ARITHMETIC MEAN - Quantitative Techniques for management

Before the discussion of arithmetic mean, we shall introduce certain notations. It will be assumed that there are n observations whose values are denoted by X1,X2, ..... Xn respectively. The sum of these observations X1 + X2 + ..... + Xn will be denoted in abbreviated form as

sum of these observations

where S (called sigma) denotes summation sign.

The subscript of X, i.e., 'i' is a positive integer, which indicates the serial number of the observation. Since there are n observations, variation in i will be from 1 to n. This is indicated by writing it below and above S, as written earlier. When there is no ambiguity in range of summation, this indication can be skipped and we may simply write X1 + X2 + ..... + Xn = SXi.

Arithmetic Mean is defined as the sum of observations divided by the number of observations. It can be computed in two ways:

  • Simple arithmetic mean and
  • weighted arithmetic mean.

In case of simple arithmetic mean, equal importance is given to all the observations while in weighted arithmetic mean, the importance given to various observations is not same.

Calculation of Simple Arithmetic Mean

(a) When Individual Observations are given.

Let there be n observations X1, X2 ..... Xn. Their arithmetic mean can be calculated either by direct method or by short cut method. The arithmetic mean of these observations will be denoted by X

Direct Method: Under this method, X is obtained by dividing sum of observations by number of observations, i.e.,

Direct Method

Short-cut Method: This method is used when the magnitude of individual observations is large. The use of short-cut method is helpful in the simplification of calculation work. Let A be any assumed mean. We subtract A from every observation. The difference between an observation and A, i.e., Xi - A is called the deviation of i th observation from A and is denoted by di. Thus, we can write ; d1 = X1 - A, d2 = X2 - A, ..... dn = Xn - A. On adding these deviations and dividing by n we get

Short-cut Method

This result can be used for the calculation of X .

Remarks: Theoretically we can select any value as assumed mean. However, for the purpose of simplification of calculation work, the selected value should be as nearer to the value of X as possible.

Example : The following figures relate to monthly output of cloth of a factory in a given year

monthly output of cloth

short cut method

short cut method

(c) When data are in the form of a grouped frequency distribution

In a grouped frequency distribution, there are classes along with their respective frequencies. Let li be the lower limit and ui be the upper limit of ith class. Further, let the number of classes be n, so that i = 1, 2,.....n. Also let fi be the frequency of ith class. This distribution can written in tabular form, as shown.

Note: Here u1 may or may not be equal to l2, i.e., the upper limit of a class may or may not be equal to the lower limit of its following class.

It may be recalled here that, in a grouped frequency distribution, we only know the number of observations in a particular class interval and not their individual magnitudes. Therefore, to calculate mean, we have to make a fundamental assumption that the observations in a class are uniformly distributed.

Under this assumption, the mid-value of a class will be equal to the mean of observations in that class and hence can be taken as their representative. Therefore, if Xi is the mid-value of i th class with frequency fi , the above assumption implies that there are fi observations each with magnitude Xi (i = 1 to n). Thus, the arithmetic mean of a grouped frequency distribution can also be calculated by the use of the formula, given in § below.

arithmetic mean of a grouped frequency distribution

Remarks: The accuracy of arithmetic mean calculated for a grouped frequency distribution depends upon the validity of the fundamental assumption. This assumption is rarely met in practice. Therefore, we can only get an approximate value of the arithmetic mean of a grouped frequency distribution.

Example : Calculate arithmetic mean of the following distribution :

arithmetic mean

Solution: Here only short-cut method will be used to calculate arithmetic mean but it can also be calculated by the use of direct-method

direct-method

Example : The following table gives the distribution of weekly wages of workers in a factory. Calculate the arithmetic mean of the distribution.

arithmetic mean of the distribution.

Solution: It may be noted here that the given class intervals are inclusive. However, for the computation of mean, they need not be converted into exclusive class intervals.

exclusive class intervals

Step deviation method or coding method

In a grouped frequency distribution, if all the classes are of equal width, say 'h', the successive mid-values of various classes will differ from each other by this width. This fact can be utilised for reducing the work of computations.

deviation method or coding method

deviation method or coding method

average amount of pension

accuracy of calculations

Weighted Arithmetic Mean

In the computation of simple arithmetic mean, equal importance is given to all the items. But this may not be so in all situations. If all the items are not of equal importance, then simple arithmetic mean will not be a good representative of the given data. Hence, weighing of different items becomes necessary. The weights are assigned to different items depending upon their importance, i.e., more important items are assigned more weight.

For example, to calculate mean wage of the workers of a factory, it would be wrong to compute simple arithmetic mean if there are a few workers (say managers) with very high wages while majority of the workers are at low level of wages. The simple arithmetic mean, in such a situation, will give a higher value that cannot be regarded as representative wage for the group. In order that the mean wage gives a realistic picture of the distribution, the wages of managers should be given less importance in its computation.

The mean calculated in this manner is called weighted arithmetic mean. The computation of weighted arithmetic is useful in many situations where different items are of unequal importance, e.g., the construction index numbers, computation of standardized death and birth rates, etc.

