the salaries of 15 employees in a company are 7000,6500,3300,5600,2350,2500,8500,1900,10000,7500,3500,4500,5500,3000,9000. find the mean of the data
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Answer:
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.