The one single value that reflects the nature and characteristics of the entire given data is called as central value. Central tendency refers to the middle point of a given distribution. It is other wise called as ‘measures of location’. The nature of this value is such that it always lies between the highest value and the lowest value of that series. In other wards, it lies at the centre or at the middle of the series.
Yule and Kendall have pointed out some basic characteristics which an average should satisfy to call it as good average. They are:
Different experts have defined differently to the concept of average. Gupta (2008) in his work has narrated Lawrence J. Kaplan definition as ‘one of the most widely used set of summery figures is known as measures of location, which are often referred to as averages, measures of central tendency or central location. The purpose of computing an average value for a set of observation is to obtain a single value which is representative of all the items and which the mind can grasp simply and quickly. The single value is the point of location around which the individual items cluster’. This opinion clearly narrates the basic purpose of computing an average. Similarly, Croxton and Cowden define the concept as ‘an average is a single value within the range of the data that is used to represent all of the values in the series. Since the average is somewhere within the range of data, it is sometimes called a measure of central value’.
Following five are frequently used types of an average or measure of central tendency. They are
All the above five types are discussed below in detail.
Arithmetic mean is the most simple and frequently used technique of computing central tendency. The average is also called as mean. It is other wise called as a single number representing a whole data set. It can be computed in a several ways. Commonly it can be computed by dividing the total value by the number of observations. Let ‘n’ be the number of items in a case. Each individual item in a list can be represented in a relationship as x1, x2, x3, …,xn. In this relationship, ‘x1′ is one value, ‘x2′ is another value in the series and the value extends upto a particular limit represented by ‘xn’. The dots in the relationship express that there are some values between the two extremes which are omitted in the relationship. Some people interprets the same relationship as, which can be read as ‘x-sub-i, as i runs from 1 upto n’.
In case the numbers of variable in list is more, then it requires a long space for deriving the mean. Thus the summation notation is used to describe the entire relationship. The above relationship can be derived with the help of summation as: , representing the sum of the ‘x’ values, using the index ‘i’ to enumerate from the starting value i =1 to the ending value i = n. thus we have
and the average can be represented as
The symbol ‘i’ is again nothing but a continuing covariance. The readers should not be confused while using the notation , rather they can also use or or any other similar notation which are of same meaning.
The mean of a series can be calculated in a number of ways. Following are some basic ways that are commonly used in researchers related to management and social sciences, particularly by the beginners. However, the readers should not be confused on sample mean and population mean. A sample of a population of ‘n’ observations and the mean of sample is denoted by ‘’. Where as when one measure the population mean i.e., the entire variables of a study than the mean is represented by the symbol ‘Aµ’, which is pronounced as ‘mue’ and is derived from the Greek letter ‘mu’. Below we are discussing the concepts of sample mean.
a. Direct method- Mean or average can be calculated directly in the following way
Step-1: First of all the researcher has to add all the observations of a given series. The observations are x1, x2, x3, …xn.
Step-2- Count how many observations are their in that series (n)
Step-3- the following procedure than adopted to get the average.
Thus the average or mean denoted as ‘’and can be read as ‘x bar’ is derives as:
Thus it can be said that the average mark of the final contestants in the quiz competition is 67.6 marks which can be rounded over to 70 marks.
b. Short-cut method- The average or mean can also be calculated by using short-cut method. This method is applicable when a particular series is having so many observations. In other wards, to reduce calculations this method is generally used. The steps of calculating mean by this method is as follows:
i. The research has to assume any one value from the entire series. This value is called as assumed value. Let this value be denoted here as ‘P’.
ii. Differentiate each a value from this assumed vale. That is find out individual values of each observation. Let this difference value be denoted as ‘B’. Hence B=xn-P where n= 1,2,3,…n.
iii. Add all the difference value or get sum of B and count the number of observation ‘n’.
iv. Putting the values in the following formula and get the value of mean.