Weighted Arithmetic Mean

properties of arithmetic mean

sum of squares of deviations of observations

arithmetic mean

mean of combined series

mean of obseravtions

average change in magnitude

Average monthly of teacher

combined average monthly salary

missing frequency

Formula for AM

correct value of mean

average profit of ballon

Merits and Demerits of Arithmetic Mean

Merits:

  • It is rigidly defined.
  • It is easy to calculate and simple to follow.
  • It is based on all the observations.
  • It is determined for almost every kind of data.
  • It is finite and not indefinite.
  • It is readily put to algebraic treatment.
  • It is least affected by fluctuations of sampling.

Demerits:

  • The arithmetic mean is highly affected by extreme values.
  • It cannot average the ratios and percentages properly.
  • It is not an appropriate average for highly skewed distributions.
  • It cannot be computed accurately if any item is missing.
  • The mean sometimes does not coincide with any of the observed value.

Exercise with Hints

  1. The frequency distribution of weights in grams of mangoes of a given variety is given below. Calculate the arithmetic mean.

    frequency distribution of weights in grams

    Hint : Take the mid-value of a class as the mean of its limits and find arithmetic mean by the step-deviation method.

  2. The following table gives the monthly income (in rupees) of families in a certain locality. By stating the necessary assumptions, calculate arithmetic mean of the distribution.

    arithmetic mean by the step-deviation method.

    Hint : This distribution is with open end classes. To calculate mean, it is to be assumed that the width of first class is same as the width of second class. On this assumption the lower limit of the first class will be 0. Similarly, it is assumed that the width of last class is equal to the width of last but one class. Therefore, the upper limit of the last class can be taken as 6,000.

  3. Compute arithmetic mean of the following distribution of marks in Economics of 50 Students

    arithmetic mean

    Hint: First convert the distribution into class intervals and then calculate X .

  4. The monthly profits, in '000 rupees, of 100 shops are distributed as follows:

    distribution into class intervals

    Find average profit per shop.

    Hint: This is a less than type cumulative frequency distribution.

  5. Typist A can type a letter in five minutes, typist B in ten minutes and typist C in fifteen minutes. What is the average number of letters typed per hour per typist?
    Hint: In one hour, A will type 12 letters, B will type 6 letters and C will type 4 letters.
  6. A taxi ride in Delhi costs Rs 5 for the first kilometre and Rs 3 for every additional kilometre travelled. The cost of each kilometre is incurred at the beginning of the kilometre so that the rider pays for the whole kilometre. What is the average cost of travelling 2 3/ 4 kilometres?
    Hint: Total cost of travelling 2*3/ 4 kilometres = Rs 5 + 3 + 3 = Rs 11.
  7. A company gave bonus to its employees. The rates of bonus in various salary groups are :

    cumulative frequency distribution

    The actual salaries of staff members are as given below :

    1120, 1200, 1500, 4500, 4250, 3900, 3700, 3950, 3750, 2900, 2500, 1650, 1350, 4800, 3300, 3500, 1100, 1800, 2450, 2700, 3550, 2400, 2900, 2600, 2750, 2900, 2100, 2600, 2350, 2450, 2500, 2700, 3200, 3800, 3100.

    Determine (i) Total amount of bonus paid and (ii) Average bonus paid per employee.

    Hint: Find the frequencies of the classes from the given information.

  8. Calculate arithmetic mean from the following distribution of weights of 100 students of a college. It is given that there is no student having weight below 90 lbs. and the total weight of persons in the highest class interval is 350 lbs.

    arithmetic mean

    Hint: Rearrange this in the form of frequency distribution by taking class intervals as 90 - 100, 100 - 110, etc.

  9. By arranging the following information in the form of a frequency distribution, find arithmetic mean.
    "In a group of companies 15%, 25%, 40% and 75% of them get profits less than Rs 6 lakhs, 10 lakhs, 14 lakhs and 20 lakhs respectively and 10% get Rs 30 lakhs or more but less than 40 lakhs."

    Hint: Take class intervals as 0 - 6, 6 - 10, 10 - 14, 14 - 20, etc.

  10. Find class intervals if the arithmetic mean of the following distribution is 38.2 and the assumed mean is equal to 40.

    arithmetic mean

    Hint: Use the formula X = A + å fu/ N× h to find the class width h.

  11. From the following data, calculate the mean rate of dividend obtainable to an investor holding shares of various companies as shown :

    mean rate of dividend obtainable

    Hint: The no. of shares of each type = no. of companies ´ average no. of shares.

  12. The mean weight of 150 students in a certain class is 60 kgs. The mean weight of boys in the class is 70 kgs and that of girls is 55 kgs. Find the number of girls and boys in the class.

    Hint: Take n1 as the no. of boys and 150 - n1 as the no. of girls.

  13. The mean wage of 100 labourers working in a factory, running two shifts of 60 and 40 workers respectively, is Rs 38. The mean wage of 60 labourers working in the morning shift is Rs 40. Find the mean wage of 40 laboures working in the evening shift.

