Statistics is the branch of science which deals with collection of information, organising and analysing. Once the data is collected, data are tabulated in a condensed form. To analyse numerical data, it is essential to know about measures of central tendency i.e..mean, median and mode.
Arithmetic mean:
The arithmetic mean of a set of “n” observations x_{1}, x_{2}, x_{3},…x_{n }is defined as
`barx = (x_1+x_2+x_3+…+x_n)/n = (sumx)/n`
Weighted Arithmetic mean:
In arithmetic maen ,the given data contribute equal points to get the value of average. In weighted arithmetic mean, some data could contribute more points or more weights than other data. We use weighted arithmetic mean in the field of descriptive statistics.
Note:
The weights cannot bear a negative value
The value of any weights can be zero but not all the value of the weights can be zero.
Problems Using Weighted Mean Formula:
Weighted arithmetic mean formula:
If x_{1}, x_{2}, x_{3},…x_{n} are n values and w_{1}, w_{2}, w_{3}…w_{n} are their weights(frequencies) respectively, then
`barx = (w_1x_1 +w_2x_2 + w_3x_3+…+w_nx_n)/(w_1+w_2+w_3+…+w_n)=(sumwx)/(sum w)`
Ex 1: Find the weighted arithmetic mean for teh given data.
ITEM  WEIGHT  COST PER KG (in dollars) 
Wheat  2  20 
Oil  4  80 
Potato  2  8 
Onion  3  7.50 
Sol:
Step 1:
We have to calculate wx and find the sum of wx and sum of w.
ITEM  WEIGHT (w)  FREQUENCY (x)  wx 
Wheat  2  20.00  40 
Oil  4  80.00  320 
Potato  2  8.00  16 
Onion  3  7.50  22.50 
Σw = 11  Σwx = 398.50 
Step 2: Write the weighted mean formula and plug the values.
Weighted Arithmetic mean = `(sumwx)/sumw = 398.50/11 = $ 36.23`
Practice Problems on Weighted Mean:
 Find the weighted arithmetic mean for the given data.

ITEM WEIGHT COST PER KG (in dollars) Powder 2 40 Soap 4 12 Pen 5 15 Instrument box 4 25.50