Type-2: In case of discrete observations or series of data:
Discrete series are the variables whose values can be identified and isolated. In such a case the variant is a whole number, but is form frequency distribution. The data set derived in case-1 above is called as ungrouped data. The computations in case of these data are not difficult. Where as, if the data set is having frequencies are called as groped data.
a. Direct method: Following are some steps of calculating mean by using the direct method
i. In the first step, the values of each row (X) are to be multiplied by its respective frequencies (f).
ii. Calculate the sum of the frequencies (column-2 in our example) at the end of the column denoted as
iii. Calculate the sum of the X*f values at the end of the column (column-3 in our below derived example) denoted as
iv. Mean () can be calculated by using the formula
b. Short cut method: Arithmetic mean can also be calculated by using the short cut method or assumed mean method. This method is generally used by the researchers to avoid the time requirements and calculation complexities. Following are the steps of calculating mean by this method.
i. The first step is to assume a value from the ‘X’ values of the series (denoted as A= assumed value)
ii. In this step in another column we have to calculate the deviation value (denoted as D) of ‘X’ to that of assumed value (A) i.e., D = X-A
iii. Multiply each D with f i.e., find our Df
iv. Calculate the value of sum of at the end of respective columns.
v. Mean can be calculated by using the formula as
Type-3: In case of continuous observations or series of data:
Another type of frequency distributions is there which consists of data that are grouped by classes. In such case each value of an observation falls somewhere in one of the classes. Calculation of arithmetic mean in case of grouped data is some what different from that of ungrouped data. To find out the arithmetic mean of continuous series, one has to calculate the midpoint of each class interval. To make midpoints come out in whole cents, one has to round up the value. Mean in continuous series can be calculated in two ways as derived below:
a. Direct method: In this method, mean can be calculated by using the steps as
i. First step is to calculate the mid point of each class interval. The mid point is denoted by ‘m’ and can be calculated as .
ii. Multiply the mid points of each class interval (m) with its respective frequencies (f) i.e., find out mf
iii. Calculate the value of sum of at the end of respective columns.
iv. Mean can be calculated by using the formula as
b. Short cut method: Mean can also be calculated by using short cut method. Following are the steps to calculate mean by this method.
i. First step is to calculate the mid point of each class interval. The mid point is denoted by ‘m’ and can be calculated as .
ii. Assume a value from the ‘m’ values of the series (denoted as A= assumed value)
iii. In this step in another column we have to calculate the deviation value (denoted as D) of ‘m’ to that of assumed value (A) i.e., D = m – A
iv. Multiply each D with f i.e., find our Df
v. Calculate the value of sum of at the end of respective columns.
vi. Put the values in the following formula to get mean of the series
In real life situation in management studies and social sciences, some items need more importance than that of the other items of that series. Hence, importance assigned to different items with the help of numerical value as per the priority basis in a series as called as weights. The arithmetic mean on the other hand, gives equal weightage or importance to each observation of the series. In such a case, the weighted mean acts as the most important tool for studying the behaviour of the entire set of study. Here use of weighted mean is the only measure of central tendency for getting correct and accurate result. Following is the procedures of computing mean of a weighted series. By the way, an important problem that arises while using weighted mean is regarding selection of weights. Weights may be either actual or arbitrary, i.e., estimated. The researcher will not face any difficulty, if the actual weights are assigned to the set of data. But in case, if actual data is not assigned than it is advisable to assign arbitrary or imaginary weights.
Following are some steps of calculating weighted mean:
i. In the first step, the values of each row (X) are to be multiplied by its respective weights (W)
ii. Calculate the sum of the weights (column-2 in our example) at the end of the column denoted as
iii. Calculate the sum of the X*W values at the end of the column (column-3 in our below derived example) denoted as
iv. Mean () can be calculated by using the formula
Advantages of Arithmetic mean: Following are some advantages of arithmetic mean.
i. The concept is more familiar concept among the people. It is unique because each data set has only one mean.
ii. It is very easy to compute and requires fewer calculations. As every data set has a mean, hence, as a measure mean can be calculated.
iii. Mean represents a single value to the entire data set. Thus easily one can interpret a data set its characteristics.
iv. An average can be calculated of any type of series.