    Hint: See example above.

  14. The mean of 25 items was calculated by a student as 20. If an item 13 is replaced by 30, find the changed value of mean.

    Hint: See example above.

  15. The average daily price of share of a company from Monday to Friday was Rs 130. If the highest and lowest price during the week were Rs 200 and Rs 100 respectively, find average daily price when the highest and lowest price are not included.

    Hint: See previous example.

  16. The mean salary paid to 1000 employees of an establishment was found to be Rs 180.40. Later on, after disbursement of the salary, it was discovered that the salaries of two employees were wrongly recorded as Rs 297 and Rs 165 instead of Rs 197 and Rs 185. Find the correct arithmetic mean.

    Hint: See previous example.

  17. Find the missing frequencies of the following frequency distribution:

    missing frequencies

    Hint: See above example.

  18. Marks obtained by students who passed a given examination are given below:

    Marks obtained by students

    If 100 students took the examination and their mean marks were 51, calculate the mean marks of students who failed.

    Hint: See above example .

  19. A appeared in three tests of the value of 20, 50 and 30 marks respectively. He obtained 75% marks in the first and 60% marks in the second test. What should be his percentage of marks in the third test in order that his aggregate is 60%?

    Hint: Let x be the percentage of marks in third test. Then the weighted average of 75, 60 and x should be 60, where weights are 20, 50 and 30 respectively.

  20. Price of a banana is 80 paise and the price of an orange is Rs 1.20. If a person purchases two dozens of bananas and one dozen of oranges, show by stating reasons that the average price per piece of fruit is 93 paise and not one rupee.

    Hint: Correct average is weighted arithmetic average.

  21. The average marks of 39 students of a class is 50. The marks obtained by 40th student are 39 more than the average marks of all the 40 students. Find mean marks of all the 40 students

    mean marks

    Percentage increase
  1. The following table gives the distribution of the number of kilometres travelled per salesman, of a pharmaceutical company, per day and their rates of conveyance allowance:

    number of kilometres travelled per salesman

    Calculate the average rate of conveyance allowance given to each salesman per kilometre by the company.

    Hint: Obtain total number of kilometre travelled for each rate of conveyance allowance by multiplying mid-values of column 1 with column 2. Treat this as frequency 'f' and third column as 'X' and find X .

  2. The details of monthly income and expenditure of a group of five families are given in the following table

    monthly income and expenditure

    Find: (i) Average income per member for the entire group of families.
    (ii) Average expenditure per family.
    (iii) The difference between actual and average expenditure for each family.
    Hint: (i) Average income per member = Total income of the group of families/Total no of members in the group

    (ii) Average expenditure per family = Total expenditure of the group/No of families

  3. The following table gives distribution of monthly incomes of 200 employees of a firm:

    distribution of monthly incomes

    Estimate:

    (i) Mean income of an employee per month.
    (ii) Monthly contribution to welfare fund if every employee belonging to the top 80% of the earners is supposed to contribute 2% of his income to this fund.

    Hint: The distribution of top 80% of the wage earners can be written as

    Mean income of an employee

    By taking mid-values of class intervals find Sfx, i.e., total salary and take 2% of this.

  4. The number of patients visiting diabetic clinic and protein urea clinic in a hospital during April 1991, are given below :

    number of patients visiting

    Which of these two diseases has more incidence in April 1991? Justify your conclusion.

    Hint: The more incidence of disease is given by higher average number of patients.

  5. A company has three categories of workers A, B and C. During 1994, the number of workers in respective category were 40, 240 and 120 with monthly wages Rs 1,000, Rs 1,300 and Rs 1,500. During the following year, the monthly wages of all the workers were increased by 15% and their number, in each category, were 130, 150 and 20, respectively.
    (a) Compute the average monthly wages of workers for the two years.
    (b) Compute the percentage change of average wage in 1995 as compared with 1994. Is it equal to 15%? Explain.

    Hint: Since the weight of the largest wage is less in 1995, the increase in average wage will be less than 15%.

  6. (a) The average cost of producing 10 units is Rs 6 and the average cost of producing 11 units is Rs 6.5. Find the marginal cost of the 11th unit.
    (b) A salesman is entitled to bonus in a year if his average quarterly sales are at least Rs 40,000. If his average sales of the first three quarters is Rs 35,000, find his minimum level of sales in the fourth quarter so that he becomes eligible for bonus.

    Hint: See above example .

  7. (a) The monthly salaries of five persons were Rs 5,000, Rs 5,500, Rs 6,000, Rs 7,000 and Rs 20,000. Compute their mean salary. Would you regard this mean as typical of the salaries? Explain.
    (b) There are 100 workers in a company out of which 70 are males and 30 females. If a male worker earns Rs 100 per day and a female worker earns Rs. 70 per day, find average wage. Would you regard this as a typical wage? Explain
    Hint: An average that is representative of most of the observations is said to be a typical average.

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