 | 111 | 1973 | 136 | 1974 | 412 | 1975 | 720 | 1976 | 900 | 1977 | 1206 | 1978 | 1322 | Original line

Trend line

2) Method of Semi Averages:
In this method the whole data is divided in two equal parts with respect to time. For example if we are given data from 1979 to 1996 i.e. over a period of 18 years the two equal parts will be first nine years i.e. from 1979 to 1987 and 1988 to 1996. In case of odd number of years like 9, 13, 17 etc. two equal parts can be made simply by omitting the middle year. For example if the data are given for 19 years from 1978 to 1996 the two equal parts would be from 1978 to 1986 and from 1988 to 1996, the middle year 1987 will be omitted. After the data have been divided into two parts, an average (arithmetic mean) of each part is obtained. We thus get two points. Each point is plotted against the mid year of the each part. Then these two points are joined by a straight line which gives us the trend line. The line can be extended downwards or upwards to get intermediate values or to predict future values. year | production | Semi averages | 1971 | 40 | | 1972 | 45 | | 1973 | 40 | | 1974 | 42 | | 1975 | 46 | | 1976 | 52 | | 1977 | 56 | | 1978 | 61 | |

3) Method of Moving Average: It is a method for computing trend values in a time series which eliminates the short term and random fluctuations from the time series by means of moving average. Moving average of a period m is a series of successive arithmetic means of m terms at a time starting with 1st, 2nd, 3rd so on. The first average is the mean of first m terms; the second average is the mean of 2nd term to (m+1)th term and 3rd average is the mean of 3rd term to (m+2)th term and so on. If m is odd then the moving average is placed against the mid value of the time interval it covers. But if m is even then the moving average lies between the two middle periods which does not correspond to any time period. So further steps has to be taken to place the moving average to a particular period of time. For that we take 2-yearly moving average of the moving averages which correspond to a particular time period. The resultant moving averages are the trend values. Seasonal Variations: Seasonal variations occur in the time series due to the rhythmic forces which occurs in a regular and a periodic manner with in a period of less than one year. Seasonal variations occur during a period of one year and have the same pattern year after year. Here the period of time may be monthly, weekly or hourly. But if the figure is given in yearly terms then seasonal fluctuations does not exist. There occur seasonal fluctuations in a time series due to two factors. 1) Due to natural forces 2) Man made convention. The most important factor causing seasonal variations is the climate changes in the climate and weather conditions such as rain fall, humidity, heat etc. act on different products and industries differently. For example during winter there is greater demand for woolen clothes, hot drinks etc. Where as in summer cotton clothes, cold drinks have a greater sale and in rainy season umbrellas and rain coats have greater demand. Though nature is primarily responsible for seasonal variation in time series, customs, traditions and habits also have their impact. For example on occasions like dipawali, dusserah, Christmas etc. there is a big demand for sweets and clothes etc., there is a large demand for books and stationary in the first few months of the opening of schools and colleges. There are different devices to measure the seasonal variations. These are 1. Method of simple averages. 2. Ratio to trend method 3. Ratio to moving average method 4. Link relative method. 3. Ratio to moving average method: The ratio to moving average method is also known as percentage of moving average method and is the most widely used method of measuring seasonal variations. The steps necessary for determining seasonal variations by this method are 1. Calculate the centered 12-monthly moving average (or 4-quarterly moving average) of the given data. These moving averages values will eliminate S and I leaving us T and C components. 2. Express the original data as percentages of the centered moving average values. 3. The seasonal indices are now obtained by eliminating the irregular or random components by averaging these percentages using A.M or median. 4. The sum of these indices will not in general be equal to 1200 (for monthly) or 400 (for quarterly). Finally the adjustment is done to make the sum of the indices to a total of 1200 for monthly and 400 for quarterly data by multiplying them through out by a constant K which is given by for monthly

for quarterly Merits: 1. Of all the methods of measuring seasonal variations, the ratio to moving average method is the most satisfactory, flexible and widely used method. 2. The fluctuations of indices based on ratio to moving average method is less than based on other methods. Demerits: 1. This method does not completely utilize the data. For example in case of 12-monthly moving average seasonal indices cannot be obtained for the first 6 months and last 6 months.

4. Link relative method: This method is slightly more complicated than other methods. This method is also known as Pearson’s method. This method consists in the following steps. 1. The link relatives for each period are calculated by using the below formula

2. Calculate the average of the link relatives for each period for all the years using mean or median. 3. Convert the average link relatives into chain relatives on the basis of the first season. Chain relative for any period can be obtained by

the chain relative for the first period is assumed to be 100. 4. Now the adjusted chain relatives are calculated by subtracting correction factor ‘kd’ from (k+1)th chain relative respectively.
Where k = 1,2,…….11 for monthly and k = 1,2,3 for quarterly data. and where N denotes the number of periods i.e. N = 12 for monthly N = 4 for quarterly 5. Finally calculate the average of the corrected chain relatives and convert the corrected chain relatives as the percentages of this average. These percentages are seasonal indices calculated by the link relative method. Merits: 1. As compared to the method of moving average the link relative method uses data more. completely. Demerits: 1. The link relative method needs extensive calculations compared to other methods and is not as simple as the method of moving average. 2. The average of link relatives contains both trend and cyclical components and these components are eliminated by applying correction.