Disadvantages of Arithmetic mean: The disadvantages are as follows.
i. One of the greatest disadvantages of average is that it is mostly affected by the extreme values. For example let consider Sachin Tendulkar’s score in last three matches. Let it be, 100 in first match, 2 in second match and 10 in third match. The average score of these three matches will me 100+2+10/3=37. Thus it implies that Tendulkar’s average score is 37 which is not correct. Hence lead to wrong conclusion.
ii. It is not possible to compute mean for a data set that has open-ended classes at either the high or low end of the scale.
iii. The arithmetic average sometimes gives such value which cannot be found from the data series from which it is calculated.
iv. It is unrealistic.
v. It cannot be identified observation or graphic method of representing the data and interpretation.
Another one technique to measure central tendency of a series of observation is the median. Median is generally that value of the entire series which divides the entire series into two equal parts from the middle. In other wards, it is the exactly middle value of the series. Hence, fifty percent of the observations in the series are above the median value and other fifty or half observations are remains below the median value. However, if the series are having odd numbers of observations like 3,5,7,9,11,13 etc., then the median value will be equal to one of the exact value from the series. On the other hand, if the series is having even observations, then median value can be calculated by getting the arithmetic mean of the two middle values of the observations of the series. Median an a technique of measuring central tendency can be best used in cases where the problem sought for more qualitative or psychological in nature such as health, intelligence, satisfaction etc.
The concept of median can be clearer from the definitions derived below. Connor defined it as ‘the median is the value which divides the distribution into two equal parts, one part comprising all values greater, and the other values less than the median’. Where as Croxton and Cowden defined it as ‘the median is that value which divides a series so that one half or more of the items are equal to or less than it and one half or more of the items are equal to or greater than it’. Median can be computed in three different series separately. All the cases are discussed separately below.
Following are some steps to calculate the median in individual series.
The first and the most important requirement is that the data should be arranged in an ascending (increasing) or descending (decreasing) order.
Than the median value can be calculated by using the formula th value or item from the series. Where, N= Number of observation in that series. When N is odd number (like 5, 7,9,11,13 etc.) median value is one of the item within that series, but in case N will be a even number than median is the arithmetic mean of the two middle value after applying the above formula. The following problem can make the concept clear.
Discrete series are those where the data set is assigned with frequencies or repetitions. Following are the steps of computing the median when the series is discrete.
Continuous series are the series of data where the data ranges are in class intervals. Each class is having an upper limit and a lower limit. In such cases the computation of median is little bit different from that of the other two cases discussed above. Following are some steps to get median in continuous series of data.
We have to get the median class first. For this, median class is N/2 th value or 70/2= 35. The value 35 lies in the third row of the table against the class 30-40. Thus 30-40 is the median class and it shows that the median value lies in this class only. After getting the median class, to get the median value we have to apply the formula .
Advantages of Median: Median as a measure of central tendency has following advantages of its own.
Disadvantages of Median: Median as a means to calculate central tendency is also not free from draw backs. Following are some important draw backs that are leveled against median.
Mode is defined as the value which occurs most often in the series or other wise called as the value having the highest frequencies. It is, hence, the value which has maximum concentration around it. Like that of median, mode is also more useful in case of qualitative data analysis. It can be used in problems generally having the discrete series of data and particularly, problems involving the expression of psychological determinants.
The concept of mode can be clearer from the definitions derived below. Croxten and Cowden defined it as ‘the mode of a distribution is the value at the point around which the items tend to be most heavily concentrated. It may be regarded as the most typical of a series of value’. Similarly, in the words of Prof. Kenny ‘the value of the variable which occurs most frequently in a distribution is called the mode’. Mode can be computed in three different series separately. All the cases are discussed separately below.
Calculation of mode in individual series is very easy. The data is to be arranged in a sequential order and that value which occurs maximum times in that series is the value mode. The following example will make the concept clear.
Discrete series are those where the data set is assigned with frequencies or repetitions. Hence directly, mode will be that value which is having maximum frequency. By the way, for accuracy in calculation, there is a method called as groping method which is frequently used for calculating mode. Following is the illustration to calculate mode of a series by using grouping method.
