Mean [Discrete Series (Tabular format)]

MEAN Discrete Series Tabular format

What if the data is in tabular format (in a table)?

The same general principles apply.

Let us consider the revenue ($) generated by students in a class:

Serial No.                             Amount ($)                          Number of students

1.                                             110                                                       2

2.                                            112                                                        2

3.                                            118                                                        4

4.                                            120                                                       6

5.                                            122                                                       2

What is the mean revenue generated by students in the class?

We know from the post on Mean (Discrete Series) that the mean can be calculated by taking the sum of all observations, and dividing it by the total number of observations.

How do we do that here?

In the problem, we have been given the amount generated, and the number of students who generated that amount. For instance, we know from the table that 2 students generated $110. So how much did they generate between them? The answer is $110 x 2, or $220.

In order to calculate the mean, we need to know the total amount generated by all the students in the class.

How do we obtain the total amount? We simply multiply each value under the “Amount($)” column with the corresponding value in the “Number of students” column. That will give us the total for each row. To obtain the total for the entire table, we merely add the row totals.

When we do that, we will have a table that looks like the one below:

Serial No.          Amount ($)               Number of students       Total Amount

1.                                110                                                 2                       110 x 2 = 220

2.                                112                                                 2                       112 x 2 = 224

3.                                118                                                 4                       118 x 4 = 472

4.                                120                                                 6                      120 x 6 = 720

5.                                122                                                 2                      122 x 2 = 244

                                                Total             16                                    1880

You would have noticed that I have obtained the total number of students by adding the number of students in each row.

So, now we have all the values needed to calculate the mean- the sum of all observations (1880); and the total number of observations (16).

Thus, we have:

Mean =  Sum of all observations/ Total number of observations

Putting the values in the equation, we have:

Mean = 1880/ 16

= 117.5

Rounding off, we could say that the average (mean) amount generated by the students was $118.

(Of course, one could be very precise and say that the mean amount generated was $117.5, but rounding off is also an acceptable practice.)

Statisticians use short forms to represent the various columns in the table.

The column “Amount ($)” would be labelled “x“.

The column “Number of students” would be labelled “f“.

The last column would be labelled “f(x)” [since we are multiplying one with the other].

If we did that, the table would look like this:

Serial No.          Amount ($)               Number of students       Total Amount

                                     (x)                                               f                                f(x)

1.                                110                                                 2                       110 x 2 = 220

2.                                112                                                 2                       112 x 2 = 224

3.                                118                                                 4                       118 x 4 = 472

4.                                120                                                 6                      120 x 6 = 720

5.                                122                                                 2                      122 x 2 = 244

                                           Total (n) =       16                     ∑f(x)=  1880

The Mean would then be given by the equation:

Mean = ∑f(x)/n          { since we are adding f(x) from each row in the table, the sum of all observations is denoted as  ∑f(x)}

Summary (Steps):

In tabulated data, the data is arranged in columns and rows.

Identify the column with the observations (amount in the example given above). Label this column “x”.

Identify the column giving the number of times each observations was recorded (number of students in the above example). Label this column “f”.

Create a new (third) column. Label it “ f(x)”.

Multiply each value in column “x” with the corresponding value in column “f” to obtain the value for column “f(x)

Add all the values in column “f” and column “f(x)” separately.

The sum of column “f” is the total number of observations, and is denoted by “n”.

The sum of column “f(x)” is the sum of all observations, and is denoted mathematically by “∑ f(x)

Mean is given by the equation Mean =  ∑ f(x)/ n .

Substitute the values in the equation to obtain the Mean.

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