Summarising Numerical Data - Mean

Mean is the most commonly used measure of central tendency. Mean is also called average. It is relied on as a measure of central tendency when there are no outliers. In this post, we will see how to calculate the mean of a given dataset depending on the kind of data we have.

Case 1: A series of discrete data

[2,1,3,4,5,2,3,3,3,4,4,1,2,3,4]

If the data is in the form above, the mean can be calculated using the formula

$$\bar{x}=\frac{(x_1+x_2+...+x_n)}{n}.$$

where n is the number of items in the data set

ie, the mean in the above case will be:

$$\bar{x} =\frac{(2+1+3+4+5+2+3+3+3+4+4+1+2+3+4)}{15}$$

$$ie, \bar{x} = 2.9333$$

Case 2: When we have grouped data with discrete elements as in the below case.

We calculate the average in the case of grouped data using the formula:

$$\bar{x} = \dfrac{\sum_{i=1}^{n}f_ix_i }{\sum_{i=1}^{n}f_i}$$

From the above calculation, we get

$$\sum_{i=1}^{n}f_ix_i = 44\: and \: \sum_{i=1}^{n}f_i = 15$$

$$Therefore,\, the \, mean \,in \,the\, above\, case\, is\, \frac{44}{15} = 2.9333$$

Note that, the mean is the same as in case 1.

Case 3: Mean in the case of grouped continuous data as intervals as in the below example.

The mean is calculated using the formula:

$$\bar{x} = \frac{\sum_{i=1}^{n}f_im_i}{\sum_{i=1}^{n}f_i}$$

In the above formula, m is the midpoint of the interval and f is the frequency of the data in that interval

Some notes on mean:

1. Mean of a column of values in a spreadsheet can be calculated using the formula average

1. If all the values in a data set is increased by a fixed constant c, the mean of the data set will also get added by the same constant c