Consider the following data set and calculate mode by using the grouping method. The calculation carried out in different steps is derived as:
Step-1: Sum of two frequencies including the first one i.e., 1+2=3, then 4+3=7, then 2+1=3 etc.
Step-2: Sum of two frequencies excluding the first one i.e., 2+4=7, then 3+2=5, then 1+2=3 etc.
Step-3: Sum of three frequencies including the first one i.e., 1+2+4=7, then 3+2+1=6 etc.
Step-4: Sum of two frequencies excluding the first one i.e., 2+4+3=9, then 2+1+2=5 etc.
Step-5: Sum of three frequencies excluding the first and second i.e., 4+3+2=9, then 1+2+1=4.
As already discussed, continuous series are the series of data where the data ranges are in class intervals. Each class is having an upper limit and a lower limit. In such cases the computation of mode is little bit different from that of the other two cases discussed above. Following are some steps to get mode in continuous series of data.
Advantages of Mode: Following are some important advantages of mode as a measure of central tendency.
Disadvantages of Mode: Following are some important disadvantages of mode.
Geometric mean, as another measure of central tendency is very much useful in social science and business related problems. It is an average which is most suitable when large weights have to be assigned to small values of observations and small weights to large values of observation. Geometric mean best suits to the problems where a particular situation changes over time in percentage terms. Hence it is basically used to find the average percent increase or decrease in sales, production, population etc. Again it is also considered to be the best average in the construction of index numbers.
Geometric mean is defined as the Nth root of the product where there are N observations of a given series of data. For example, if a series is having only two observations then N will be two or we will take square root of the observations. Similarly, when series is having three observations then we have to take cube root and the process will continue like wise. Geometric mean can be calculated separately for two sets of data. Both are discussed below.
In case of ungrouped series of observations, GM can be calculated by using the following formula: where X1 , X2 , X3, …XN various observations of a series and N is the Nth observation of the data. But it is very difficult to calculate GM by using the above formula. Hence the above formula needs to be simplified. To simplify the formula, both side of the above formula is to be taken logarithms.
To calculate the G.M. of an ungrouped data, following steps are to be adopted.
Calculation of geometric mean in case of grouped data is little bit different from that of calculation of G.M. in case of ungrouped series. Following are some steps to calculate the G.M. in case of grouped data series.
To calculate the G.M. of a grouped data, following steps are to be adopted.
Advantages of G.M.: Following are some advantages of G.M.
i. One of the greatest advantages of G.M. is that it can be possible for further algebraic treatment i.e., combined G.M., can be calculated when there is availability of G.M., of two or more series along with their corresponding number of observations.
ii. It is a very useful method of getting average when the series of observation possess rates of growth i.e., increase or decrease over a period of time.
iii. Since it is useful in averaging ratios and percentages, hence, are more useful in social science and business related problems.
Disadvantages of G.M.: G.M., as a technique of calculating central value is also not free from defects. Following are some disadvantages of G.M.
i. It is very difficult to calculate the value of log and antilog and hence, compared to other methods of central tendency, G.M., is very difficult to compute.
ii. The greatest disadvantage of G.M., is that it cannot be used when the series is having both negative or positive observations and observations having more zero values.
The last technique of getting the central tendency of a series of data is the Harmonic mean (H.M.). Harmonic mean, like the other methods of central tendency is not clearly defined. It is the reciprocal of the arithmetic mean of the reciprocal of the individual observations. H.M., is very much useful in those cases of observations where the nature of data is such that it express the average rate of growth of any events. For example, the average rate of increase of sales or profits, the average speed of a train or bus or a journey can be completed etc. Following is the general formula to calculate H.M.:
When the observations of the series are ungrouped, H.M., can be calculated as:
The step for calculating H.M., of ungrouped data by using the derived formula is very simple. In such a case, one has to find out the values of 1/X and then sum of 1/X.
When the data is grouped: In case of grouped data, the formula for calculating H.M., is discussed as below:
Advantages of H.M.: Harmonic mean as a measure of central tendency is having following advantages.
i. Harmonic mean considers each and every observation of the series.
ii. It is simple to compute when compared to G.M.
iii. It is very useful for averaging rates.
Disadvantages of H.M.: Following are some disadvantages of H.M.
i. It is rarely used as a technique of measuring central tendency.
ii. It is not defined clearly like that of other techniques of measuring central value mean, median and mode.
iii. Like that of G.M., H.M., cannot be used when the series is having both negative or positive observations and observations having more zero values.
An average is a single value representing a group of values. Each type of averages has their own advantages and disadvantages and hence, they are having their own usefulness. But it is always confusing among the researchers that which average is the best among the five different techniques that we have discussed above? The answer to this question is very simple and says that no single average can be considered as best for all types of data. However, experts opine two considerations that the researchers must be kept in mind while going for selecting a technique to determine the average. The first consideration is that of determining the nature of data. If the data is more skewed it is better to avoid arithmetic mean, if the data is having gap around the middle value of the series, then median should be avoided and on the other hand, if the nature of series is such that they are unequal in class-intervals, then mode is to be avoided. The second consideration is on the type of value required. When there is need of composite average of all absolute or relative values, then arithmetic mean or geometric mean is to be selected, in case the researcher is in need of a middle value of the series, then median may be the best choice, but in case the most common value is needed, then will not be any alternative except mode. Similarly, Harmonic mean is useful in averaging ratios and percentages.
1. Different experts have defined differently to the concept of average.
2. Arithmetic mean is the most simple and frequently used technique of computing central tendency. The average is also called as mean. It is other wise called as a single number representing a whole data set.
3. The best use of arithmetic mean is at the time of correcting some wrong entered data. For example in a group of 10 students, scoring an average of 60 marks, in a paper it was wrongly marked 70 instead of 65. the solution in such a cases is derived below:
4. In such a case, the weighted mean acts as the most important tool for studying the behaviour of the entire set of study. Here use of weighted mean is the only measure of central tendency for getting correct and accurate result.
5. Median is generally that value of the entire series which divides the entire series into two equal parts from the middle.
6. Mode is defined as the value which occurs most often in the series or other wise called as the value having the highest frequencies. It is, hence, the value which has maximum concentration around it.
7. Geometric mean is defined as the Nth root of the product where there are N observations of a given series of data.
8. Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the individual observations.
1. In a class containing 90 students following heights (in inches) has been observed. Based on the data calculate the mean, median and mode of the class.
2. In a physical test camp meant for selection of army solders the following heights of the candidates have been observed. Find the mean, median and mode of the distribution.
3. From the distribution derived below, calculate mean and standard deviation of the series.
4. The following table derives the marks obtained in Indian Economy paper by 90 students in a class. Calculate the mean, median and mode of the following distribution.
5. The monthly profits of 180 shop keepers selling different commodities in a city footpath is derived below. Calculate the mean and median of the distribution.
6. The daily wage of 130 labourers working in a cotton mill in Ahmadabad cith is derived below. Calculate the mean, median and mode.
7. There is always controversy before the BCCI before selection of batsmen between Rahul Dravid and V.V.S. Laxman. Runs of 10 test matches of both the players are given below. Suggest who the better run getter is and who the consistent player is.
8. Calculate the mean, median and mode of the following distribution.
9. What do you mean by measure of central tendency? How far it helpful to a decision-maker in the process of decision making?
10. Define measure of central tendency? What are the basic criteria of a good average?
11. What do you mean by measure of central tendency? Compare and contrast arithmetic mean, median and mode by pointing out the advantages and disadvantages.
12. The expenditure on purchase of snacks by a group of hosteller per week is give below. Calculate the mean, median and mode of the series.
13. The mean, median and mode of a group of 85 persons were calculated as 28, 31 and 36 respectively. It was later found that while calculating these values, one value was wrongly calculated as 46 instead of the correct value 56. What will be the effect on the correction of this value on the observation?
14. Mr Sachdeva has been heading the computer department of an organization since last 7 years. Following are the year wise expenditure in Rupee for 17 years that has been spent for the maintenance of the computers.
15. Yesh Travels Limited, a travel agent is having 20 cars which are used as taxi in Greater Noida of Uttar Pradesh. The owner of the Travel agent in a surprise check asked the manager the weekly mileage records of all the 20 cars. Being the owner of the travel agent, calculate: (a) the median miles of a car traveled during the week, (b) mean mileage of the cars
16. Delhi Transport Corporation (DTC) is in news since last three months because of repeated cases of fire in its low line buses that is running from different destinations in Delhi city. The high level committee set up of by the chief minister of Delhi is in the process of investigation about the cause of the fire in buses. One of the important causes that the driver of a bus explained is the excessive speed of the buses. It is estimated that in all the routes that the buses are running requires 45 minutes. The sample data derived below shows the arrival time that had taken by some buses to their destination. Conclude from the data the reality.
17. The ages of the students pursuing their master degree in a class is given by following distribution. Estimate the modal value.
18. Calculate the arithmetic mean of the following set of data by using (a) direct method and (b) short-cut method:
19. Following is the daily wage structure of some employees who are working in M/s. Ansul Food Process on daily basis. Calculate the arithmetic mean by using (a) direct method and (b) short-cut method.
20. A candidate has attended three papers like Indian Economics (IE), Statistics (S) and econometrics (E) to clear his M.Phil degree. In each subject he has to appear a oral tests of 40 marks and written test of 60 marks. He secured 25, 21 and 18 marks in oral tests and 52, 35 and 31 marks in written tests in subjects IE, S and E respectively. You have to calculate the weighted average of marks obtained in written test taking the weights percentage of marks obtained in corresponding oral test.
21. A candidate has obtained the following percentage of marks in an examination: Business Law 65, Statistics for Managers 70, Managerial Economics 62, Business Communication 55 and Organisation Behaviour 58. The weights allocated to each subject are as 4, 1, 2, 3, and 3 respectively. Calculate the weighted mean.
22. Tata Motors Limited wanted to offer a cash gift of 7 per cent on the number of cars sold by its sales managers in Northern region of India. Calculate the mode and the median taking the average value.
23. Obtain the median and mode from the following records of a school.
24. Calculate the mean and median of the following data series:
25. Following is the temperature that is maintained in a cold storage in different seasons to preserve the vegetables. Calculate the mean and mode of the series.
26. Find the median from the following data:
27. The distribution of 2000 houses of a locality according to their distance from a petrol pump is given in the following table:
28. A housewife saves Rs. 1/- on the first day, Rs. 2/- on the second day, Rs. 3/- on third day and Rs. 31/- on the 31st day in a particular month. Calculate the mean and median of per day savings. In the amount, her husband contributes Rs. 100 on the 32nd day and Rs. 600/- on the 33rd day. Calculate the new mean and median of savings per day.
29. Compute the mode value of the following data:
30.Calculate the modal value of the following distribution.
31. The distribution of the marks obtained by 70 students of a class in a class is given below:
32. The average rainfall for a week, excluding Sunday, was 12 cms. Sunday was observed heavy rainfall for which when Sunday was included on the other days the average rose to 18 cms. Get how much rainfall was on Sunday?
33. The mean age of the combined group of men and women is 31.5 years. If the mean age of the sub-group of men is 36 years and that of the sub-group of women is 24 years, find out percentage of men and women in the group.
34. The arithmetic mean of 60 items of a series was estimated by an entrepreneur as 22. However, it was latter calculated by the auditor that an item 26 was wrongly calculated as 62. Calculate the correct mean.
35. The sales of a street ice cream seller on seven days of a week during summer season are given below. If the profit is 15% of the sales, find his average profits per day.
36.Calculate the mode of the following data:
37. Calculate the mode of the following distribution:
38. The distribution derived below reveals monthly expenditure on food items incurred by a sample of 135 families in Jalbau Bihar, a residential colony at Greater Noida. Calculate the modal value of the distribution.
39. Calculate the geometric mean of the following data:
40. A distribution is derived below. Calculate the geometric mean.
41. Calculate the geometric mean of the following distribution:
42. In between the years 2005-2009, precious metals changes rapidly in their value in the market. The total rate of return (in %) data is derived in the table below:
43. The data derived below represents the battery life (in minutes) for mobile phones of different brand available in the market.
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